id	sid	tid	token	lemma	pos
ejpam-3659	1	1	european	european	PROPN
ejpam-3659	1	2	journal	journal	PROPN
ejpam-3659	1	3	of	of	ADP
ejpam-3659	1	4	pure	pure	ADJ
ejpam-3659	1	5	and	and	CCONJ
ejpam-3659	1	6	applied	apply	VERB
ejpam-3659	1	7	mathematics	mathematic	NOUN
ejpam-3659	1	8	vol	vol	NOUN
ejpam-3659	1	9	.	.	PROPN
ejpam-3659	2	1	13	13	NUM
ejpam-3659	2	2	,	,	PUNCT
ejpam-3659	2	3	no	no	INTJ
ejpam-3659	2	4	.	.	NOUN
ejpam-3659	2	5	2	2	NUM
ejpam-3659	2	6	,	,	PUNCT
ejpam-3659	2	7	2020	2020	NUM
ejpam-3659	2	8	,	,	PUNCT
ejpam-3659	2	9	303	303	NUM
ejpam-3659	2	10	-	-	SYM
ejpam-3659	2	11	313	313	NUM
ejpam-3659	2	12	issn	issn	PROPN
ejpam-3659	2	13	1307	1307	NUM
ejpam-3659	2	14	-	-	SYM
ejpam-3659	2	15	5543	5543	NUM
ejpam-3659	2	16	–	–	PUNCT
ejpam-3659	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3659	2	18	published	publish	VERB
ejpam-3659	2	19	by	by	ADP
ejpam-3659	2	20	new	new	PROPN
ejpam-3659	2	21	york	york	PROPN
ejpam-3659	2	22	business	business	PROPN
ejpam-3659	2	23	global	global	PROPN
ejpam-3659	2	24	mcshane	mcshane	PROPN
ejpam-3659	2	25	integrability	integrability	PROPN
ejpam-3659	2	26	using	use	VERB
ejpam-3659	2	27	variational	variational	ADJ
ejpam-3659	2	28	measure	measure	NOUN
ejpam-3659	2	29	felipe	felipe	PROPN
ejpam-3659	2	30	r.	r.	PROPN
ejpam-3659	2	31	sumalpong	sumalpong	PROPN
ejpam-3659	2	32	,	,	PUNCT
ejpam-3659	2	33	jr.1,∗	jr.1,∗	PROPN
ejpam-3659	2	34	,	,	PUNCT
ejpam-3659	2	35	julius	julius	PROPN
ejpam-3659	2	36	v.	v.	ADP
ejpam-3659	2	37	benitez1	benitez1	PROPN
ejpam-3659	2	38	1	1	NUM
ejpam-3659	2	39	department	department	NOUN
ejpam-3659	2	40	of	of	ADP
ejpam-3659	2	41	mathematics	mathematic	NOUN
ejpam-3659	2	42	and	and	CCONJ
ejpam-3659	2	43	statistics	statistic	NOUN
ejpam-3659	2	44	,	,	PUNCT
ejpam-3659	2	45	college	college	NOUN
ejpam-3659	2	46	of	of	ADP
ejpam-3659	2	47	sciences	science	NOUN
ejpam-3659	2	48	and	and	CCONJ
ejpam-3659	2	49	mathematics	mathematic	NOUN
ejpam-3659	2	50	,	,	PUNCT
ejpam-3659	2	51	mindanao	mindanao	PROPN
ejpam-3659	2	52	state	state	PROPN
ejpam-3659	2	53	university	university	PROPN
ejpam-3659	2	54	-	-	PUNCT
ejpam-3659	2	55	iligan	iligan	PROPN
ejpam-3659	2	56	institute	institute	PROPN
ejpam-3659	2	57	of	of	ADP
ejpam-3659	2	58	technology	technology	PROPN
ejpam-3659	2	59	,	,	PUNCT
ejpam-3659	2	60	tibanga	tibanga	PROPN
ejpam-3659	2	61	,	,	PUNCT
ejpam-3659	2	62	iligan	iligan	ADJ
ejpam-3659	2	63	city	city	PROPN
ejpam-3659	2	64	,	,	PUNCT
ejpam-3659	2	65	philippines	philippine	NOUN
ejpam-3659	2	66	abstract	abstract	ADJ
ejpam-3659	2	67	.	.	PUNCT
ejpam-3659	3	1	if	if	SCONJ
ejpam-3659	3	2	f	f	X
ejpam-3659	3	3	:	:	PUNCT
ejpam-3659	4	1	[	[	X
ejpam-3659	4	2	a	a	X
ejpam-3659	4	3	,	,	PUNCT
ejpam-3659	4	4	b]→	b]→	ADJ
ejpam-3659	4	5	r	r	NOUN
ejpam-3659	4	6	is	be	AUX
ejpam-3659	4	7	mcshane	mcshane	NOUN
ejpam-3659	4	8	integrable	integrable	ADJ
ejpam-3659	4	9	on	on	ADP
ejpam-3659	4	10	[	[	X
ejpam-3659	4	11	a	a	X
ejpam-3659	4	12	,	,	PUNCT
ejpam-3659	4	13	b	b	NOUN
ejpam-3659	4	14	]	]	X
ejpam-3659	4	15	,	,	PUNCT
ejpam-3659	4	16	then	then	ADV
ejpam-3659	4	17	f	f	PROPN
ejpam-3659	4	18	is	be	AUX
ejpam-3659	4	19	mcshane	mcshane	PROPN
ejpam-3659	4	20	integrable	integrable	ADJ
ejpam-3659	4	21	on	on	ADP
ejpam-3659	4	22	every	every	DET
ejpam-3659	4	23	lebesgue	lebesgue	ADJ
ejpam-3659	4	24	measurable	measurable	NOUN
ejpam-3659	4	25	subset	subset	NOUN
ejpam-3659	4	26	of	of	ADP
ejpam-3659	4	27	[	[	X
ejpam-3659	4	28	a	a	X
ejpam-3659	4	29	,	,	PUNCT
ejpam-3659	4	30	b	b	NOUN
ejpam-3659	4	31	]	]	X
ejpam-3659	4	32	.	.	PUNCT
ejpam-3659	5	1	however	however	ADV
ejpam-3659	5	2	,	,	PUNCT
ejpam-3659	5	3	integrability	integrability	NOUN
ejpam-3659	5	4	of	of	ADP
ejpam-3659	5	5	a	a	DET
ejpam-3659	5	6	real	real	ADV
ejpam-3659	5	7	-	-	PUNCT
ejpam-3659	5	8	valued	value	VERB
ejpam-3659	5	9	function	function	NOUN
ejpam-3659	5	10	on	on	ADP
ejpam-3659	5	11	[	[	X
ejpam-3659	5	12	a	a	X
ejpam-3659	5	13	,	,	PUNCT
ejpam-3659	5	14	b	b	NOUN
ejpam-3659	5	15	]	]	X
ejpam-3659	5	16	does	do	AUX
ejpam-3659	5	17	not	not	PART
ejpam-3659	5	18	imply	imply	VERB
ejpam-3659	5	19	mcshane	mcshane	PROPN
ejpam-3659	5	20	integrability	integrability	NOUN
ejpam-3659	5	21	on	on	ADP
ejpam-3659	5	22	any	any	DET
ejpam-3659	5	23	e	e	NOUN
ejpam-3659	5	24	⊆	⊆	NUM
ejpam-3659	5	25	[	[	X
ejpam-3659	5	26	a	a	X
ejpam-3659	5	27	,	,	PUNCT
ejpam-3659	5	28	b	b	NOUN
ejpam-3659	5	29	]	]	X
ejpam-3659	5	30	.	.	PUNCT
ejpam-3659	6	1	in	in	ADP
ejpam-3659	6	2	this	this	DET
ejpam-3659	6	3	paper	paper	NOUN
ejpam-3659	6	4	,	,	PUNCT
ejpam-3659	6	5	we	we	PRON
ejpam-3659	6	6	give	give	VERB
ejpam-3659	6	7	a	a	DET
ejpam-3659	6	8	characterization	characterization	NOUN
ejpam-3659	6	9	for	for	ADP
ejpam-3659	6	10	the	the	DET
ejpam-3659	6	11	mcshane	mcshane	PROPN
ejpam-3659	6	12	integrability	integrability	NOUN
ejpam-3659	6	13	of	of	ADP
ejpam-3659	6	14	f	f	NOUN
ejpam-3659	6	15	:	:	PUNCT
ejpam-3659	7	1	[	[	X
ejpam-3659	7	2	a	a	X
ejpam-3659	7	3	,	,	PUNCT
ejpam-3659	7	4	b]→	b]→	ADJ
ejpam-3659	7	5	r	r	NOUN
ejpam-3659	7	6	over	over	ADP
ejpam-3659	7	7	e	e	NOUN
ejpam-3659	7	8	⊆	⊆	NUM
ejpam-3659	7	9	[	[	X
ejpam-3659	7	10	a	a	X
ejpam-3659	7	11	,	,	PUNCT
ejpam-3659	7	12	b	b	NOUN
ejpam-3659	7	13	]	]	PUNCT
ejpam-3659	7	14	using	use	VERB
ejpam-3659	7	15	concept	concept	NOUN
ejpam-3659	7	16	of	of	ADP
ejpam-3659	7	17	variational	variational	ADJ
ejpam-3659	7	18	measure	measure	NOUN
ejpam-3659	7	19	.	.	PUNCT
ejpam-3659	8	1	2020	2020	NUM
ejpam-3659	8	2	mathematics	mathematic	NOUN
ejpam-3659	8	3	subject	subject	NOUN
ejpam-3659	8	4	classifications	classification	NOUN
ejpam-3659	8	5	:	:	PUNCT
ejpam-3659	8	6	26a39	26a39	NUM
ejpam-3659	8	7	,	,	PUNCT
ejpam-3659	8	8	26a42	26a42	NUM
ejpam-3659	8	9	key	key	ADJ
ejpam-3659	8	10	words	word	NOUN
ejpam-3659	8	11	and	and	CCONJ
ejpam-3659	8	12	phrases	phrase	NOUN
ejpam-3659	8	13	:	:	PUNCT
ejpam-3659	8	14	mcshane	mcshane	PROPN
ejpam-3659	8	15	integral	integral	ADJ
ejpam-3659	8	16	,	,	PUNCT
ejpam-3659	8	17	integrable	integrable	ADJ
ejpam-3659	8	18	set	set	NOUN
ejpam-3659	8	19	,	,	PUNCT
ejpam-3659	8	20	mcshane	mcshane	PROPN
ejpam-3659	8	21	δ	δ	PROPN
ejpam-3659	8	22	-	-	PUNCT
ejpam-3659	8	23	variation	variation	NOUN
ejpam-3659	8	24	,	,	PUNCT
ejpam-3659	8	25	mcshane	mcshane	PROPN
ejpam-3659	8	26	variational	variational	ADJ
ejpam-3659	8	27	measure	measure	NOUN
ejpam-3659	8	28	,	,	PUNCT
ejpam-3659	8	29	variation	variation	NOUN
ejpam-3659	8	30	zero	zero	NUM
ejpam-3659	8	31	,	,	PUNCT
ejpam-3659	8	32	cauchy	cauchy	ADJ
ejpam-3659	8	33	extension	extension	NOUN
ejpam-3659	8	34	.	.	PUNCT
ejpam-3659	9	1	1	1	X
ejpam-3659	9	2	.	.	X
ejpam-3659	9	3	introduction	introduction	NOUN
ejpam-3659	9	4	let	let	VERB
ejpam-3659	9	5	f	f	NOUN
ejpam-3659	9	6	:	:	PUNCT
ejpam-3659	10	1	[	[	X
ejpam-3659	10	2	a	a	X
ejpam-3659	10	3	,	,	PUNCT
ejpam-3659	10	4	b]→	b]→	ADJ
ejpam-3659	10	5	r	r	NOUN
ejpam-3659	10	6	be	be	AUX
ejpam-3659	10	7	a	a	DET
ejpam-3659	10	8	function	function	NOUN
ejpam-3659	10	9	.	.	PUNCT
ejpam-3659	11	1	it	it	PRON
ejpam-3659	11	2	is	be	AUX
ejpam-3659	11	3	well	well	ADV
ejpam-3659	11	4	-	-	PUNCT
ejpam-3659	11	5	known	know	VERB
ejpam-3659	11	6	that	that	SCONJ
ejpam-3659	11	7	f	f	PROPN
ejpam-3659	11	8	is	be	AUX
ejpam-3659	11	9	mcshane	mcshane	PROPN
ejpam-3659	11	10	integrable	integrable	ADJ
ejpam-3659	11	11	on	on	ADP
ejpam-3659	11	12	[	[	X
ejpam-3659	11	13	a	a	X
ejpam-3659	11	14	,	,	PUNCT
ejpam-3659	11	15	b	b	NOUN
ejpam-3659	11	16	]	]	X
ejpam-3659	11	17	if	if	SCONJ
ejpam-3659	12	1	and	and	CCONJ
ejpam-3659	12	2	only	only	ADV
ejpam-3659	12	3	if	if	SCONJ
ejpam-3659	12	4	f	f	PROPN
ejpam-3659	12	5	is	be	AUX
ejpam-3659	12	6	lebesgue	lebesgue	NOUN
ejpam-3659	12	7	integrable	integrable	ADJ
ejpam-3659	12	8	on	on	ADP
ejpam-3659	12	9	[	[	X
ejpam-3659	12	10	a	a	DET
ejpam-3659	12	11	,	,	PUNCT
ejpam-3659	12	12	b	b	NOUN
ejpam-3659	12	13	]	]	PUNCT
ejpam-3659	12	14	and	and	CCONJ
ejpam-3659	12	15	the	the	DET
ejpam-3659	12	16	values	value	NOUN
ejpam-3659	12	17	of	of	ADP
ejpam-3659	12	18	the	the	DET
ejpam-3659	12	19	integrals	integral	NOUN
ejpam-3659	12	20	are	be	AUX
ejpam-3659	12	21	equal	equal	ADJ
ejpam-3659	12	22	,	,	PUNCT
ejpam-3659	12	23	see	see	VERB
ejpam-3659	12	24	[	[	X
ejpam-3659	12	25	5	5	NUM
ejpam-3659	12	26	,	,	PUNCT
ejpam-3659	12	27	theorem	theorem	VERB
ejpam-3659	12	28	10.11	10.11	NUM
ejpam-3659	12	29	]	]	PUNCT
ejpam-3659	12	30	.	.	PUNCT
ejpam-3659	13	1	if	if	SCONJ
ejpam-3659	13	2	f	f	PROPN
ejpam-3659	13	3	is	be	AUX
ejpam-3659	13	4	mcshane	mcshane	PROPN
ejpam-3659	13	5	integrable	integrable	ADJ
ejpam-3659	13	6	on	on	ADP
ejpam-3659	13	7	[	[	X
ejpam-3659	13	8	a	a	X
ejpam-3659	13	9	,	,	PUNCT
ejpam-3659	13	10	b	b	NOUN
ejpam-3659	13	11	]	]	X
ejpam-3659	13	12	,	,	PUNCT
ejpam-3659	13	13	then	then	ADV
ejpam-3659	13	14	f	f	PROPN
ejpam-3659	13	15	is	be	AUX
ejpam-3659	13	16	henstock	henstock	NOUN
ejpam-3659	13	17	-	-	PUNCT
ejpam-3659	13	18	kurzweil	kurzweil	NOUN
ejpam-3659	13	19	integrable	integrable	ADJ
ejpam-3659	13	20	on	on	ADP
ejpam-3659	13	21	[	[	X
ejpam-3659	13	22	a	a	DET
ejpam-3659	13	23	,	,	PUNCT
ejpam-3659	13	24	b	b	NOUN
ejpam-3659	13	25	]	]	X
ejpam-3659	13	26	(	(	PUNCT
ejpam-3659	13	27	with	with	ADP
ejpam-3659	13	28	the	the	DET
ejpam-3659	13	29	same	same	ADJ
ejpam-3659	13	30	value	value	NOUN
ejpam-3659	13	31	of	of	ADP
ejpam-3659	13	32	integrals	integral	NOUN
ejpam-3659	13	33	)	)	PUNCT
ejpam-3659	13	34	but	but	CCONJ
ejpam-3659	13	35	the	the	DET
ejpam-3659	13	36	converse	converse	NOUN
ejpam-3659	13	37	is	be	AUX
ejpam-3659	13	38	not	not	PART
ejpam-3659	13	39	true	true	ADJ
ejpam-3659	13	40	,	,	PUNCT
ejpam-3659	13	41	as	as	SCONJ
ejpam-3659	13	42	seen	see	VERB
ejpam-3659	13	43	in	in	ADP
ejpam-3659	13	44	example	example	NOUN
ejpam-3659	13	45	1	1	NUM
ejpam-3659	13	46	below	below	ADV
ejpam-3659	13	47	.	.	PUNCT
ejpam-3659	14	1	if	if	SCONJ
ejpam-3659	14	2	f	f	X
ejpam-3659	14	3	:	:	PUNCT
ejpam-3659	15	1	[	[	X
ejpam-3659	15	2	a	a	X
ejpam-3659	15	3	,	,	PUNCT
ejpam-3659	15	4	b	b	NOUN
ejpam-3659	15	5	]	]	X
ejpam-3659	15	6	→	→	PUNCT
ejpam-3659	15	7	r	r	NOUN
ejpam-3659	15	8	is	be	AUX
ejpam-3659	15	9	mcshane	mcshane	NOUN
ejpam-3659	15	10	integrable	integrable	ADJ
ejpam-3659	15	11	on	on	ADP
ejpam-3659	15	12	[	[	X
ejpam-3659	15	13	a	a	X
ejpam-3659	15	14	,	,	PUNCT
ejpam-3659	15	15	b	b	NOUN
ejpam-3659	15	16	]	]	X
ejpam-3659	15	17	,	,	PUNCT
ejpam-3659	15	18	then	then	ADV
ejpam-3659	15	19	f	f	PROPN
ejpam-3659	15	20	is	be	AUX
ejpam-3659	15	21	mcshane	mcshane	PROPN
ejpam-3659	15	22	integrable	integrable	ADJ
ejpam-3659	15	23	on	on	ADP
ejpam-3659	15	24	any	any	DET
ejpam-3659	15	25	sub	sub	NOUN
ejpam-3659	15	26	-	-	NOUN
ejpam-3659	15	27	interval	interval	NOUN
ejpam-3659	15	28	[	[	X
ejpam-3659	15	29	c	c	X
ejpam-3659	15	30	,	,	PUNCT
ejpam-3659	15	31	d	d	X
ejpam-3659	15	32	]	]	X
ejpam-3659	15	33	of	of	ADP
ejpam-3659	15	34	[	[	X
ejpam-3659	15	35	a	a	X
ejpam-3659	15	36	,	,	PUNCT
ejpam-3659	15	37	b	b	NOUN
ejpam-3659	15	38	]	]	X
ejpam-3659	15	39	,	,	PUNCT
ejpam-3659	15	40	see	see	VERB
ejpam-3659	15	41	[	[	X
ejpam-3659	15	42	5	5	NUM
ejpam-3659	15	43	,	,	PUNCT
ejpam-3659	15	44	theorem	theorem	VERB
ejpam-3659	15	45	10.4	10.4	NUM
ejpam-3659	15	46	]	]	PUNCT
ejpam-3659	15	47	,	,	PUNCT
ejpam-3659	15	48	[	[	X
ejpam-3659	15	49	6	6	NUM
ejpam-3659	15	50	]	]	PUNCT
ejpam-3659	15	51	,	,	PUNCT
ejpam-3659	15	52	[	[	X
ejpam-3659	15	53	7	7	X
ejpam-3659	15	54	]	]	PUNCT
ejpam-3659	15	55	and	and	CCONJ
ejpam-3659	15	56	[	[	X
ejpam-3659	15	57	10	10	NUM
ejpam-3659	15	58	]	]	PUNCT
ejpam-3659	15	59	.	.	PUNCT
ejpam-3659	16	1	it	it	PRON
ejpam-3659	16	2	is	be	AUX
ejpam-3659	16	3	shown	show	VERB
ejpam-3659	16	4	in	in	ADP
ejpam-3659	16	5	[	[	X
ejpam-3659	16	6	8	8	NUM
ejpam-3659	16	7	]	]	PUNCT
ejpam-3659	16	8	that	that	SCONJ
ejpam-3659	16	9	f	f	X
ejpam-3659	16	10	:	:	PUNCT
ejpam-3659	17	1	[	[	X
ejpam-3659	17	2	a	a	X
ejpam-3659	17	3	,	,	PUNCT
ejpam-3659	17	4	b]→	b]→	ADJ
ejpam-3659	17	5	r	r	NOUN
ejpam-3659	17	6	is	be	AUX
ejpam-3659	17	7	henstock	henstock	NOUN
ejpam-3659	17	8	-	-	PUNCT
ejpam-3659	17	9	kurzweil	kurzweil	NOUN
ejpam-3659	17	10	integrable	integrable	ADJ
ejpam-3659	17	11	on	on	ADP
ejpam-3659	17	12	[	[	X
ejpam-3659	17	13	a	a	X
ejpam-3659	17	14	,	,	PUNCT
ejpam-3659	17	15	b	b	NOUN
ejpam-3659	17	16	]	]	X
ejpam-3659	17	17	if	if	SCONJ
ejpam-3659	17	18	and	and	CCONJ
ejpam-3659	17	19	only	only	ADV
ejpam-3659	17	20	if	if	SCONJ
ejpam-3659	17	21	for	for	ADP
ejpam-3659	17	22	each	each	DET
ejpam-3659	17	23	c	c	NOUN
ejpam-3659	17	24	∈	∈	PROPN
ejpam-3659	17	25	(	(	PUNCT
ejpam-3659	17	26	a	a	DET
ejpam-3659	17	27	,	,	PUNCT
ejpam-3659	17	28	b	b	NOUN
ejpam-3659	17	29	)	)	PUNCT
ejpam-3659	17	30	the	the	DET
ejpam-3659	17	31	function	function	NOUN
ejpam-3659	17	32	f	f	PROPN
ejpam-3659	17	33	·	·	PUNCT
ejpam-3659	17	34	χ[a	χ[a	PROPN
ejpam-3659	17	35	,	,	PUNCT
ejpam-3659	17	36	c	c	X
ejpam-3659	17	37	]	]	PUNCT
ejpam-3659	17	38	is	be	AUX
ejpam-3659	17	39	henstock	henstock	NOUN
ejpam-3659	17	40	-	-	PUNCT
ejpam-3659	17	41	kurzweil	kurzweil	NOUN
ejpam-3659	17	42	integrable	integrable	ADJ
ejpam-3659	17	43	on	on	ADP
ejpam-3659	17	44	[	[	X
ejpam-3659	17	45	a	a	X
ejpam-3659	17	46	,	,	PUNCT
ejpam-3659	17	47	c	c	NOUN
ejpam-3659	17	48	]	]	PUNCT
ejpam-3659	17	49	and	and	CCONJ
ejpam-3659	17	50	lim	lim	PROPN
ejpam-3659	17	51	c→b−	c→b−	VERB
ejpam-3659	18	1	∫	∫	PROPN
ejpam-3659	18	2	c	c	PROPN
ejpam-3659	18	3	a	a	DET
ejpam-3659	18	4	f	f	PROPN
ejpam-3659	18	5	exists	exist	VERB
ejpam-3659	18	6	.	.	PUNCT
ejpam-3659	19	1	in	in	ADP
ejpam-3659	19	2	this	this	DET
ejpam-3659	19	3	case	case	NOUN
ejpam-3659	19	4	,	,	PUNCT
ejpam-3659	19	5	lim	lim	PROPN
ejpam-3659	19	6	c→b−	c→b−	VERB
ejpam-3659	19	7	∫	∫	PROPN
ejpam-3659	19	8	c	c	PROPN
ejpam-3659	19	9	a	a	DET
ejpam-3659	19	10	f	f	X
ejpam-3659	19	11	=	=	SYM
ejpam-3659	19	12	∫	∫	PROPN
ejpam-3659	19	13	b	b	PROPN
ejpam-3659	19	14	a	a	DET
ejpam-3659	19	15	f.	f.	PROPN
ejpam-3659	19	16	∗corresponding	∗corresponde	VERB
ejpam-3659	19	17	author	author	NOUN
ejpam-3659	19	18	.	.	PUNCT
ejpam-3659	20	1	doi	doi	NOUN
ejpam-3659	20	2	:	:	PUNCT
ejpam-3659	20	3	https://doi.org/10.29020/nybg.ejpam.v13i2.3659	https://doi.org/10.29020/nybg.ejpam.v13i2.3659	NOUN
ejpam-3659	20	4	email	email	NOUN
ejpam-3659	20	5	addresses	address	NOUN
ejpam-3659	20	6	:	:	PUNCT
ejpam-3659	20	7	felipejr.sumalpong@g.msuiit.edu.ph	felipejr.sumalpong@g.msuiit.edu.ph	PROPN
ejpam-3659	20	8	(	(	PUNCT
ejpam-3659	20	9	f.	f.	PROPN
ejpam-3659	20	10	sumalpong	sumalpong	PROPN
ejpam-3659	20	11	jr	jr	PROPN
ejpam-3659	20	12	.	.	PROPN
ejpam-3659	20	13	)	)	PUNCT
ejpam-3659	20	14	,	,	PUNCT
ejpam-3659	20	15	julius.benitez@g.msuiit	julius.benitez@g.msuiit	PROPN
ejpam-3659	20	16	,	,	PUNCT
ejpam-3659	20	17	edu.ph	edu.ph	PROPN
ejpam-3659	20	18	(	(	PUNCT
ejpam-3659	20	19	j.	j.	PROPN
ejpam-3659	20	20	benitez	benitez	PROPN
ejpam-3659	20	21	)	)	PUNCT
ejpam-3659	20	22	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3659	21	1	303	303	NUM
ejpam-3659	21	2	c	c	NOUN
ejpam-3659	21	3	©	©	NOUN
ejpam-3659	21	4	2020	2020	NUM
ejpam-3659	21	5	ejpam	ejpam	VERB
ejpam-3659	21	6	all	all	DET
ejpam-3659	21	7	rights	right	NOUN
ejpam-3659	21	8	reserved	reserve	VERB
ejpam-3659	21	9	.	.	PUNCT
ejpam-3659	22	1	f.	f.	PROPN
ejpam-3659	22	2	sumalpong	sumalpong	PROPN
ejpam-3659	22	3	jr	jr	PROPN
ejpam-3659	22	4	.	.	PROPN
ejpam-3659	22	5	,	,	PUNCT
ejpam-3659	22	6	j.	j.	PROPN
ejpam-3659	22	7	benitez	benitez	PROPN
ejpam-3659	22	8	/	/	PUNCT
ejpam-3659	22	9	eur	eur	PROPN
ejpam-3659	22	10	.	.	PUNCT
ejpam-3659	23	1	j.	j.	PROPN
ejpam-3659	23	2	pure	pure	PROPN
ejpam-3659	23	3	appl	appl	PROPN
ejpam-3659	23	4	.	.	PROPN
ejpam-3659	23	5	math	math	PROPN
ejpam-3659	23	6	,	,	PUNCT
ejpam-3659	23	7	13	13	NUM
ejpam-3659	23	8	(	(	PUNCT
ejpam-3659	23	9	2	2	NUM
ejpam-3659	23	10	)	)	PUNCT
ejpam-3659	23	11	(	(	PUNCT
ejpam-3659	23	12	2020	2020	NUM
ejpam-3659	23	13	)	)	PUNCT
ejpam-3659	23	14	,	,	PUNCT
ejpam-3659	23	15	303	303	NUM
ejpam-3659	23	16	-	-	SYM
ejpam-3659	23	17	313	313	NUM
ejpam-3659	23	18	304	304	NUM
ejpam-3659	23	19	this	this	PRON
ejpam-3659	23	20	is	be	AUX
ejpam-3659	23	21	known	know	VERB
ejpam-3659	23	22	as	as	ADP
ejpam-3659	23	23	the	the	DET
ejpam-3659	23	24	cauchy	cauchy	PROPN
ejpam-3659	23	25	extension	extension	NOUN
ejpam-3659	23	26	.	.	PUNCT
ejpam-3659	24	1	example	example	NOUN
ejpam-3659	24	2	1	1	NUM
ejpam-3659	24	3	also	also	ADV
ejpam-3659	24	4	shows	show	VERB
ejpam-3659	24	5	that	that	SCONJ
ejpam-3659	24	6	cauchy	cauchy	ADJ
ejpam-3659	24	7	extension	extension	NOUN
ejpam-3659	24	8	does	do	AUX
ejpam-3659	24	9	not	not	PART
ejpam-3659	24	10	hold	hold	VERB
ejpam-3659	24	11	for	for	ADP
ejpam-3659	24	12	mcshane	mcshane	PROPN
ejpam-3659	24	13	integral	integral	PROPN
ejpam-3659	24	14	.	.	PUNCT
ejpam-3659	25	1	indeed	indeed	ADV
ejpam-3659	25	2	,	,	PUNCT
ejpam-3659	25	3	f	f	PROPN
ejpam-3659	25	4	is	be	AUX
ejpam-3659	25	5	mcshane	mcshane	PROPN
ejpam-3659	25	6	integrable	integrable	ADJ
ejpam-3659	25	7	on	on	ADP
ejpam-3659	25	8	[	[	X
ejpam-3659	25	9	ε	ε	PROPN
ejpam-3659	25	10	,	,	PUNCT
ejpam-3659	25	11	1	1	NUM
ejpam-3659	25	12	]	]	PUNCT
ejpam-3659	25	13	for	for	ADP
ejpam-3659	25	14	every	every	DET
ejpam-3659	25	15	0	0	NUM
ejpam-3659	25	16	<	<	X
ejpam-3659	25	17	ε	ε	X
ejpam-3659	25	18	<	<	X
ejpam-3659	25	19	1	1	NUM
ejpam-3659	25	20	and	and	CCONJ
ejpam-3659	25	21	lim	lim	PROPN
ejpam-3659	25	22	ε→0	ε→0	NOUN
ejpam-3659	26	1	+	+	CCONJ
ejpam-3659	26	2	∫	∫	PROPN
ejpam-3659	26	3	1	1	NUM
ejpam-3659	26	4	ε	ε	PROPN
ejpam-3659	26	5	f	f	PROPN
ejpam-3659	26	6	=	=	PROPN
ejpam-3659	26	7	lim	lim	PROPN
ejpam-3659	26	8	ε→0	ε→0	VERB
ejpam-3659	26	9	+	+	CCONJ
ejpam-3659	26	10	(	(	PUNCT
ejpam-3659	26	11	f	f	PROPN
ejpam-3659	26	12	(	(	PUNCT
ejpam-3659	26	13	1)−	1)−	PROPN
ejpam-3659	26	14	f	f	PROPN
ejpam-3659	26	15	(	(	PUNCT
ejpam-3659	26	16	ε	ε	PROPN
ejpam-3659	26	17	)	)	PUNCT
ejpam-3659	26	18	)	)	PUNCT
ejpam-3659	26	19	=	=	PRON
ejpam-3659	26	20	sin	sin	NOUN
ejpam-3659	26	21	1	1	NUM
ejpam-3659	26	22	,	,	PUNCT
ejpam-3659	26	23	but	but	CCONJ
ejpam-3659	26	24	f	f	PROPN
ejpam-3659	26	25	is	be	AUX
ejpam-3659	26	26	not	not	PART
ejpam-3659	26	27	mcshane	mcshane	NOUN
ejpam-3659	26	28	integrable	integrable	ADJ
ejpam-3659	26	29	on	on	ADP
ejpam-3659	26	30	[	[	X
ejpam-3659	26	31	0	0	NUM
ejpam-3659	26	32	,	,	PUNCT
ejpam-3659	26	33	1	1	NUM
ejpam-3659	26	34	]	]	PUNCT
ejpam-3659	26	35	.	.	PUNCT
ejpam-3659	27	1	example	example	NOUN
ejpam-3659	28	1	1	1	X
ejpam-3659	28	2	.	.	X
ejpam-3659	28	3	consider	consider	VERB
ejpam-3659	28	4	the	the	DET
ejpam-3659	28	5	function	function	NOUN
ejpam-3659	28	6	f	f	NOUN
ejpam-3659	28	7	:	:	PUNCT
ejpam-3659	29	1	[	[	X
ejpam-3659	29	2	0	0	NUM
ejpam-3659	29	3	,	,	PUNCT
ejpam-3659	29	4	1]→	1]→	ADJ
ejpam-3659	29	5	r	r	NOUN
ejpam-3659	29	6	defined	define	VERB
ejpam-3659	29	7	as	as	SCONJ
ejpam-3659	29	8	follows	follow	VERB
ejpam-3659	29	9	:	:	PUNCT
ejpam-3659	29	10	f	f	PROPN
ejpam-3659	29	11	(	(	PUNCT
ejpam-3659	29	12	x	x	X
ejpam-3659	29	13	)	)	PUNCT
ejpam-3659	29	14	=	=	PRON
ejpam-3659	29	15	{	{	PUNCT
ejpam-3659	29	16	x2	x2	NOUN
ejpam-3659	29	17	sin	sin	VERB
ejpam-3659	29	18	1	1	NUM
ejpam-3659	29	19	x2	x2	NOUN
ejpam-3659	29	20	,	,	PUNCT
ejpam-3659	29	21	if	if	SCONJ
ejpam-3659	29	22	x	x	PROPN
ejpam-3659	29	23	6=	6=	ADP
ejpam-3659	29	24	0	0	NUM
ejpam-3659	29	25	;	;	PUNCT
ejpam-3659	29	26	0	0	NUM
ejpam-3659	29	27	,	,	PUNCT
ejpam-3659	29	28	if	if	SCONJ
ejpam-3659	29	29	x	x	ADP
ejpam-3659	29	30	=	=	NOUN
ejpam-3659	29	31	0	0	X
ejpam-3659	29	32	.	.	PUNCT
ejpam-3659	30	1	let	let	VERB
ejpam-3659	30	2	f	f	NOUN
ejpam-3659	30	3	:	:	PUNCT
ejpam-3659	31	1	[	[	X
ejpam-3659	31	2	0	0	NUM
ejpam-3659	31	3	,	,	PUNCT
ejpam-3659	31	4	1]→	1]→	ADJ
ejpam-3659	31	5	r	r	NOUN
ejpam-3659	31	6	be	be	VERB
ejpam-3659	31	7	the	the	DET
ejpam-3659	31	8	function	function	NOUN
ejpam-3659	31	9	defined	define	VERB
ejpam-3659	31	10	by	by	ADP
ejpam-3659	31	11	f(x	f(x	PROPN
ejpam-3659	31	12	)	)	PUNCT
ejpam-3659	32	1	=	=	PRON
ejpam-3659	32	2	{	{	PUNCT
ejpam-3659	32	3	f	f	NOUN
ejpam-3659	32	4	′(x	′(x	PROPN
ejpam-3659	32	5	)	)	PUNCT
ejpam-3659	32	6	,	,	PUNCT
ejpam-3659	32	7	if	if	SCONJ
ejpam-3659	32	8	0	0	NUM
ejpam-3659	32	9	<	<	X
ejpam-3659	32	10	x	x	SYM
ejpam-3659	32	11	≤	≤	NUM
ejpam-3659	32	12	1	1	NUM
ejpam-3659	32	13	;	;	PUNCT
ejpam-3659	32	14	0	0	NUM
ejpam-3659	32	15	,	,	PUNCT
ejpam-3659	32	16	if	if	SCONJ
ejpam-3659	32	17	x	x	ADP
ejpam-3659	32	18	=	=	SYM
ejpam-3659	32	19	0	0	NUM
ejpam-3659	32	20	,	,	PUNCT
ejpam-3659	32	21	where	where	SCONJ
ejpam-3659	32	22	f	f	PROPN
ejpam-3659	32	23	′(x	′(x	NOUN
ejpam-3659	32	24	)	)	PUNCT
ejpam-3659	32	25	=	=	PUNCT
ejpam-3659	33	1	2x	2x	NUM
ejpam-3659	33	2	sin	sin	VERB
ejpam-3659	33	3	1	1	NUM
ejpam-3659	33	4	x2	x2	NOUN
ejpam-3659	33	5	−	−	NOUN
ejpam-3659	33	6	2	2	NUM
ejpam-3659	33	7	x	x	SYM
ejpam-3659	33	8	cos	cos	PROPN
ejpam-3659	33	9	1	1	NUM
ejpam-3659	33	10	x2	x2	NOUN
ejpam-3659	33	11	,	,	PUNCT
ejpam-3659	33	12	for	for	ADP
ejpam-3659	33	13	0	0	NUM
ejpam-3659	33	14	<	<	X
ejpam-3659	33	15	x	x	SYM
ejpam-3659	33	16	≤	≤	NUM
ejpam-3659	33	17	1	1	NUM
ejpam-3659	33	18	.	.	PUNCT
ejpam-3659	34	1	we	we	PRON
ejpam-3659	34	2	observe	observe	VERB
ejpam-3659	34	3	that	that	SCONJ
ejpam-3659	34	4	f	f	PROPN
ejpam-3659	34	5	is	be	AUX
ejpam-3659	34	6	continuous	continuous	ADJ
ejpam-3659	34	7	on	on	ADP
ejpam-3659	34	8	[	[	X
ejpam-3659	34	9	0	0	NUM
ejpam-3659	34	10	,	,	PUNCT
ejpam-3659	34	11	1	1	NUM
ejpam-3659	34	12	]	]	PUNCT
ejpam-3659	34	13	.	.	PUNCT
ejpam-3659	35	1	now	now	ADV
ejpam-3659	35	2	,	,	PUNCT
ejpam-3659	35	3	we	we	PRON
ejpam-3659	35	4	show	show	VERB
ejpam-3659	35	5	that	that	SCONJ
ejpam-3659	35	6	f	f	PROPN
ejpam-3659	35	7	is	be	AUX
ejpam-3659	35	8	not	not	PART
ejpam-3659	35	9	of	of	ADP
ejpam-3659	35	10	bounded	bounded	ADJ
ejpam-3659	35	11	variation	variation	NOUN
ejpam-3659	35	12	on	on	ADP
ejpam-3659	35	13	[	[	X
ejpam-3659	35	14	0	0	NUM
ejpam-3659	35	15	,	,	PUNCT
ejpam-3659	35	16	1	1	NUM
ejpam-3659	35	17	]	]	PUNCT
ejpam-3659	35	18	;	;	PUNCT
ejpam-3659	35	19	that	that	PRON
ejpam-3659	35	20	is	is	ADV
ejpam-3659	35	21	,	,	PUNCT
ejpam-3659	35	22	v	v	INTJ
ejpam-3659	35	23	(	(	PUNCT
ejpam-3659	35	24	f	f	NOUN
ejpam-3659	35	25	;	;	PUNCT
ejpam-3659	36	1	[	[	X
ejpam-3659	36	2	0	0	NUM
ejpam-3659	36	3	,	,	PUNCT
ejpam-3659	36	4	1	1	NUM
ejpam-3659	36	5	]	]	PUNCT
ejpam-3659	36	6	)	)	PUNCT
ejpam-3659	36	7	=	=	SYM
ejpam-3659	36	8	∞	∞	PROPN
ejpam-3659	36	9	,	,	PUNCT
ejpam-3659	36	10	where	where	SCONJ
ejpam-3659	36	11	v	v	X
ejpam-3659	36	12	(	(	PUNCT
ejpam-3659	36	13	f	f	X
ejpam-3659	36	14	;	;	PUNCT
ejpam-3659	36	15	[	[	X
ejpam-3659	36	16	0	0	NUM
ejpam-3659	36	17	,	,	PUNCT
ejpam-3659	36	18	1	1	NUM
ejpam-3659	36	19	]	]	PUNCT
ejpam-3659	36	20	)	)	PUNCT
ejpam-3659	36	21	=	=	SYM
ejpam-3659	36	22	sup	sup	NOUN
ejpam-3659	36	23	d∈d	d∈d	NOUN
ejpam-3659	36	24	(	(	PUNCT
ejpam-3659	36	25	d	d	NOUN
ejpam-3659	36	26	)	)	PUNCT
ejpam-3659	36	27	∑	∑	PROPN
ejpam-3659	36	28	|f	|f	PROPN
ejpam-3659	36	29	(	(	PUNCT
ejpam-3659	36	30	v)−	v)−	PROPN
ejpam-3659	36	31	f	f	X
ejpam-3659	36	32	(	(	PUNCT
ejpam-3659	36	33	u)|	u)|	NOUN
ejpam-3659	36	34	and	and	CCONJ
ejpam-3659	36	35	d	d	PROPN
ejpam-3659	36	36	is	be	AUX
ejpam-3659	36	37	the	the	DET
ejpam-3659	36	38	class	class	NOUN
ejpam-3659	36	39	of	of	ADP
ejpam-3659	36	40	all	all	DET
ejpam-3659	36	41	partition	partition	NOUN
ejpam-3659	37	1	d	d	NOUN
ejpam-3659	37	2	=	=	PRON
ejpam-3659	37	3	{	{	PUNCT
ejpam-3659	37	4	[	[	X
ejpam-3659	37	5	u	u	NOUN
ejpam-3659	37	6	,	,	PUNCT
ejpam-3659	37	7	v	v	NOUN
ejpam-3659	37	8	]	]	X
ejpam-3659	37	9	}	}	PUNCT
ejpam-3659	37	10	of	of	ADP
ejpam-3659	37	11	[	[	X
ejpam-3659	37	12	0	0	NUM
ejpam-3659	37	13	,	,	PUNCT
ejpam-3659	37	14	1	1	NUM
ejpam-3659	37	15	]	]	PUNCT
ejpam-3659	37	16	.	.	PUNCT
ejpam-3659	38	1	note	note	VERB
ejpam-3659	38	2	that	that	SCONJ
ejpam-3659	38	3	f	f	PROPN
ejpam-3659	38	4	(	(	PUNCT
ejpam-3659	38	5	x	x	X
ejpam-3659	38	6	)	)	PUNCT
ejpam-3659	38	7	=	=	VERB
ejpam-3659	38	8	±x2	±x2	ADJ
ejpam-3659	38	9	if	if	SCONJ
ejpam-3659	38	10	and	and	CCONJ
ejpam-3659	38	11	only	only	ADV
ejpam-3659	38	12	if	if	SCONJ
ejpam-3659	38	13	x2	x2	PROPN
ejpam-3659	38	14	=	=	SYM
ejpam-3659	38	15	2	2	NUM
ejpam-3659	38	16	2nπ±π	2nπ±π	NUM
ejpam-3659	38	17	,	,	PUNCT
ejpam-3659	38	18	where	where	SCONJ
ejpam-3659	38	19	n	n	PRON
ejpam-3659	38	20	is	be	AUX
ejpam-3659	38	21	a	a	DET
ejpam-3659	38	22	positive	positive	ADJ
ejpam-3659	38	23	integer	integer	NOUN
ejpam-3659	38	24	.	.	PUNCT
ejpam-3659	39	1	for	for	ADP
ejpam-3659	39	2	each	each	DET
ejpam-3659	39	3	n	n	PRON
ejpam-3659	39	4	∈	∈	PROPN
ejpam-3659	39	5	n	n	CCONJ
ejpam-3659	39	6	,	,	PUNCT
ejpam-3659	39	7	let	let	VERB
ejpam-3659	39	8	xn	xn	PUNCT
ejpam-3659	40	1	=	=	PUNCT
ejpam-3659	40	2	√	√	PROPN
ejpam-3659	40	3	2	2	NUM
ejpam-3659	40	4	2nπ+π	2nπ+π	NUM
ejpam-3659	40	5	.	.	PUNCT
ejpam-3659	41	1	then	then	ADV
ejpam-3659	41	2	for	for	ADP
ejpam-3659	41	3	each	each	DET
ejpam-3659	41	4	n	n	PRON
ejpam-3659	41	5	∈	∈	PROPN
ejpam-3659	41	6	n	n	CCONJ
ejpam-3659	41	7	f	f	X
ejpam-3659	41	8	(	(	PUNCT
ejpam-3659	41	9	xn	xn	PROPN
ejpam-3659	41	10	)	)	PUNCT
ejpam-3659	41	11	=	=	SYM
ejpam-3659	41	12	2	2	X
ejpam-3659	41	13	·	·	PUNCT
ejpam-3659	41	14	(	(	PUNCT
ejpam-3659	41	15	−1)n	−1)n	PROPN
ejpam-3659	41	16	2nπ	2nπ	NOUN
ejpam-3659	41	17	+	+	CCONJ
ejpam-3659	41	18	π	π	X
ejpam-3659	41	19	.	.	PUNCT
ejpam-3659	42	1	thus	thus	ADV
ejpam-3659	42	2	,	,	PUNCT
ejpam-3659	42	3	∞∑	∞∑	PROPN
ejpam-3659	42	4	n=0	n=0	NUM
ejpam-3659	42	5	|f	|f	PROPN
ejpam-3659	42	6	(	(	PUNCT
ejpam-3659	42	7	xn)−	xn)−	NOUN
ejpam-3659	42	8	f	f	X
ejpam-3659	42	9	(	(	PUNCT
ejpam-3659	42	10	xn+1)|	xn+1)|	PROPN
ejpam-3659	42	11	=	=	PUNCT
ejpam-3659	42	12	∞∑	∞∑	NUM
ejpam-3659	42	13	n=0	n=0	ADJ
ejpam-3659	42	14	∣∣∣∣2	∣∣∣∣2	NOUN
ejpam-3659	42	15	·	·	PUNCT
ejpam-3659	42	16	(	(	PUNCT
ejpam-3659	42	17	−1)n	−1)n	PROPN
ejpam-3659	42	18	2nπ	2nπ	NOUN
ejpam-3659	42	19	+	+	CCONJ
ejpam-3659	42	20	π	π	PROPN
ejpam-3659	42	21	−	−	PROPN
ejpam-3659	42	22	2	2	NUM
ejpam-3659	42	23	·	·	PUNCT
ejpam-3659	42	24	(	(	PUNCT
ejpam-3659	42	25	−1)n+1	−1)n+1	VERB
ejpam-3659	42	26	2nπ	2nπ	NOUN
ejpam-3659	43	1	+	+	CCONJ
ejpam-3659	43	2	3π	3π	NOUN
ejpam-3659	43	3	∣∣∣∣	∣∣∣∣	NOUN
ejpam-3659	43	4	=	=	SYM
ejpam-3659	43	5	2	2	NUM
ejpam-3659	43	6	π	π	PROPN
ejpam-3659	43	7	∞∑	∞∑	VERB
ejpam-3659	43	8	n=0	n=0	NUM
ejpam-3659	43	9	4n+	4n+	NUM
ejpam-3659	43	10	4	4	NUM
ejpam-3659	43	11	4n2	4n2	NUM
ejpam-3659	44	1	+	+	CCONJ
ejpam-3659	44	2	8n+	8n+	NUM
ejpam-3659	44	3	3	3	NUM
ejpam-3659	44	4	≥	≥	NOUN
ejpam-3659	44	5	2	2	NUM
ejpam-3659	44	6	π	π	PROPN
ejpam-3659	44	7	∞∑	∞∑	NUM
ejpam-3659	44	8	n=0	n=0	NUM
ejpam-3659	44	9	1	1	NUM
ejpam-3659	44	10	n+	n+	SYM
ejpam-3659	44	11	1	1	NUM
ejpam-3659	44	12	=	=	SYM
ejpam-3659	44	13	2	2	NUM
ejpam-3659	44	14	π	π	PROPN
ejpam-3659	44	15	∞∑	∞∑	NUM
ejpam-3659	44	16	n=1	n=1	ADP
ejpam-3659	44	17	1	1	NUM
ejpam-3659	44	18	n	n	NOUN
ejpam-3659	44	19	=	=	NOUN
ejpam-3659	44	20	∞	∞	PROPN
ejpam-3659	44	21	that	that	PRON
ejpam-3659	44	22	is	be	AUX
ejpam-3659	44	23	,	,	PUNCT
ejpam-3659	44	24	∞∑	∞∑	PROPN
ejpam-3659	44	25	n=0	n=0	NUM
ejpam-3659	44	26	|f	|f	PROPN
ejpam-3659	44	27	(	(	PUNCT
ejpam-3659	44	28	xn)−	xn)−	NOUN
ejpam-3659	44	29	f	f	X
ejpam-3659	44	30	(	(	PUNCT
ejpam-3659	44	31	xn+1)|	xn+1)|	PROPN
ejpam-3659	44	32	=	=	AUX
ejpam-3659	44	33	∞.	∞.	PROPN
ejpam-3659	44	34	hence	hence	ADV
ejpam-3659	44	35	,	,	PUNCT
ejpam-3659	44	36	v	v	X
ejpam-3659	44	37	(	(	PUNCT
ejpam-3659	44	38	f	f	NOUN
ejpam-3659	44	39	;	;	PUNCT
ejpam-3659	45	1	[	[	X
ejpam-3659	45	2	0	0	NUM
ejpam-3659	45	3	,	,	PUNCT
ejpam-3659	45	4	1	1	NUM
ejpam-3659	45	5	]	]	PUNCT
ejpam-3659	45	6	)	)	PUNCT
ejpam-3659	45	7	≥	≥	NOUN
ejpam-3659	46	1	∞∑	∞∑	PROPN
ejpam-3659	46	2	n=0	n=0	NUM
ejpam-3659	46	3	|f	|f	PROPN
ejpam-3659	46	4	(	(	PUNCT
ejpam-3659	46	5	xn)−	xn)−	NOUN
ejpam-3659	46	6	f	f	X
ejpam-3659	46	7	(	(	PUNCT
ejpam-3659	46	8	xn+1)|	xn+1)|	PROPN
ejpam-3659	46	9	,	,	PUNCT
ejpam-3659	46	10	f.	f.	PROPN
ejpam-3659	46	11	sumalpong	sumalpong	PROPN
ejpam-3659	46	12	jr	jr	PROPN
ejpam-3659	46	13	.	.	PROPN
ejpam-3659	46	14	,	,	PUNCT
ejpam-3659	46	15	j.	j.	PROPN
ejpam-3659	46	16	benitez	benitez	PROPN
ejpam-3659	46	17	/	/	PUNCT
ejpam-3659	46	18	eur	eur	PROPN
ejpam-3659	46	19	.	.	PUNCT
ejpam-3659	47	1	j.	j.	PROPN
ejpam-3659	47	2	pure	pure	PROPN
ejpam-3659	47	3	appl	appl	PROPN
ejpam-3659	47	4	.	.	PROPN
ejpam-3659	47	5	math	math	PROPN
ejpam-3659	47	6	,	,	PUNCT
ejpam-3659	47	7	13	13	NUM
ejpam-3659	47	8	(	(	PUNCT
ejpam-3659	47	9	2	2	NUM
ejpam-3659	47	10	)	)	PUNCT
ejpam-3659	47	11	(	(	PUNCT
ejpam-3659	47	12	2020	2020	NUM
ejpam-3659	47	13	)	)	PUNCT
ejpam-3659	47	14	,	,	PUNCT
ejpam-3659	47	15	303	303	NUM
ejpam-3659	47	16	-	-	SYM
ejpam-3659	47	17	313	313	NUM
ejpam-3659	47	18	305	305	NUM
ejpam-3659	47	19	and	and	CCONJ
ejpam-3659	47	20	so	so	ADV
ejpam-3659	47	21	,	,	PUNCT
ejpam-3659	47	22	v	v	INTJ
ejpam-3659	47	23	(	(	PUNCT
ejpam-3659	47	24	f	f	NOUN
ejpam-3659	47	25	;	;	PUNCT
ejpam-3659	48	1	[	[	X
ejpam-3659	48	2	0	0	NUM
ejpam-3659	48	3	,	,	PUNCT
ejpam-3659	48	4	1	1	NUM
ejpam-3659	48	5	]	]	PUNCT
ejpam-3659	48	6	)	)	PUNCT
ejpam-3659	49	1	=	=	SYM
ejpam-3659	49	2	∞.	∞.	PROPN
ejpam-3659	49	3	thus	thus	ADV
ejpam-3659	49	4	,	,	PUNCT
ejpam-3659	49	5	f	f	PROPN
ejpam-3659	49	6	is	be	AUX
ejpam-3659	49	7	not	not	PART
ejpam-3659	49	8	of	of	ADP
ejpam-3659	49	9	bounded	bounded	ADJ
ejpam-3659	49	10	variation	variation	NOUN
ejpam-3659	49	11	.	.	PUNCT
ejpam-3659	50	1	recall	recall	VERB
ejpam-3659	50	2	that	that	PRON
ejpam-3659	50	3	in	in	ADP
ejpam-3659	50	4	[	[	X
ejpam-3659	50	5	4	4	NUM
ejpam-3659	50	6	,	,	PUNCT
ejpam-3659	50	7	p.19	p.19	NOUN
ejpam-3659	50	8	]	]	PUNCT
ejpam-3659	50	9	,	,	PUNCT
ejpam-3659	50	10	if	if	SCONJ
ejpam-3659	50	11	f	f	X
ejpam-3659	50	12	:	:	PUNCT
ejpam-3659	51	1	[	[	X
ejpam-3659	51	2	a	a	X
ejpam-3659	51	3	,	,	PUNCT
ejpam-3659	51	4	b]→	b]→	ADJ
ejpam-3659	51	5	r	r	NOUN
ejpam-3659	51	6	is	be	AUX
ejpam-3659	51	7	mcshane	mcshane	NOUN
ejpam-3659	51	8	integrable	integrable	ADJ
ejpam-3659	51	9	on	on	ADP
ejpam-3659	51	10	[	[	X
ejpam-3659	51	11	a	a	X
ejpam-3659	51	12	,	,	PUNCT
ejpam-3659	51	13	b	b	NOUN
ejpam-3659	51	14	]	]	X
ejpam-3659	51	15	,	,	PUNCT
ejpam-3659	51	16	then	then	ADV
ejpam-3659	51	17	the	the	DET
ejpam-3659	51	18	primitive	primitive	NOUN
ejpam-3659	51	19	of	of	ADP
ejpam-3659	51	20	f	f	PROPN
ejpam-3659	51	21	is	be	AUX
ejpam-3659	51	22	of	of	ADP
ejpam-3659	51	23	bounded	bounded	ADJ
ejpam-3659	51	24	variation	variation	NOUN
ejpam-3659	51	25	.	.	PUNCT
ejpam-3659	52	1	hence	hence	ADV
ejpam-3659	52	2	,	,	PUNCT
ejpam-3659	52	3	f	f	PROPN
ejpam-3659	52	4	is	be	AUX
ejpam-3659	52	5	not	not	PART
ejpam-3659	52	6	mcshane	mcshane	NOUN
ejpam-3659	52	7	integrable	integrable	ADJ
ejpam-3659	52	8	on	on	ADP
ejpam-3659	52	9	[	[	X
ejpam-3659	52	10	0	0	NUM
ejpam-3659	52	11	,	,	PUNCT
ejpam-3659	52	12	1	1	NUM
ejpam-3659	52	13	]	]	PUNCT
ejpam-3659	52	14	.	.	PUNCT
ejpam-3659	53	1	moreover	moreover	ADV
ejpam-3659	53	2	,	,	PUNCT
ejpam-3659	53	3	for	for	ADP
ejpam-3659	53	4	each	each	DET
ejpam-3659	53	5	ε	ε	PROPN
ejpam-3659	53	6	∈	∈	PROPN
ejpam-3659	53	7	(	(	PUNCT
ejpam-3659	53	8	0	0	NUM
ejpam-3659	53	9	,	,	PUNCT
ejpam-3659	53	10	1	1	NUM
ejpam-3659	53	11	)	)	PUNCT
ejpam-3659	53	12	,	,	PUNCT
ejpam-3659	53	13	f	f	PROPN
ejpam-3659	53	14	is	be	AUX
ejpam-3659	53	15	continuous	continuous	ADJ
ejpam-3659	53	16	on	on	ADP
ejpam-3659	53	17	[	[	X
ejpam-3659	53	18	ε	ε	PROPN
ejpam-3659	53	19	,	,	PUNCT
ejpam-3659	53	20	1	1	NUM
ejpam-3659	53	21	]	]	PUNCT
ejpam-3659	53	22	which	which	PRON
ejpam-3659	53	23	implies	imply	VERB
ejpam-3659	53	24	that	that	SCONJ
ejpam-3659	53	25	f	f	PROPN
ejpam-3659	53	26	is	be	AUX
ejpam-3659	53	27	mcshane	mcshane	PROPN
ejpam-3659	53	28	integrable	integrable	ADJ
ejpam-3659	53	29	on	on	ADP
ejpam-3659	53	30	every	every	DET
ejpam-3659	53	31	closed	closed	ADJ
ejpam-3659	53	32	interval	interval	NOUN
ejpam-3659	54	1	[	[	X
ejpam-3659	54	2	ε	ε	PROPN
ejpam-3659	54	3	,	,	PUNCT
ejpam-3659	54	4	1	1	NUM
ejpam-3659	54	5	]	]	PUNCT
ejpam-3659	54	6	.	.	PUNCT
ejpam-3659	55	1	however	however	ADV
ejpam-3659	55	2	,	,	PUNCT
ejpam-3659	55	3	by	by	ADP
ejpam-3659	55	4	theorem	theorem	NOUN
ejpam-3659	55	5	16	16	NUM
ejpam-3659	55	6	of	of	ADP
ejpam-3659	55	7	[	[	X
ejpam-3659	55	8	11	11	NUM
ejpam-3659	55	9	]	]	PUNCT
ejpam-3659	55	10	,	,	PUNCT
ejpam-3659	55	11	f	f	PROPN
ejpam-3659	55	12	is	be	AUX
ejpam-3659	55	13	henstock	henstock	NOUN
ejpam-3659	55	14	integrable	integrable	ADJ
ejpam-3659	55	15	on	on	ADP
ejpam-3659	55	16	[	[	X
ejpam-3659	55	17	0	0	NUM
ejpam-3659	55	18	,	,	PUNCT
ejpam-3659	55	19	1	1	NUM
ejpam-3659	55	20	]	]	PUNCT
ejpam-3659	55	21	.	.	PUNCT
ejpam-3659	56	1	�	�	PROPN
ejpam-3659	56	2	if	if	SCONJ
ejpam-3659	56	3	f	f	PROPN
ejpam-3659	56	4	is	be	AUX
ejpam-3659	56	5	henstock	henstock	NOUN
ejpam-3659	56	6	-	-	PUNCT
ejpam-3659	56	7	kurzweil	kurzweil	NOUN
ejpam-3659	56	8	integrable	integrable	ADJ
ejpam-3659	56	9	on	on	ADP
ejpam-3659	56	10	every	every	DET
ejpam-3659	56	11	measurable	measurable	ADJ
ejpam-3659	56	12	subset	subset	NOUN
ejpam-3659	56	13	of	of	ADP
ejpam-3659	56	14	[	[	X
ejpam-3659	56	15	a	a	X
ejpam-3659	56	16	,	,	PUNCT
ejpam-3659	56	17	b	b	NOUN
ejpam-3659	56	18	]	]	X
ejpam-3659	56	19	,	,	PUNCT
ejpam-3659	56	20	then	then	ADV
ejpam-3659	56	21	f	f	PROPN
ejpam-3659	56	22	is	be	AUX
ejpam-3659	56	23	lebesgue	lebesgue	NOUN
ejpam-3659	56	24	integrable	integrable	ADJ
ejpam-3659	56	25	on	on	ADP
ejpam-3659	56	26	[	[	X
ejpam-3659	56	27	a	a	X
ejpam-3659	56	28	,	,	PUNCT
ejpam-3659	56	29	b	b	NOUN
ejpam-3659	56	30	]	]	X
ejpam-3659	56	31	,	,	PUNCT
ejpam-3659	56	32	see	see	VERB
ejpam-3659	56	33	[	[	X
ejpam-3659	56	34	5	5	NUM
ejpam-3659	56	35	,	,	PUNCT
ejpam-3659	56	36	theorem	theorem	VERB
ejpam-3659	56	37	9.13	9.13	NUM
ejpam-3659	56	38	]	]	PUNCT
ejpam-3659	56	39	.	.	PUNCT
ejpam-3659	57	1	in	in	ADP
ejpam-3659	57	2	particular	particular	ADJ
ejpam-3659	57	3	,	,	PUNCT
ejpam-3659	57	4	if	if	SCONJ
ejpam-3659	57	5	f	f	PROPN
ejpam-3659	57	6	is	be	AUX
ejpam-3659	57	7	mcshane	mcshane	PROPN
ejpam-3659	57	8	integrable	integrable	ADJ
ejpam-3659	57	9	on	on	ADP
ejpam-3659	57	10	every	every	DET
ejpam-3659	57	11	measurable	measurable	ADJ
ejpam-3659	57	12	subset	subset	NOUN
ejpam-3659	57	13	of	of	ADP
ejpam-3659	57	14	[	[	X
ejpam-3659	57	15	a	a	X
ejpam-3659	57	16	,	,	PUNCT
ejpam-3659	57	17	b	b	NOUN
ejpam-3659	57	18	]	]	X
ejpam-3659	57	19	,	,	PUNCT
ejpam-3659	57	20	then	then	ADV
ejpam-3659	57	21	f	f	PROPN
ejpam-3659	57	22	is	be	AUX
ejpam-3659	57	23	mcshane	mcshane	PROPN
ejpam-3659	57	24	integrable	integrable	ADJ
ejpam-3659	57	25	on	on	ADP
ejpam-3659	57	26	[	[	X
ejpam-3659	57	27	a	a	X
ejpam-3659	57	28	,	,	PUNCT
ejpam-3659	57	29	b	b	NOUN
ejpam-3659	57	30	]	]	X
ejpam-3659	57	31	.	.	PUNCT
ejpam-3659	58	1	it	it	PRON
ejpam-3659	58	2	was	be	AUX
ejpam-3659	58	3	pointed	point	VERB
ejpam-3659	58	4	out	out	ADP
ejpam-3659	58	5	in	in	ADP
ejpam-3659	58	6	[	[	X
ejpam-3659	58	7	3	3	X
ejpam-3659	58	8	]	]	PUNCT
ejpam-3659	58	9	that	that	SCONJ
ejpam-3659	58	10	if	if	SCONJ
ejpam-3659	58	11	f	f	X
ejpam-3659	58	12	:	:	PUNCT
ejpam-3659	59	1	[	[	X
ejpam-3659	59	2	a	a	X
ejpam-3659	59	3	,	,	PUNCT
ejpam-3659	59	4	b]→	b]→	ADJ
ejpam-3659	59	5	r	r	NOUN
ejpam-3659	59	6	is	be	AUX
ejpam-3659	59	7	mcshane	mcshane	NOUN
ejpam-3659	59	8	integrable	integrable	ADJ
ejpam-3659	59	9	on	on	ADP
ejpam-3659	59	10	[	[	X
ejpam-3659	59	11	a	a	X
ejpam-3659	59	12	,	,	PUNCT
ejpam-3659	59	13	b	b	NOUN
ejpam-3659	59	14	]	]	X
ejpam-3659	59	15	,	,	PUNCT
ejpam-3659	59	16	then	then	ADV
ejpam-3659	59	17	f	f	PROPN
ejpam-3659	59	18	is	be	AUX
ejpam-3659	59	19	mcshane	mcshane	PROPN
ejpam-3659	59	20	integrable	integrable	ADJ
ejpam-3659	59	21	on	on	ADP
ejpam-3659	59	22	[	[	X
ejpam-3659	59	23	a	a	DET
ejpam-3659	59	24	,	,	PUNCT
ejpam-3659	59	25	b	b	NOUN
ejpam-3659	59	26	]	]	X
ejpam-3659	59	27	on	on	ADP
ejpam-3659	59	28	every	every	DET
ejpam-3659	59	29	integrable	integrable	ADJ
ejpam-3659	59	30	(	(	PUNCT
ejpam-3659	59	31	equivalently	equivalently	ADV
ejpam-3659	59	32	,	,	PUNCT
ejpam-3659	59	33	lebesgue	lebesgue	ADJ
ejpam-3659	59	34	measurable	measurable	NOUN
ejpam-3659	59	35	)	)	PUNCT
ejpam-3659	59	36	subset	subset	NOUN
ejpam-3659	59	37	of	of	ADP
ejpam-3659	59	38	[	[	X
ejpam-3659	59	39	a	a	X
ejpam-3659	59	40	,	,	PUNCT
ejpam-3659	59	41	b	b	NOUN
ejpam-3659	59	42	]	]	X
ejpam-3659	59	43	.	.	PUNCT
ejpam-3659	60	1	if	if	SCONJ
ejpam-3659	60	2	f	f	PROPN
ejpam-3659	60	3	is	be	AUX
ejpam-3659	60	4	henstock	henstock	NOUN
ejpam-3659	60	5	-	-	PUNCT
ejpam-3659	60	6	kurzweil	kurzweil	NOUN
ejpam-3659	60	7	integrable	integrable	ADJ
ejpam-3659	60	8	on	on	ADP
ejpam-3659	60	9	[	[	X
ejpam-3659	60	10	a	a	X
ejpam-3659	60	11	,	,	PUNCT
ejpam-3659	60	12	b	b	NOUN
ejpam-3659	60	13	]	]	X
ejpam-3659	60	14	,	,	PUNCT
ejpam-3659	60	15	then	then	ADV
ejpam-3659	60	16	[	[	X
ejpam-3659	60	17	a	a	X
ejpam-3659	60	18	,	,	PUNCT
ejpam-3659	60	19	b	b	X
ejpam-3659	60	20	]	]	PUNCT
ejpam-3659	60	21	contains	contain	VERB
ejpam-3659	60	22	a	a	DET
ejpam-3659	60	23	subinterval	subinterval	NOUN
ejpam-3659	60	24	on	on	ADP
ejpam-3659	60	25	which	which	PRON
ejpam-3659	60	26	f	f	PROPN
ejpam-3659	60	27	is	be	AUX
ejpam-3659	60	28	lebesgue	lebesgue	NOUN
ejpam-3659	60	29	(	(	PUNCT
ejpam-3659	60	30	mcshane	mcshane	NOUN
ejpam-3659	60	31	)	)	PUNCT
ejpam-3659	60	32	integrable	integrable	ADJ
ejpam-3659	60	33	,	,	PUNCT
ejpam-3659	60	34	see	see	VERB
ejpam-3659	60	35	[	[	X
ejpam-3659	60	36	5	5	NUM
ejpam-3659	60	37	,	,	PUNCT
ejpam-3659	60	38	corollary	corollary	ADJ
ejpam-3659	60	39	9.19	9.19	NUM
ejpam-3659	60	40	]	]	PUNCT
ejpam-3659	60	41	and	and	CCONJ
ejpam-3659	60	42	[	[	X
ejpam-3659	60	43	8	8	NUM
ejpam-3659	60	44	]	]	PUNCT
ejpam-3659	60	45	.	.	PUNCT
ejpam-3659	61	1	thus	thus	ADV
ejpam-3659	61	2	,	,	PUNCT
ejpam-3659	61	3	a	a	DET
ejpam-3659	61	4	natural	natural	ADJ
ejpam-3659	61	5	problem	problem	NOUN
ejpam-3659	61	6	is	be	AUX
ejpam-3659	61	7	:	:	PUNCT
ejpam-3659	61	8	if	if	SCONJ
ejpam-3659	61	9	f	f	X
ejpam-3659	61	10	:	:	PUNCT
ejpam-3659	62	1	[	[	X
ejpam-3659	62	2	a	a	X
ejpam-3659	62	3	,	,	PUNCT
ejpam-3659	62	4	b]→	b]→	ADJ
ejpam-3659	62	5	r	r	NOUN
ejpam-3659	62	6	is	be	AUX
ejpam-3659	62	7	mcshane	mcshane	NOUN
ejpam-3659	62	8	integrable	integrable	ADJ
ejpam-3659	62	9	on	on	ADP
ejpam-3659	62	10	[	[	X
ejpam-3659	62	11	a	a	PRON
ejpam-3659	62	12	,	,	PUNCT
ejpam-3659	62	13	b	b	NOUN
ejpam-3659	62	14	]	]	X
ejpam-3659	62	15	and	and	CCONJ
ejpam-3659	62	16	x	x	SYM
ejpam-3659	62	17	⊆	⊆	NUM
ejpam-3659	62	18	[	[	X
ejpam-3659	62	19	a	a	X
ejpam-3659	62	20	,	,	PUNCT
ejpam-3659	62	21	b	b	NOUN
ejpam-3659	62	22	]	]	X
ejpam-3659	62	23	,	,	PUNCT
ejpam-3659	62	24	find	find	VERB
ejpam-3659	62	25	a	a	DET
ejpam-3659	62	26	condition	condition	NOUN
ejpam-3659	62	27	satisfied	satisfy	VERB
ejpam-3659	62	28	by	by	ADP
ejpam-3659	62	29	x	x	PUNCT
ejpam-3659	62	30	which	which	PRON
ejpam-3659	62	31	is	be	AUX
ejpam-3659	62	32	necessary	necessary	ADJ
ejpam-3659	62	33	and	and	CCONJ
ejpam-3659	62	34	sufficient	sufficient	ADJ
ejpam-3659	62	35	for	for	ADP
ejpam-3659	62	36	the	the	DET
ejpam-3659	62	37	mcshane	mcshane	PROPN
ejpam-3659	62	38	integrability	integrability	NOUN
ejpam-3659	62	39	of	of	ADP
ejpam-3659	62	40	f	f	PROPN
ejpam-3659	62	41	on	on	ADP
ejpam-3659	62	42	x.	x.	NOUN
ejpam-3659	62	43	in	in	ADP
ejpam-3659	62	44	this	this	DET
ejpam-3659	62	45	paper	paper	NOUN
ejpam-3659	62	46	,	,	PUNCT
ejpam-3659	62	47	we	we	PRON
ejpam-3659	62	48	give	give	VERB
ejpam-3659	62	49	a	a	DET
ejpam-3659	62	50	characterization	characterization	NOUN
ejpam-3659	62	51	for	for	ADP
ejpam-3659	62	52	the	the	DET
ejpam-3659	62	53	mcshane	mcshane	PROPN
ejpam-3659	62	54	integrability	integrability	NOUN
ejpam-3659	62	55	of	of	ADP
ejpam-3659	62	56	f	f	NOUN
ejpam-3659	62	57	:	:	PUNCT
ejpam-3659	63	1	[	[	X
ejpam-3659	63	2	a	a	X
ejpam-3659	63	3	,	,	PUNCT
ejpam-3659	63	4	b]→	b]→	ADJ
ejpam-3659	63	5	r	r	NOUN
ejpam-3659	63	6	on	on	ADP
ejpam-3659	63	7	x	x	X
ejpam-3659	63	8	⊆	⊆	NUM
ejpam-3659	63	9	[	[	X
ejpam-3659	63	10	a	a	X
ejpam-3659	63	11	,	,	PUNCT
ejpam-3659	63	12	b	b	NOUN
ejpam-3659	63	13	]	]	PUNCT
ejpam-3659	63	14	using	use	VERB
ejpam-3659	63	15	concept	concept	NOUN
ejpam-3659	63	16	of	of	ADP
ejpam-3659	63	17	mcshane	mcshane	PROPN
ejpam-3659	63	18	variational	variational	ADJ
ejpam-3659	63	19	measure	measure	NOUN
ejpam-3659	63	20	.	.	PUNCT
ejpam-3659	64	1	2	2	X
ejpam-3659	64	2	.	.	X
ejpam-3659	64	3	preliminary	preliminary	ADJ
ejpam-3659	64	4	concepts	concept	NOUN
ejpam-3659	64	5	and	and	CCONJ
ejpam-3659	64	6	known	know	VERB
ejpam-3659	64	7	results	result	NOUN
ejpam-3659	64	8	a	a	DET
ejpam-3659	64	9	gauge	gauge	NOUN
ejpam-3659	64	10	on	on	ADP
ejpam-3659	64	11	[	[	X
ejpam-3659	64	12	a	a	X
ejpam-3659	64	13	,	,	PUNCT
ejpam-3659	64	14	b	b	X
ejpam-3659	64	15	]	]	X
ejpam-3659	64	16	is	be	AUX
ejpam-3659	64	17	a	a	DET
ejpam-3659	64	18	positive	positive	ADJ
ejpam-3659	64	19	function	function	NOUN
ejpam-3659	64	20	δ	δ	NOUN
ejpam-3659	64	21	:	:	PUNCT
ejpam-3659	65	1	[	[	X
ejpam-3659	65	2	a	a	X
ejpam-3659	65	3	,	,	PUNCT
ejpam-3659	65	4	b	b	NOUN
ejpam-3659	65	5	]	]	X
ejpam-3659	65	6	→	→	SYM
ejpam-3659	65	7	r+	r+	X
ejpam-3659	65	8	.	.	PUNCT
ejpam-3659	66	1	a	a	DET
ejpam-3659	66	2	henstock	henstock	NOUN
ejpam-3659	66	3	δ	δ	PROPN
ejpam-3659	66	4	-	-	PUNCT
ejpam-3659	66	5	fine	fine	ADJ
ejpam-3659	66	6	division	division	NOUN
ejpam-3659	66	7	of	of	ADP
ejpam-3659	66	8	[	[	X
ejpam-3659	66	9	a	a	X
ejpam-3659	66	10	,	,	PUNCT
ejpam-3659	66	11	b	b	X
ejpam-3659	66	12	]	]	X
ejpam-3659	66	13	is	be	AUX
ejpam-3659	66	14	a	a	DET
ejpam-3659	66	15	finite	finite	ADJ
ejpam-3659	66	16	collection	collection	NOUN
ejpam-3659	66	17	d	d	NOUN
ejpam-3659	66	18	=	=	PRON
ejpam-3659	66	19	{	{	PUNCT
ejpam-3659	66	20	(	(	PUNCT
ejpam-3659	66	21	[	[	X
ejpam-3659	66	22	xi−1	xi−1	PROPN
ejpam-3659	66	23	,	,	PUNCT
ejpam-3659	66	24	xi	xi	ADP
ejpam-3659	66	25	]	]	PUNCT
ejpam-3659	66	26	,	,	PUNCT
ejpam-3659	66	27	ξi)}ni=1	ξi)}ni=1	PROPN
ejpam-3659	66	28	of	of	ADP
ejpam-3659	66	29	non	non	ADJ
ejpam-3659	66	30	-	-	ADJ
ejpam-3659	66	31	overlapping	overlapping	ADJ
ejpam-3659	66	32	interval	interval	NOUN
ejpam-3659	66	33	-	-	PUNCT
ejpam-3659	66	34	point	point	NOUN
ejpam-3659	66	35	pairs	pair	NOUN
ejpam-3659	66	36	such	such	ADJ
ejpam-3659	66	37	that	that	PRON
ejpam-3659	66	38	for	for	ADP
ejpam-3659	66	39	all	all	DET
ejpam-3659	66	40	i	i	PRON
ejpam-3659	66	41	=	=	NOUN
ejpam-3659	66	42	1	1	NUM
ejpam-3659	66	43	,	,	PUNCT
ejpam-3659	66	44	2	2	NUM
ejpam-3659	66	45	,	,	PUNCT
ejpam-3659	66	46	.	.	PUNCT
ejpam-3659	66	47	.	.	PUNCT
ejpam-3659	67	1	.	.	PUNCT
ejpam-3659	68	1	,	,	PUNCT
ejpam-3659	68	2	n	n	PRON
ejpam-3659	68	3	ξi	ξi	NOUN
ejpam-3659	68	4	∈	∈	PROPN
ejpam-3659	68	5	[	[	X
ejpam-3659	68	6	xi−1	xi−1	PROPN
ejpam-3659	68	7	,	,	PUNCT
ejpam-3659	68	8	xi	xi	X
ejpam-3659	68	9	]	]	X
ejpam-3659	68	10	⊆	⊆	NUM
ejpam-3659	68	11	(	(	PUNCT
ejpam-3659	68	12	ξi	ξi	NOUN
ejpam-3659	68	13	−	−	NOUN
ejpam-3659	68	14	δ(ξi	δ(ξi	ADP
ejpam-3659	68	15	)	)	PUNCT
ejpam-3659	68	16	,	,	PUNCT
ejpam-3659	68	17	ξi	ξi	NOUN
ejpam-3659	68	18	+	+	CCONJ
ejpam-3659	68	19	δ(ξi	δ(ξi	NOUN
ejpam-3659	68	20	)	)	PUNCT
ejpam-3659	68	21	)	)	PUNCT
ejpam-3659	68	22	and	and	CCONJ
ejpam-3659	68	23	n⋃	n⋃	PRON
ejpam-3659	68	24	i=1	i=1	PROPN
ejpam-3659	69	1	[	[	X
ejpam-3659	69	2	xi−1	xi−1	PROPN
ejpam-3659	69	3	,	,	PUNCT
ejpam-3659	69	4	xi	xi	X
ejpam-3659	69	5	]	]	PUNCT
ejpam-3659	69	6	=	=	PUNCT
ejpam-3659	70	1	[	[	X
ejpam-3659	70	2	a	a	X
ejpam-3659	70	3	,	,	PUNCT
ejpam-3659	70	4	b	b	NOUN
ejpam-3659	70	5	]	]	X
ejpam-3659	70	6	.	.	PUNCT
ejpam-3659	71	1	we	we	PRON
ejpam-3659	71	2	say	say	VERB
ejpam-3659	71	3	that	that	SCONJ
ejpam-3659	71	4	d	d	PROPN
ejpam-3659	71	5	=	=	PRON
ejpam-3659	71	6	{	{	PUNCT
ejpam-3659	71	7	(	(	PUNCT
ejpam-3659	71	8	[	[	X
ejpam-3659	71	9	xi−1	xi−1	PROPN
ejpam-3659	71	10	,	,	PUNCT
ejpam-3659	71	11	xi	xi	ADP
ejpam-3659	71	12	]	]	PUNCT
ejpam-3659	71	13	,	,	PUNCT
ejpam-3659	71	14	ξi)}ni=1	ξi)}ni=1	PROPN
ejpam-3659	71	15	is	be	AUX
ejpam-3659	71	16	a	a	DET
ejpam-3659	71	17	mcshane	mcshane	PROPN
ejpam-3659	71	18	δ	δ	PROPN
ejpam-3659	71	19	-	-	PUNCT
ejpam-3659	71	20	fine	fine	ADJ
ejpam-3659	71	21	division	division	NOUN
ejpam-3659	71	22	of	of	ADP
ejpam-3659	71	23	[	[	X
ejpam-3659	71	24	a	a	X
ejpam-3659	71	25	,	,	PUNCT
ejpam-3659	71	26	b	b	NOUN
ejpam-3659	71	27	]	]	X
ejpam-3659	71	28	if	if	SCONJ
ejpam-3659	71	29	for	for	ADP
ejpam-3659	71	30	all	all	DET
ejpam-3659	71	31	i	i	PRON
ejpam-3659	71	32	=	=	NOUN
ejpam-3659	71	33	1	1	NUM
ejpam-3659	71	34	,	,	PUNCT
ejpam-3659	71	35	2	2	NUM
ejpam-3659	71	36	,	,	PUNCT
ejpam-3659	71	37	.	.	PUNCT
ejpam-3659	71	38	.	.	PUNCT
ejpam-3659	72	1	.	.	PUNCT
ejpam-3659	73	1	,	,	PUNCT
ejpam-3659	74	1	n	n	PROPN
ejpam-3659	74	2	[	[	X
ejpam-3659	74	3	xi−1	xi−1	PROPN
ejpam-3659	74	4	,	,	PUNCT
ejpam-3659	74	5	xi	xi	X
ejpam-3659	74	6	]	]	X
ejpam-3659	74	7	⊆	⊆	NUM
ejpam-3659	74	8	(	(	PUNCT
ejpam-3659	74	9	ξi	ξi	NOUN
ejpam-3659	74	10	−	−	NOUN
ejpam-3659	74	11	δ(ξi	δ(ξi	ADP
ejpam-3659	74	12	)	)	PUNCT
ejpam-3659	74	13	,	,	PUNCT
ejpam-3659	74	14	ξi	ξi	NOUN
ejpam-3659	74	15	+	+	CCONJ
ejpam-3659	74	16	δ(ξi	δ(ξi	NOUN
ejpam-3659	74	17	)	)	PUNCT
ejpam-3659	74	18	)	)	PUNCT
ejpam-3659	74	19	,	,	PUNCT
ejpam-3659	74	20	ξi	ξi	NOUN
ejpam-3659	74	21	∈	∈	PROPN
ejpam-3659	75	1	[	[	X
ejpam-3659	75	2	a	a	X
ejpam-3659	75	3	,	,	PUNCT
ejpam-3659	75	4	b	b	NOUN
ejpam-3659	75	5	]	]	X
ejpam-3659	75	6	,	,	PUNCT
ejpam-3659	75	7	and	and	CCONJ
ejpam-3659	75	8	n⋃	n⋃	VERB
ejpam-3659	75	9	i=1	i=1	PROPN
ejpam-3659	76	1	[	[	X
ejpam-3659	76	2	xi−1	xi−1	PROPN
ejpam-3659	76	3	,	,	PUNCT
ejpam-3659	76	4	xi	xi	X
ejpam-3659	76	5	]	]	PUNCT
ejpam-3659	76	6	=	=	PUNCT
ejpam-3659	77	1	[	[	X
ejpam-3659	77	2	a	a	X
ejpam-3659	77	3	,	,	PUNCT
ejpam-3659	77	4	b	b	NOUN
ejpam-3659	77	5	]	]	X
ejpam-3659	77	6	.	.	PUNCT
ejpam-3659	78	1	this	this	PRON
ejpam-3659	78	2	means	mean	VERB
ejpam-3659	78	3	that	that	SCONJ
ejpam-3659	78	4	every	every	DET
ejpam-3659	78	5	henstock	henstock	NOUN
ejpam-3659	78	6	δ	δ	NOUN
ejpam-3659	78	7	-	-	PUNCT
ejpam-3659	78	8	fine	fine	ADJ
ejpam-3659	78	9	divisions	division	NOUN
ejpam-3659	78	10	of	of	ADP
ejpam-3659	78	11	[	[	X
ejpam-3659	78	12	a	a	X
ejpam-3659	78	13	,	,	PUNCT
ejpam-3659	78	14	b	b	NOUN
ejpam-3659	78	15	]	]	X
ejpam-3659	78	16	are	be	AUX
ejpam-3659	78	17	mcshane	mcshane	PROPN
ejpam-3659	78	18	δ	δ	PROPN
ejpam-3659	78	19	-	-	PUNCT
ejpam-3659	78	20	fine	fine	PROPN
ejpam-3659	78	21	.	.	PUNCT
ejpam-3659	79	1	for	for	ADP
ejpam-3659	79	2	brevity	brevity	NOUN
ejpam-3659	79	3	,	,	PUNCT
ejpam-3659	79	4	we	we	PRON
ejpam-3659	79	5	use	use	VERB
ejpam-3659	79	6	(	(	PUNCT
ejpam-3659	79	7	[	[	X
ejpam-3659	79	8	u	u	NOUN
ejpam-3659	79	9	,	,	PUNCT
ejpam-3659	79	10	v	v	ADP
ejpam-3659	79	11	]	]	X
ejpam-3659	79	12	,	,	PUNCT
ejpam-3659	79	13	ξ	ξ	X
ejpam-3659	79	14	)	)	PUNCT
ejpam-3659	79	15	to	to	PART
ejpam-3659	79	16	represent	represent	VERB
ejpam-3659	79	17	a	a	DET
ejpam-3659	79	18	typical	typical	ADJ
ejpam-3659	79	19	interval	interval	NOUN
ejpam-3659	79	20	-	-	PUNCT
ejpam-3659	79	21	point	point	NOUN
ejpam-3659	79	22	pair	pair	NOUN
ejpam-3659	79	23	(	(	PUNCT
ejpam-3659	79	24	[	[	X
ejpam-3659	79	25	xi−1	xi−1	PROPN
ejpam-3659	79	26	,	,	PUNCT
ejpam-3659	79	27	xi	xi	ADP
ejpam-3659	79	28	]	]	PUNCT
ejpam-3659	79	29	,	,	PUNCT
ejpam-3659	79	30	ξi	ξi	NUM
ejpam-3659	79	31	)	)	PUNCT
ejpam-3659	79	32	∈	∈	PROPN
ejpam-3659	79	33	d.	d.	PROPN
ejpam-3659	79	34	a	a	DET
ejpam-3659	79	35	finite	finite	ADJ
ejpam-3659	79	36	collection	collection	NOUN
ejpam-3659	79	37	p	p	NOUN
ejpam-3659	79	38	=	=	X
ejpam-3659	79	39	{	{	PUNCT
ejpam-3659	79	40	(	(	PUNCT
ejpam-3659	79	41	[	[	X
ejpam-3659	79	42	u	u	NOUN
ejpam-3659	79	43	,	,	PUNCT
ejpam-3659	79	44	v	v	ADP
ejpam-3659	79	45	]	]	X
ejpam-3659	79	46	,	,	PUNCT
ejpam-3659	79	47	ξ	ξ	X
ejpam-3659	79	48	)	)	PUNCT
ejpam-3659	79	49	}	}	PUNCT
ejpam-3659	79	50	of	of	ADP
ejpam-3659	79	51	interval	interval	NOUN
ejpam-3659	79	52	-	-	PUNCT
ejpam-3659	79	53	point	point	NOUN
ejpam-3659	79	54	pairs	pair	NOUN
ejpam-3659	79	55	is	be	AUX
ejpam-3659	79	56	a	a	DET
ejpam-3659	79	57	partial	partial	ADJ
ejpam-3659	79	58	division	division	NOUN
ejpam-3659	79	59	of	of	ADP
ejpam-3659	79	60	[	[	X
ejpam-3659	79	61	a	a	X
ejpam-3659	79	62	,	,	PUNCT
ejpam-3659	79	63	b	b	NOUN
ejpam-3659	79	64	]	]	X
ejpam-3659	79	65	if	if	SCONJ
ejpam-3659	79	66	∪[u	∪[u	NOUN
ejpam-3659	79	67	,	,	PUNCT
ejpam-3659	79	68	v	v	NOUN
ejpam-3659	79	69	]	]	PUNCT
ejpam-3659	79	70	⊆	⊆	NUM
ejpam-3659	79	71	[	[	X
ejpam-3659	79	72	a	a	X
ejpam-3659	79	73	,	,	PUNCT
ejpam-3659	79	74	b	b	NOUN
ejpam-3659	79	75	]	]	PUNCT
ejpam-3659	79	76	.	.	PUNCT
ejpam-3659	80	1	interested	interested	ADJ
ejpam-3659	80	2	readers	reader	NOUN
ejpam-3659	80	3	may	may	AUX
ejpam-3659	80	4	refer	refer	VERB
ejpam-3659	80	5	to	to	ADP
ejpam-3659	80	6	[	[	X
ejpam-3659	80	7	5–7	5–7	NOUN
ejpam-3659	80	8	,	,	PUNCT
ejpam-3659	80	9	10	10	NUM
ejpam-3659	80	10	]	]	PUNCT
ejpam-3659	80	11	for	for	ADP
ejpam-3659	80	12	more	more	ADJ
ejpam-3659	80	13	details	detail	NOUN
ejpam-3659	80	14	on	on	ADP
ejpam-3659	80	15	the	the	DET
ejpam-3659	80	16	basic	basic	ADJ
ejpam-3659	80	17	concepts	concept	NOUN
ejpam-3659	80	18	introduced	introduce	VERB
ejpam-3659	80	19	.	.	PUNCT
ejpam-3659	81	1	f.	f.	PROPN
ejpam-3659	81	2	sumalpong	sumalpong	PROPN
ejpam-3659	81	3	jr	jr	PROPN
ejpam-3659	81	4	.	.	PROPN
ejpam-3659	81	5	,	,	PUNCT
ejpam-3659	81	6	j.	j.	PROPN
ejpam-3659	81	7	benitez	benitez	PROPN
ejpam-3659	81	8	/	/	PUNCT
ejpam-3659	81	9	eur	eur	PROPN
ejpam-3659	81	10	.	.	PUNCT
ejpam-3659	82	1	j.	j.	PROPN
ejpam-3659	82	2	pure	pure	PROPN
ejpam-3659	82	3	appl	appl	PROPN
ejpam-3659	82	4	.	.	PROPN
ejpam-3659	82	5	math	math	PROPN
ejpam-3659	82	6	,	,	PUNCT
ejpam-3659	82	7	13	13	NUM
ejpam-3659	82	8	(	(	PUNCT
ejpam-3659	82	9	2	2	NUM
ejpam-3659	82	10	)	)	PUNCT
ejpam-3659	82	11	(	(	PUNCT
ejpam-3659	82	12	2020	2020	NUM
ejpam-3659	82	13	)	)	PUNCT
ejpam-3659	82	14	,	,	PUNCT
ejpam-3659	82	15	303	303	NUM
ejpam-3659	82	16	-	-	SYM
ejpam-3659	82	17	313	313	NUM
ejpam-3659	82	18	306	306	NUM
ejpam-3659	82	19	lemma	lemma	PROPN
ejpam-3659	82	20	1	1	NUM
ejpam-3659	82	21	(	(	PUNCT
ejpam-3659	82	22	cousin	cousin	PROPN
ejpam-3659	82	23	’s	’s	PART
ejpam-3659	82	24	lemma	lemma	PROPN
ejpam-3659	82	25	)	)	PUNCT
ejpam-3659	82	26	.	.	PUNCT
ejpam-3659	83	1	[	[	X
ejpam-3659	83	2	6	6	X
ejpam-3659	83	3	]	]	PUNCT
ejpam-3659	83	4	if	if	SCONJ
ejpam-3659	83	5	δ	δ	PROPN
ejpam-3659	83	6	is	be	AUX
ejpam-3659	83	7	a	a	DET
ejpam-3659	83	8	gauge	gauge	NOUN
ejpam-3659	83	9	on	on	ADP
ejpam-3659	83	10	[	[	X
ejpam-3659	83	11	a	a	X
ejpam-3659	83	12	,	,	PUNCT
ejpam-3659	83	13	b	b	NOUN
ejpam-3659	83	14	]	]	X
ejpam-3659	83	15	,	,	PUNCT
ejpam-3659	83	16	then	then	ADV
ejpam-3659	83	17	there	there	PRON
ejpam-3659	83	18	exists	exist	VERB
ejpam-3659	83	19	a	a	DET
ejpam-3659	83	20	δ	δ	NOUN
ejpam-3659	83	21	-	-	PUNCT
ejpam-3659	83	22	fine	fine	ADJ
ejpam-3659	83	23	division	division	NOUN
ejpam-3659	83	24	of	of	ADP
ejpam-3659	83	25	[	[	X
ejpam-3659	83	26	a	a	X
ejpam-3659	83	27	,	,	PUNCT
ejpam-3659	83	28	b	b	NOUN
ejpam-3659	83	29	]	]	PUNCT
ejpam-3659	83	30	.	.	PUNCT
ejpam-3659	84	1	definition	definition	NOUN
ejpam-3659	84	2	1	1	NUM
ejpam-3659	84	3	.	.	PUNCT
ejpam-3659	85	1	[	[	X
ejpam-3659	85	2	5–7	5–7	NUM
ejpam-3659	85	3	,	,	PUNCT
ejpam-3659	85	4	10	10	NUM
ejpam-3659	85	5	]	]	PUNCT
ejpam-3659	85	6	a	a	DET
ejpam-3659	85	7	function	function	NOUN
ejpam-3659	86	1	f	f	NOUN
ejpam-3659	86	2	:	:	PUNCT
ejpam-3659	87	1	[	[	X
ejpam-3659	87	2	a	a	X
ejpam-3659	87	3	,	,	PUNCT
ejpam-3659	87	4	b	b	NOUN
ejpam-3659	87	5	]	]	X
ejpam-3659	87	6	→	→	PUNCT
ejpam-3659	87	7	r	r	NOUN
ejpam-3659	87	8	is	be	AUX
ejpam-3659	87	9	said	say	VERB
ejpam-3659	87	10	to	to	PART
ejpam-3659	87	11	be	be	AUX
ejpam-3659	87	12	mcshane	mcshane	NOUN
ejpam-3659	87	13	integrable	integrable	ADJ
ejpam-3659	87	14	to	to	ADP
ejpam-3659	87	15	a	a	DET
ejpam-3659	87	16	real	real	ADJ
ejpam-3659	87	17	number	number	NOUN
ejpam-3659	87	18	a	a	PRON
ejpam-3659	87	19	on	on	ADP
ejpam-3659	87	20	[	[	X
ejpam-3659	87	21	a	a	X
ejpam-3659	87	22	,	,	PUNCT
ejpam-3659	87	23	b	b	NOUN
ejpam-3659	87	24	]	]	X
ejpam-3659	87	25	if	if	SCONJ
ejpam-3659	87	26	for	for	ADP
ejpam-3659	87	27	any	any	DET
ejpam-3659	87	28	ε	ε	PROPN
ejpam-3659	87	29	>	>	X
ejpam-3659	87	30	0	0	PROPN
ejpam-3659	87	31	,	,	PUNCT
ejpam-3659	87	32	there	there	PRON
ejpam-3659	87	33	exists	exist	VERB
ejpam-3659	87	34	a	a	DET
ejpam-3659	87	35	gauge	gauge	NOUN
ejpam-3659	87	36	δ	δ	NOUN
ejpam-3659	87	37	:	:	PUNCT
ejpam-3659	88	1	[	[	X
ejpam-3659	88	2	a	a	PRON
ejpam-3659	88	3	,	,	PUNCT
ejpam-3659	88	4	b]→	b]→	PUNCT
ejpam-3659	88	5	r+	r+	PUNCT
ejpam-3659	88	6	such	such	ADJ
ejpam-3659	88	7	that	that	PRON
ejpam-3659	88	8	for	for	ADP
ejpam-3659	88	9	any	any	DET
ejpam-3659	88	10	mcshane	mcshane	PROPN
ejpam-3659	88	11	δ	δ	PROPN
ejpam-3659	88	12	-	-	PUNCT
ejpam-3659	88	13	fine	fine	ADJ
ejpam-3659	88	14	division	division	NOUN
ejpam-3659	88	15	d	d	NOUN
ejpam-3659	88	16	=	=	PRON
ejpam-3659	88	17	{	{	PUNCT
ejpam-3659	88	18	(	(	PUNCT
ejpam-3659	88	19	[	[	X
ejpam-3659	88	20	u	u	NOUN
ejpam-3659	88	21	,	,	PUNCT
ejpam-3659	88	22	v	v	ADP
ejpam-3659	88	23	]	]	X
ejpam-3659	88	24	,	,	PUNCT
ejpam-3659	88	25	ξ	ξ	X
ejpam-3659	88	26	)	)	PUNCT
ejpam-3659	88	27	}	}	PUNCT
ejpam-3659	88	28	of	of	ADP
ejpam-3659	88	29	[	[	X
ejpam-3659	88	30	a	a	X
ejpam-3659	88	31	,	,	PUNCT
ejpam-3659	88	32	b	b	NOUN
ejpam-3659	88	33	]	]	X
ejpam-3659	88	34	,	,	PUNCT
ejpam-3659	88	35	we	we	PRON
ejpam-3659	88	36	have∣∣∣(d	have∣∣∣(d	VERB
ejpam-3659	88	37	)	)	PUNCT
ejpam-3659	88	38	∑	∑	PUNCT
ejpam-3659	88	39	f(ξ)(v	f(ξ)(v	NUM
ejpam-3659	88	40	−	−	PROPN
ejpam-3659	88	41	u)−a	u)−a	SYM
ejpam-3659	88	42	∣∣∣	∣∣∣	NOUN
ejpam-3659	88	43	<	<	X
ejpam-3659	88	44	ε	ε	PROPN
ejpam-3659	88	45	.	.	PUNCT
ejpam-3659	89	1	if	if	SCONJ
ejpam-3659	89	2	f	f	X
ejpam-3659	89	3	:	:	PUNCT
ejpam-3659	90	1	[	[	X
ejpam-3659	90	2	a	a	X
ejpam-3659	90	3	,	,	PUNCT
ejpam-3659	90	4	b]→	b]→	ADJ
ejpam-3659	90	5	r	r	NOUN
ejpam-3659	90	6	is	be	AUX
ejpam-3659	90	7	mcshane	mcshane	NOUN
ejpam-3659	90	8	integrable	integrable	ADJ
ejpam-3659	90	9	to	to	ADP
ejpam-3659	90	10	a	a	PRON
ejpam-3659	90	11	on	on	ADP
ejpam-3659	90	12	[	[	X
ejpam-3659	90	13	a	a	DET
ejpam-3659	90	14	,	,	PUNCT
ejpam-3659	90	15	b	b	NOUN
ejpam-3659	90	16	]	]	X
ejpam-3659	90	17	,	,	PUNCT
ejpam-3659	90	18	then	then	ADV
ejpam-3659	90	19	we	we	PRON
ejpam-3659	90	20	write	write	VERB
ejpam-3659	90	21	a	a	DET
ejpam-3659	90	22	=	=	X
ejpam-3659	90	23	∫	∫	PROPN
ejpam-3659	90	24	b	b	PROPN
ejpam-3659	90	25	a	a	DET
ejpam-3659	90	26	f.	f.	NOUN
ejpam-3659	90	27	for	for	ADP
ejpam-3659	90	28	e	e	PROPN
ejpam-3659	90	29	⊆	⊆	NUM
ejpam-3659	90	30	[	[	X
ejpam-3659	90	31	a	a	X
ejpam-3659	90	32	,	,	PUNCT
ejpam-3659	90	33	b	b	NOUN
ejpam-3659	90	34	]	]	X
ejpam-3659	90	35	,	,	PUNCT
ejpam-3659	90	36	we	we	PRON
ejpam-3659	90	37	say	say	VERB
ejpam-3659	90	38	that	that	SCONJ
ejpam-3659	90	39	f	f	X
ejpam-3659	90	40	:	:	PUNCT
ejpam-3659	91	1	[	[	X
ejpam-3659	91	2	a	a	X
ejpam-3659	91	3	,	,	PUNCT
ejpam-3659	91	4	b	b	NOUN
ejpam-3659	91	5	]	]	X
ejpam-3659	91	6	→	→	PUNCT
ejpam-3659	91	7	r	r	NOUN
ejpam-3659	91	8	is	be	AUX
ejpam-3659	91	9	mcshane	mcshane	NOUN
ejpam-3659	91	10	integrable	integrable	ADJ
ejpam-3659	91	11	on	on	ADP
ejpam-3659	91	12	e	e	PROPN
ejpam-3659	91	13	if	if	SCONJ
ejpam-3659	91	14	f	f	PROPN
ejpam-3659	91	15	·	·	PUNCT
ejpam-3659	91	16	χe	χe	PROPN
ejpam-3659	91	17	is	be	AUX
ejpam-3659	91	18	mcshane	mcshane	PROPN
ejpam-3659	91	19	integrable	integrable	ADJ
ejpam-3659	91	20	on	on	ADP
ejpam-3659	91	21	[	[	X
ejpam-3659	91	22	a	a	X
ejpam-3659	91	23	,	,	PUNCT
ejpam-3659	91	24	b	b	NOUN
ejpam-3659	91	25	]	]	X
ejpam-3659	91	26	,	,	PUNCT
ejpam-3659	91	27	where	where	SCONJ
ejpam-3659	91	28	χe	χe	PROPN
ejpam-3659	91	29	is	be	AUX
ejpam-3659	91	30	the	the	DET
ejpam-3659	91	31	characteristic	characteristic	ADJ
ejpam-3659	91	32	function	function	NOUN
ejpam-3659	91	33	of	of	ADP
ejpam-3659	91	34	e	e	PROPN
ejpam-3659	91	35	and	and	CCONJ
ejpam-3659	91	36	write∫	write∫	VERB
ejpam-3659	92	1	e	e	PROPN
ejpam-3659	92	2	f	f	PROPN
ejpam-3659	92	3	=	=	SYM
ejpam-3659	92	4	∫	∫	PROPN
ejpam-3659	93	1	b	b	PROPN
ejpam-3659	94	1	a	a	PRON
ejpam-3659	94	2	(	(	PUNCT
ejpam-3659	94	3	f	f	PROPN
ejpam-3659	94	4	·	·	PUNCT
ejpam-3659	94	5	χe	χe	PROPN
ejpam-3659	94	6	)	)	PUNCT
ejpam-3659	94	7	.	.	PUNCT
ejpam-3659	95	1	the	the	DET
ejpam-3659	95	2	next	next	ADJ
ejpam-3659	95	3	theorem	theorem	NOUN
ejpam-3659	95	4	tells	tell	VERB
ejpam-3659	95	5	us	we	PRON
ejpam-3659	95	6	that	that	SCONJ
ejpam-3659	95	7	the	the	DET
ejpam-3659	95	8	mcshane	mcshane	PROPN
ejpam-3659	95	9	integral	integral	NOUN
ejpam-3659	95	10	is	be	AUX
ejpam-3659	95	11	an	an	DET
ejpam-3659	95	12	absolute	absolute	ADJ
ejpam-3659	95	13	integral	integral	ADJ
ejpam-3659	95	14	.	.	PUNCT
ejpam-3659	96	1	theorem	theorem	NOUN
ejpam-3659	96	2	1	1	NUM
ejpam-3659	96	3	.	.	PUNCT
ejpam-3659	97	1	[	[	X
ejpam-3659	97	2	7	7	X
ejpam-3659	97	3	]	]	X
ejpam-3659	97	4	if	if	SCONJ
ejpam-3659	97	5	f	f	PROPN
ejpam-3659	97	6	:	:	PUNCT
ejpam-3659	98	1	[	[	X
ejpam-3659	98	2	a	a	X
ejpam-3659	98	3	,	,	PUNCT
ejpam-3659	98	4	b]→	b]→	ADJ
ejpam-3659	98	5	r	r	NOUN
ejpam-3659	98	6	is	be	AUX
ejpam-3659	98	7	mcshane	mcshane	NOUN
ejpam-3659	98	8	integrable	integrable	ADJ
ejpam-3659	98	9	on	on	ADP
ejpam-3659	98	10	[	[	X
ejpam-3659	98	11	a	a	X
ejpam-3659	98	12	,	,	PUNCT
ejpam-3659	98	13	b	b	NOUN
ejpam-3659	98	14	]	]	X
ejpam-3659	98	15	,	,	PUNCT
ejpam-3659	98	16	then	then	ADV
ejpam-3659	98	17	so	so	ADV
ejpam-3659	98	18	is	be	AUX
ejpam-3659	98	19	|f	|f	PROPN
ejpam-3659	98	20	|	|	INTJ
ejpam-3659	98	21	.	.	PUNCT
ejpam-3659	99	1	if	if	SCONJ
ejpam-3659	99	2	f	f	X
ejpam-3659	99	3	:	:	PUNCT
ejpam-3659	100	1	[	[	X
ejpam-3659	100	2	a	a	X
ejpam-3659	100	3	,	,	PUNCT
ejpam-3659	100	4	b]→	b]→	ADJ
ejpam-3659	100	5	r	r	NOUN
ejpam-3659	100	6	is	be	AUX
ejpam-3659	100	7	mcshane	mcshane	NOUN
ejpam-3659	100	8	integrable	integrable	ADJ
ejpam-3659	100	9	on	on	ADP
ejpam-3659	100	10	[	[	X
ejpam-3659	100	11	a	a	X
ejpam-3659	100	12	,	,	PUNCT
ejpam-3659	100	13	b	b	NOUN
ejpam-3659	100	14	]	]	X
ejpam-3659	100	15	,	,	PUNCT
ejpam-3659	100	16	then	then	ADV
ejpam-3659	100	17	the	the	DET
ejpam-3659	100	18	function	function	NOUN
ejpam-3659	100	19	f	f	NOUN
ejpam-3659	100	20	:	:	PUNCT
ejpam-3659	101	1	[	[	X
ejpam-3659	101	2	a	a	X
ejpam-3659	101	3	,	,	PUNCT
ejpam-3659	101	4	b]→	b]→	ADJ
ejpam-3659	101	5	r	r	NOUN
ejpam-3659	101	6	defined	define	VERB
ejpam-3659	101	7	by	by	ADP
ejpam-3659	101	8	f	f	PROPN
ejpam-3659	101	9	(	(	PUNCT
ejpam-3659	101	10	x	x	X
ejpam-3659	101	11	)	)	PUNCT
ejpam-3659	101	12	=	=	SYM
ejpam-3659	102	1	∫	∫	PROPN
ejpam-3659	102	2	x	x	X
ejpam-3659	102	3	a	a	DET
ejpam-3659	102	4	f	f	X
ejpam-3659	102	5	,	,	PUNCT
ejpam-3659	102	6	for	for	ADP
ejpam-3659	102	7	all	all	DET
ejpam-3659	102	8	x	x	SYM
ejpam-3659	102	9	∈	∈	PROPN
ejpam-3659	102	10	[	[	X
ejpam-3659	102	11	a	a	X
ejpam-3659	102	12	,	,	PUNCT
ejpam-3659	102	13	b	b	NOUN
ejpam-3659	102	14	]	]	PUNCT
ejpam-3659	102	15	is	be	AUX
ejpam-3659	102	16	called	call	VERB
ejpam-3659	102	17	the	the	DET
ejpam-3659	102	18	primitive	primitive	NOUN
ejpam-3659	102	19	of	of	ADP
ejpam-3659	102	20	f	f	PROPN
ejpam-3659	102	21	.	.	PUNCT
ejpam-3659	103	1	almost	almost	ADV
ejpam-3659	103	2	all	all	PRON
ejpam-3659	103	3	significant	significant	ADJ
ejpam-3659	103	4	results	result	NOUN
ejpam-3659	103	5	in	in	ADP
ejpam-3659	103	6	the	the	DET
ejpam-3659	103	7	henstock	henstock	NOUN
ejpam-3659	103	8	integration	integration	NOUN
ejpam-3659	103	9	theory	theory	NOUN
ejpam-3659	103	10	rely	rely	VERB
ejpam-3659	103	11	on	on	ADP
ejpam-3659	103	12	the	the	DET
ejpam-3659	103	13	following	follow	VERB
ejpam-3659	103	14	very	very	ADV
ejpam-3659	103	15	important	important	ADJ
ejpam-3659	103	16	theorem	theorem	NOUN
ejpam-3659	103	17	involving	involve	VERB
ejpam-3659	103	18	the	the	DET
ejpam-3659	103	19	primitive	primitive	ADJ
ejpam-3659	103	20	,	,	PUNCT
ejpam-3659	103	21	called	call	VERB
ejpam-3659	103	22	the	the	DET
ejpam-3659	103	23	henstock	henstock	NOUN
ejpam-3659	103	24	’s	’s	PART
ejpam-3659	103	25	lemma	lemma	PROPN
ejpam-3659	103	26	.	.	PUNCT
ejpam-3659	104	1	theorem	theorem	PROPN
ejpam-3659	104	2	2	2	NUM
ejpam-3659	104	3	(	(	PUNCT
ejpam-3659	104	4	henstock	henstock	PROPN
ejpam-3659	104	5	’s	’s	PART
ejpam-3659	104	6	lemma	lemma	PROPN
ejpam-3659	104	7	)	)	PUNCT
ejpam-3659	104	8	.	.	PUNCT
ejpam-3659	105	1	[	[	X
ejpam-3659	105	2	6	6	NUM
ejpam-3659	105	3	]	]	PUNCT
ejpam-3659	105	4	if	if	SCONJ
ejpam-3659	105	5	f	f	X
ejpam-3659	105	6	:	:	PUNCT
ejpam-3659	106	1	[	[	X
ejpam-3659	106	2	a	a	X
ejpam-3659	106	3	,	,	PUNCT
ejpam-3659	106	4	b	b	NOUN
ejpam-3659	106	5	]	]	X
ejpam-3659	106	6	→	→	PUNCT
ejpam-3659	106	7	r	r	NOUN
ejpam-3659	106	8	is	be	AUX
ejpam-3659	106	9	mcshane	mcshane	NOUN
ejpam-3659	106	10	(	(	PUNCT
ejpam-3659	106	11	resp	resp	PROPN
ejpam-3659	106	12	.	.	PUNCT
ejpam-3659	106	13	,	,	PUNCT
ejpam-3659	106	14	henstock	henstock	PROPN
ejpam-3659	106	15	)	)	PUNCT
ejpam-3659	106	16	integrable	integrable	VERB
ejpam-3659	106	17	on	on	ADP
ejpam-3659	106	18	[	[	X
ejpam-3659	106	19	a	a	X
ejpam-3659	106	20	,	,	PUNCT
ejpam-3659	106	21	b	b	NOUN
ejpam-3659	106	22	]	]	X
ejpam-3659	106	23	with	with	ADP
ejpam-3659	106	24	primitive	primitive	ADJ
ejpam-3659	106	25	f	f	NOUN
ejpam-3659	106	26	,	,	PUNCT
ejpam-3659	106	27	then	then	ADV
ejpam-3659	106	28	for	for	ADP
ejpam-3659	106	29	each	each	DET
ejpam-3659	106	30	ε	ε	PROPN
ejpam-3659	106	31	>	>	X
ejpam-3659	106	32	0	0	PROPN
ejpam-3659	106	33	,	,	PUNCT
ejpam-3659	106	34	there	there	PRON
ejpam-3659	106	35	exist	exist	VERB
ejpam-3659	106	36	δ	δ	NOUN
ejpam-3659	106	37	:	:	PUNCT
ejpam-3659	107	1	[	[	X
ejpam-3659	107	2	a	a	X
ejpam-3659	107	3	,	,	PUNCT
ejpam-3659	107	4	b	b	NOUN
ejpam-3659	107	5	]	]	X
ejpam-3659	107	6	→	→	PUNCT
ejpam-3659	107	7	r+	r+	NOUN
ejpam-3659	107	8	such	such	ADJ
ejpam-3659	107	9	that	that	SCONJ
ejpam-3659	107	10	whenever	whenever	SCONJ
ejpam-3659	107	11	d	d	NOUN
ejpam-3659	107	12	=	=	PRON
ejpam-3659	107	13	{	{	PUNCT
ejpam-3659	107	14	(	(	PUNCT
ejpam-3659	107	15	[	[	X
ejpam-3659	107	16	u	u	NOUN
ejpam-3659	107	17	,	,	PUNCT
ejpam-3659	107	18	v	v	ADP
ejpam-3659	107	19	]	]	X
ejpam-3659	107	20	,	,	PUNCT
ejpam-3659	107	21	ξ	ξ	X
ejpam-3659	107	22	)	)	PUNCT
ejpam-3659	107	23	}	}	PUNCT
ejpam-3659	107	24	is	be	AUX
ejpam-3659	107	25	a	a	DET
ejpam-3659	107	26	mcshane	mcshane	NOUN
ejpam-3659	107	27	(	(	PUNCT
ejpam-3659	107	28	resp	resp	PROPN
ejpam-3659	107	29	.	.	PUNCT
ejpam-3659	107	30	,	,	PUNCT
ejpam-3659	107	31	henstock	henstock	PROPN
ejpam-3659	107	32	)	)	PUNCT
ejpam-3659	107	33	δ	δ	NOUN
ejpam-3659	107	34	-	-	PUNCT
ejpam-3659	107	35	fine	fine	ADJ
ejpam-3659	107	36	division	division	NOUN
ejpam-3659	107	37	of	of	ADP
ejpam-3659	107	38	[	[	X
ejpam-3659	107	39	a	a	X
ejpam-3659	107	40	,	,	PUNCT
ejpam-3659	107	41	b	b	NOUN
ejpam-3659	107	42	]	]	X
ejpam-3659	107	43	,	,	PUNCT
ejpam-3659	107	44	we	we	PRON
ejpam-3659	107	45	have	have	VERB
ejpam-3659	107	46	(	(	PUNCT
ejpam-3659	108	1	d	d	X
ejpam-3659	108	2	)	)	PUNCT
ejpam-3659	108	3	∑∣∣∣f(ξ)(v	∑∣∣∣f(ξ)(v	PROPN
ejpam-3659	108	4	−	−	NOUN
ejpam-3659	109	1	u)−	u)−	PROPN
ejpam-3659	109	2	f	f	PROPN
ejpam-3659	109	3	(	(	PUNCT
ejpam-3659	109	4	v	v	NOUN
ejpam-3659	109	5	)	)	PUNCT
ejpam-3659	109	6	+	+	NUM
ejpam-3659	109	7	f	f	X
ejpam-3659	109	8	(	(	PUNCT
ejpam-3659	109	9	u	u	NOUN
ejpam-3659	109	10	)	)	PUNCT
ejpam-3659	109	11	∣∣∣	∣∣∣	NOUN
ejpam-3659	109	12	<	<	X
ejpam-3659	109	13	ε	ε	PROPN
ejpam-3659	109	14	.	.	PUNCT
ejpam-3659	110	1	the	the	DET
ejpam-3659	110	2	following	follow	VERB
ejpam-3659	110	3	concept	concept	NOUN
ejpam-3659	110	4	was	be	AUX
ejpam-3659	110	5	introduced	introduce	VERB
ejpam-3659	110	6	by	by	ADP
ejpam-3659	110	7	yang	yang	PROPN
ejpam-3659	111	1	[	[	X
ejpam-3659	111	2	12	12	NUM
ejpam-3659	111	3	]	]	PUNCT
ejpam-3659	111	4	.	.	PUNCT
ejpam-3659	112	1	definition	definition	NOUN
ejpam-3659	112	2	2	2	NUM
ejpam-3659	112	3	.	.	PUNCT
ejpam-3659	113	1	a	a	DET
ejpam-3659	113	2	set	set	NOUN
ejpam-3659	113	3	e	e	NOUN
ejpam-3659	113	4	⊆	⊆	NUM
ejpam-3659	113	5	r	r	NOUN
ejpam-3659	113	6	is	be	AUX
ejpam-3659	113	7	integrable	integrable	ADJ
ejpam-3659	113	8	if	if	SCONJ
ejpam-3659	113	9	χe∩[a	χe∩[a	PROPN
ejpam-3659	113	10	,	,	PUNCT
ejpam-3659	113	11	b	b	X
ejpam-3659	113	12	]	]	X
ejpam-3659	113	13	is	be	AUX
ejpam-3659	113	14	mcshane	mcshane	NOUN
ejpam-3659	113	15	integrable	integrable	ADJ
ejpam-3659	113	16	on	on	ADP
ejpam-3659	113	17	[	[	X
ejpam-3659	113	18	a	a	X
ejpam-3659	113	19	,	,	PUNCT
ejpam-3659	113	20	b	b	NOUN
ejpam-3659	113	21	]	]	X
ejpam-3659	113	22	,	,	PUNCT
ejpam-3659	113	23	for	for	ADP
ejpam-3659	113	24	all	all	DET
ejpam-3659	113	25	[	[	X
ejpam-3659	113	26	a	a	X
ejpam-3659	113	27	,	,	PUNCT
ejpam-3659	113	28	b	b	NOUN
ejpam-3659	113	29	]	]	X
ejpam-3659	113	30	⊆	⊆	NUM
ejpam-3659	113	31	r.	r.	PROPN
ejpam-3659	113	32	clearly	clearly	ADV
ejpam-3659	113	33	,	,	PUNCT
ejpam-3659	113	34	∅	∅	NOUN
ejpam-3659	113	35	and	and	CCONJ
ejpam-3659	113	36	r	r	NOUN
ejpam-3659	113	37	are	be	AUX
ejpam-3659	113	38	integrable	integrable	ADJ
ejpam-3659	113	39	subsets	subset	NOUN
ejpam-3659	113	40	of	of	ADP
ejpam-3659	113	41	r.	r.	PROPN
ejpam-3659	113	42	moreover	moreover	ADV
ejpam-3659	113	43	,	,	PUNCT
ejpam-3659	113	44	it	it	PRON
ejpam-3659	113	45	can	can	AUX
ejpam-3659	113	46	be	be	AUX
ejpam-3659	113	47	seen	see	VERB
ejpam-3659	113	48	that	that	SCONJ
ejpam-3659	113	49	the	the	DET
ejpam-3659	113	50	collection	collection	NOUN
ejpam-3659	113	51	i	i	PRON
ejpam-3659	113	52	of	of	ADP
ejpam-3659	113	53	all	all	DET
ejpam-3659	113	54	integrable	integrable	ADJ
ejpam-3659	113	55	subsets	subset	NOUN
ejpam-3659	113	56	of	of	ADP
ejpam-3659	113	57	r	r	NOUN
ejpam-3659	113	58	is	be	AUX
ejpam-3659	113	59	a	a	DET
ejpam-3659	113	60	σ	σ	NOUN
ejpam-3659	113	61	-	-	PUNCT
ejpam-3659	113	62	algebra	algebra	NOUN
ejpam-3659	113	63	.	.	PUNCT
ejpam-3659	114	1	in	in	ADP
ejpam-3659	114	2	[	[	X
ejpam-3659	114	3	9	9	NUM
ejpam-3659	114	4	]	]	PUNCT
ejpam-3659	114	5	,	,	PUNCT
ejpam-3659	114	6	quindala	quindala	PROPN
ejpam-3659	114	7	and	and	CCONJ
ejpam-3659	114	8	benitez	benitez	PROPN
ejpam-3659	114	9	showed	show	VERB
ejpam-3659	114	10	that	that	SCONJ
ejpam-3659	114	11	the	the	DET
ejpam-3659	114	12	set	set	NOUN
ejpam-3659	114	13	-	-	PUNCT
ejpam-3659	114	14	function	function	NOUN
ejpam-3659	114	15	µ	µ	NOUN
ejpam-3659	114	16	:	:	PUNCT
ejpam-3659	114	17	i	i	PRON
ejpam-3659	114	18	→	→	PUNCT
ejpam-3659	115	1	[	[	X
ejpam-3659	115	2	0,∞	0,∞	X
ejpam-3659	115	3	]	]	PUNCT
ejpam-3659	115	4	defined	define	VERB
ejpam-3659	115	5	by	by	ADP
ejpam-3659	115	6	µ(e	µ(e	PROPN
ejpam-3659	115	7	)	)	PUNCT
ejpam-3659	115	8	=	=	SYM
ejpam-3659	116	1	∫	∫	PROPN
ejpam-3659	116	2	∞	∞	PROPN
ejpam-3659	116	3	−∞	−∞	ADP
ejpam-3659	116	4	χe	χe	PROPN
ejpam-3659	116	5	,	,	PUNCT
ejpam-3659	116	6	for	for	ADP
ejpam-3659	116	7	all	all	DET
ejpam-3659	116	8	e	e	NOUN
ejpam-3659	116	9	∈	∈	PROPN
ejpam-3659	116	10	i	i	PRON
ejpam-3659	116	11	f.	f.	PROPN
ejpam-3659	116	12	sumalpong	sumalpong	PROPN
ejpam-3659	117	1	jr	jr	PROPN
ejpam-3659	117	2	.	.	PROPN
ejpam-3659	117	3	,	,	PUNCT
ejpam-3659	117	4	j.	j.	PROPN
ejpam-3659	117	5	benitez	benitez	PROPN
ejpam-3659	117	6	/	/	PUNCT
ejpam-3659	117	7	eur	eur	PROPN
ejpam-3659	117	8	.	.	PUNCT
ejpam-3659	118	1	j.	j.	PROPN
ejpam-3659	118	2	pure	pure	PROPN
ejpam-3659	118	3	appl	appl	PROPN
ejpam-3659	118	4	.	.	PROPN
ejpam-3659	118	5	math	math	PROPN
ejpam-3659	118	6	,	,	PUNCT
ejpam-3659	118	7	13	13	NUM
ejpam-3659	118	8	(	(	PUNCT
ejpam-3659	118	9	2	2	NUM
ejpam-3659	118	10	)	)	PUNCT
ejpam-3659	118	11	(	(	PUNCT
ejpam-3659	118	12	2020	2020	NUM
ejpam-3659	118	13	)	)	PUNCT
ejpam-3659	118	14	,	,	PUNCT
ejpam-3659	118	15	303	303	NUM
ejpam-3659	118	16	-	-	SYM
ejpam-3659	118	17	313	313	NUM
ejpam-3659	118	18	307	307	NUM
ejpam-3659	118	19	is	be	AUX
ejpam-3659	118	20	a	a	DET
ejpam-3659	118	21	measure	measure	NOUN
ejpam-3659	118	22	.	.	PUNCT
ejpam-3659	119	1	in	in	ADP
ejpam-3659	119	2	the	the	DET
ejpam-3659	119	3	same	same	ADJ
ejpam-3659	119	4	paper	paper	NOUN
ejpam-3659	119	5	,	,	PUNCT
ejpam-3659	119	6	for	for	ADP
ejpam-3659	119	7	a	a	DET
ejpam-3659	119	8	bounded	bound	VERB
ejpam-3659	119	9	integrable	integrable	ADJ
ejpam-3659	119	10	set	set	NOUN
ejpam-3659	119	11	e	e	NOUN
ejpam-3659	119	12	⊆	⊆	NUM
ejpam-3659	119	13	r	r	NOUN
ejpam-3659	119	14	µ(e	µ(e	PROPN
ejpam-3659	119	15	)	)	PUNCT
ejpam-3659	119	16	=	=	SYM
ejpam-3659	119	17	m∗(e	m∗(e	PROPN
ejpam-3659	119	18	)	)	PUNCT
ejpam-3659	119	19	where	where	SCONJ
ejpam-3659	119	20	m∗	m∗	PROPN
ejpam-3659	119	21	is	be	AUX
ejpam-3659	119	22	the	the	DET
ejpam-3659	119	23	lebesgue	lebesgue	ADJ
ejpam-3659	119	24	outer	outer	ADJ
ejpam-3659	119	25	measure	measure	NOUN
ejpam-3659	119	26	.	.	PUNCT
ejpam-3659	120	1	see	see	VERB
ejpam-3659	120	2	[	[	X
ejpam-3659	120	3	1	1	NUM
ejpam-3659	120	4	,	,	PUNCT
ejpam-3659	120	5	p.85	p.85	ADP
ejpam-3659	120	6	]	]	PUNCT
ejpam-3659	120	7	for	for	ADP
ejpam-3659	120	8	the	the	DET
ejpam-3659	120	9	proof	proof	NOUN
ejpam-3659	120	10	of	of	ADP
ejpam-3659	120	11	the	the	DET
ejpam-3659	120	12	following	follow	VERB
ejpam-3659	120	13	lemma	lemma	PROPN
ejpam-3659	120	14	which	which	PRON
ejpam-3659	120	15	utilizes	utilize	VERB
ejpam-3659	120	16	the	the	DET
ejpam-3659	120	17	heine	heine	PROPN
ejpam-3659	120	18	-	-	PUNCT
ejpam-3659	120	19	borel	borel	PROPN
ejpam-3659	120	20	covering	covering	NOUN
ejpam-3659	120	21	theorem	theorem	PROPN
ejpam-3659	120	22	.	.	PUNCT
ejpam-3659	121	1	lemma	lemma	PROPN
ejpam-3659	121	2	2	2	NUM
ejpam-3659	121	3	.	.	PUNCT
ejpam-3659	122	1	[	[	X
ejpam-3659	122	2	1	1	X
ejpam-3659	122	3	]	]	PUNCT
ejpam-3659	122	4	if	if	SCONJ
ejpam-3659	122	5	e	e	PROPN
ejpam-3659	122	6	⊆	⊆	NUM
ejpam-3659	122	7	[	[	X
ejpam-3659	122	8	a	a	X
ejpam-3659	122	9	,	,	PUNCT
ejpam-3659	122	10	b	b	NOUN
ejpam-3659	122	11	]	]	PUNCT
ejpam-3659	122	12	and	and	CCONJ
ejpam-3659	122	13	χe	χe	PROPN
ejpam-3659	122	14	is	be	AUX
ejpam-3659	122	15	mcshane	mcshane	PROPN
ejpam-3659	122	16	integrable	integrable	ADJ
ejpam-3659	122	17	on	on	ADP
ejpam-3659	122	18	[	[	X
ejpam-3659	122	19	a	a	X
ejpam-3659	122	20	,	,	PUNCT
ejpam-3659	122	21	b	b	NOUN
ejpam-3659	122	22	]	]	X
ejpam-3659	122	23	,	,	PUNCT
ejpam-3659	122	24	then	then	ADV
ejpam-3659	122	25	for	for	ADP
ejpam-3659	122	26	all	all	DET
ejpam-3659	122	27	ε	ε	PROPN
ejpam-3659	122	28	>	>	X
ejpam-3659	122	29	0	0	PUNCT
ejpam-3659	123	1	there	there	PRON
ejpam-3659	123	2	exists	exist	VERB
ejpam-3659	123	3	an	an	DET
ejpam-3659	123	4	open	open	ADJ
ejpam-3659	123	5	set	set	NOUN
ejpam-3659	123	6	o	o	NOUN
ejpam-3659	123	7	⊆	⊆	NUM
ejpam-3659	123	8	[	[	X
ejpam-3659	123	9	a	a	X
ejpam-3659	123	10	,	,	PUNCT
ejpam-3659	123	11	b	b	NOUN
ejpam-3659	123	12	]	]	X
ejpam-3659	123	13	such	such	ADJ
ejpam-3659	123	14	that	that	SCONJ
ejpam-3659	123	15	e	e	PROPN
ejpam-3659	123	16	⊆	⊆	NUM
ejpam-3659	123	17	o	o	NOUN
ejpam-3659	123	18	and∫	and∫	PROPN
ejpam-3659	123	19	b	b	PROPN
ejpam-3659	123	20	a	a	DET
ejpam-3659	123	21	χore	χore	NOUN
ejpam-3659	123	22	<	<	X
ejpam-3659	123	23	ε	ε	PROPN
ejpam-3659	123	24	.	.	PUNCT
ejpam-3659	124	1	the	the	DET
ejpam-3659	124	2	following	follow	VERB
ejpam-3659	124	3	is	be	AUX
ejpam-3659	124	4	a	a	DET
ejpam-3659	124	5	characterization	characterization	NOUN
ejpam-3659	124	6	of	of	ADP
ejpam-3659	124	7	an	an	DET
ejpam-3659	124	8	integrable	integrable	ADJ
ejpam-3659	124	9	set	set	NOUN
ejpam-3659	124	10	e	e	NOUN
ejpam-3659	124	11	⊆	⊆	NUM
ejpam-3659	124	12	[	[	X
ejpam-3659	124	13	a	a	X
ejpam-3659	124	14	,	,	PUNCT
ejpam-3659	124	15	b	b	NOUN
ejpam-3659	124	16	]	]	PUNCT
ejpam-3659	124	17	.	.	PUNCT
ejpam-3659	125	1	lemma	lemma	PROPN
ejpam-3659	125	2	3	3	X
ejpam-3659	125	3	.	.	PUNCT
ejpam-3659	126	1	[	[	X
ejpam-3659	126	2	12	12	NUM
ejpam-3659	126	3	]	]	PUNCT
ejpam-3659	126	4	let	let	VERB
ejpam-3659	126	5	e	e	NOUN
ejpam-3659	126	6	⊆	⊆	NUM
ejpam-3659	126	7	[	[	X
ejpam-3659	126	8	a	a	X
ejpam-3659	126	9	,	,	PUNCT
ejpam-3659	126	10	b	b	NOUN
ejpam-3659	126	11	]	]	X
ejpam-3659	126	12	.	.	PUNCT
ejpam-3659	127	1	the	the	DET
ejpam-3659	127	2	following	follow	VERB
ejpam-3659	127	3	statements	statement	NOUN
ejpam-3659	127	4	are	be	AUX
ejpam-3659	127	5	equivalent	equivalent	ADJ
ejpam-3659	127	6	:	:	PUNCT
ejpam-3659	127	7	(	(	PUNCT
ejpam-3659	127	8	i	i	NOUN
ejpam-3659	127	9	)	)	PUNCT
ejpam-3659	127	10	χe	χe	PROPN
ejpam-3659	127	11	is	be	AUX
ejpam-3659	127	12	mcshane	mcshane	PROPN
ejpam-3659	127	13	integrable	integrable	ADJ
ejpam-3659	127	14	on	on	ADP
ejpam-3659	127	15	[	[	X
ejpam-3659	127	16	a	a	X
ejpam-3659	127	17	,	,	PUNCT
ejpam-3659	127	18	b	b	NOUN
ejpam-3659	127	19	]	]	X
ejpam-3659	127	20	.	.	PUNCT
ejpam-3659	128	1	(	(	PUNCT
ejpam-3659	128	2	ii	ii	NOUN
ejpam-3659	128	3	)	)	PUNCT
ejpam-3659	128	4	e	e	NOUN
ejpam-3659	128	5	is	be	AUX
ejpam-3659	128	6	integrable	integrable	ADJ
ejpam-3659	128	7	.	.	PUNCT
ejpam-3659	129	1	definition	definition	NOUN
ejpam-3659	129	2	3	3	NUM
ejpam-3659	129	3	.	.	PUNCT
ejpam-3659	130	1	an	an	DET
ejpam-3659	130	2	integrable	integrable	ADJ
ejpam-3659	130	3	set	set	NOUN
ejpam-3659	130	4	e	e	NOUN
ejpam-3659	130	5	⊆	⊆	NUM
ejpam-3659	130	6	[	[	X
ejpam-3659	130	7	a	a	X
ejpam-3659	130	8	,	,	PUNCT
ejpam-3659	130	9	b	b	NOUN
ejpam-3659	130	10	]	]	PUNCT
ejpam-3659	130	11	is	be	AUX
ejpam-3659	130	12	said	say	VERB
ejpam-3659	130	13	to	to	PART
ejpam-3659	130	14	have	have	VERB
ejpam-3659	130	15	variation	variation	NOUN
ejpam-3659	130	16	zero	zero	NUM
ejpam-3659	130	17	if∫	if∫	PROPN
ejpam-3659	130	18	b	b	PROPN
ejpam-3659	130	19	a	a	DET
ejpam-3659	130	20	χe	χe	NOUN
ejpam-3659	130	21	=	=	NOUN
ejpam-3659	130	22	0	0	X
ejpam-3659	130	23	.	.	PUNCT
ejpam-3659	131	1	it	it	PRON
ejpam-3659	131	2	is	be	AUX
ejpam-3659	131	3	worth	worth	ADJ
ejpam-3659	131	4	noting	note	VERB
ejpam-3659	131	5	that	that	SCONJ
ejpam-3659	131	6	a	a	DET
ejpam-3659	131	7	subset	subset	NOUN
ejpam-3659	131	8	of	of	ADP
ejpam-3659	131	9	a	a	DET
ejpam-3659	131	10	set	set	NOUN
ejpam-3659	131	11	of	of	ADP
ejpam-3659	131	12	variation	variation	NOUN
ejpam-3659	131	13	zero	zero	NUM
ejpam-3659	131	14	is	be	AUX
ejpam-3659	131	15	again	again	ADV
ejpam-3659	131	16	of	of	ADP
ejpam-3659	131	17	variation	variation	NOUN
ejpam-3659	131	18	zero	zero	NUM
ejpam-3659	131	19	.	.	PUNCT
ejpam-3659	132	1	definition	definition	NOUN
ejpam-3659	132	2	4	4	NUM
ejpam-3659	132	3	.	.	PUNCT
ejpam-3659	133	1	a	a	DET
ejpam-3659	133	2	property	property	NOUN
ejpam-3659	133	3	is	be	AUX
ejpam-3659	133	4	said	say	VERB
ejpam-3659	133	5	to	to	PART
ejpam-3659	133	6	hold	hold	VERB
ejpam-3659	133	7	almost	almost	ADV
ejpam-3659	133	8	everywhere	everywhere	ADV
ejpam-3659	133	9	(	(	PUNCT
ejpam-3659	133	10	abbreviated	abbreviate	VERB
ejpam-3659	133	11	a.e	a.e	PROPN
ejpam-3659	133	12	.	.	PUNCT
ejpam-3659	133	13	)	)	PUNCT
ejpam-3659	134	1	on	on	ADP
ejpam-3659	134	2	a	a	PRON
ejpam-3659	134	3	if	if	SCONJ
ejpam-3659	134	4	the	the	DET
ejpam-3659	134	5	set	set	NOUN
ejpam-3659	134	6	of	of	ADP
ejpam-3659	134	7	points	point	NOUN
ejpam-3659	134	8	in	in	ADP
ejpam-3659	134	9	a	a	PRON
ejpam-3659	134	10	where	where	SCONJ
ejpam-3659	134	11	it	it	PRON
ejpam-3659	134	12	fails	fail	VERB
ejpam-3659	134	13	to	to	PART
ejpam-3659	134	14	hold	hold	VERB
ejpam-3659	134	15	is	be	AUX
ejpam-3659	134	16	an	an	DET
ejpam-3659	134	17	integrable	integrable	ADJ
ejpam-3659	134	18	set	set	NOUN
ejpam-3659	134	19	of	of	ADP
ejpam-3659	134	20	variation	variation	NOUN
ejpam-3659	134	21	zero	zero	NUM
ejpam-3659	134	22	.	.	PUNCT
ejpam-3659	135	1	it	it	PRON
ejpam-3659	135	2	was	be	AUX
ejpam-3659	135	3	proved	prove	VERB
ejpam-3659	135	4	in	in	ADP
ejpam-3659	135	5	[	[	X
ejpam-3659	135	6	9	9	NUM
ejpam-3659	135	7	]	]	PUNCT
ejpam-3659	135	8	that	that	SCONJ
ejpam-3659	135	9	the	the	DET
ejpam-3659	135	10	notions	notion	NOUN
ejpam-3659	135	11	of	of	ADP
ejpam-3659	135	12	mcshane	mcshane	PROPN
ejpam-3659	135	13	integrable	integrable	ADJ
ejpam-3659	135	14	set	set	NOUN
ejpam-3659	135	15	and	and	CCONJ
ejpam-3659	135	16	lebesgue	lebesgue	PROPN
ejpam-3659	135	17	measurable	measurable	ADJ
ejpam-3659	135	18	set	set	NOUN
ejpam-3659	135	19	are	be	AUX
ejpam-3659	135	20	equivalent	equivalent	ADJ
ejpam-3659	135	21	.	.	PUNCT
ejpam-3659	136	1	this	this	PRON
ejpam-3659	136	2	implies	imply	VERB
ejpam-3659	136	3	that	that	SCONJ
ejpam-3659	136	4	integrable	integrable	ADJ
ejpam-3659	136	5	sets	set	NOUN
ejpam-3659	136	6	of	of	ADP
ejpam-3659	136	7	variation	variation	NOUN
ejpam-3659	136	8	zero	zero	NUM
ejpam-3659	136	9	are	be	AUX
ejpam-3659	136	10	exactly	exactly	ADV
ejpam-3659	136	11	those	those	DET
ejpam-3659	136	12	subsets	subset	NOUN
ejpam-3659	136	13	of	of	ADP
ejpam-3659	136	14	[	[	X
ejpam-3659	136	15	a	a	X
ejpam-3659	136	16	,	,	PUNCT
ejpam-3659	136	17	b	b	NOUN
ejpam-3659	136	18	]	]	X
ejpam-3659	136	19	with	with	ADP
ejpam-3659	136	20	zero	zero	NUM
ejpam-3659	136	21	lebesgue	lebesgue	NOUN
ejpam-3659	136	22	measure	measure	NOUN
ejpam-3659	136	23	.	.	PUNCT
ejpam-3659	137	1	furthermore	furthermore	ADV
ejpam-3659	137	2	,	,	PUNCT
ejpam-3659	137	3	the	the	DET
ejpam-3659	137	4	concept	concept	NOUN
ejpam-3659	137	5	of	of	ADP
ejpam-3659	137	6	“	"	PUNCT
ejpam-3659	137	7	almost	almost	ADV
ejpam-3659	137	8	everywhere	everywhere	ADV
ejpam-3659	137	9	”	"	PUNCT
ejpam-3659	137	10	in	in	ADP
ejpam-3659	137	11	the	the	DET
ejpam-3659	137	12	sense	sense	NOUN
ejpam-3659	137	13	of	of	ADP
ejpam-3659	137	14	mcshane	mcshane	NOUN
ejpam-3659	137	15	,	,	PUNCT
ejpam-3659	137	16	introduced	introduce	VERB
ejpam-3659	137	17	in	in	ADP
ejpam-3659	137	18	definition	definition	NOUN
ejpam-3659	137	19	4	4	NUM
ejpam-3659	137	20	,	,	PUNCT
ejpam-3659	137	21	coincides	coincide	VERB
ejpam-3659	137	22	with	with	ADP
ejpam-3659	137	23	the	the	DET
ejpam-3659	137	24	corresponding	corresponding	ADJ
ejpam-3659	137	25	concept	concept	NOUN
ejpam-3659	137	26	from	from	ADP
ejpam-3659	137	27	lebesgue	lebesgue	PROPN
ejpam-3659	137	28	theory	theory	NOUN
ejpam-3659	137	29	.	.	PUNCT
ejpam-3659	138	1	this	this	DET
ejpam-3659	138	2	equivalences	equivalence	NOUN
ejpam-3659	138	3	do	do	AUX
ejpam-3659	138	4	not	not	PART
ejpam-3659	138	5	diminish	diminish	VERB
ejpam-3659	138	6	the	the	DET
ejpam-3659	138	7	relevance	relevance	NOUN
ejpam-3659	138	8	of	of	ADP
ejpam-3659	138	9	the	the	DET
ejpam-3659	138	10	notion	notion	NOUN
ejpam-3659	138	11	of	of	ADP
ejpam-3659	138	12	mcshane	mcshane	PROPN
ejpam-3659	138	13	integrable	integrable	ADJ
ejpam-3659	138	14	set	set	NOUN
ejpam-3659	138	15	.	.	PUNCT
ejpam-3659	139	1	if	if	SCONJ
ejpam-3659	139	2	functions	function	NOUN
ejpam-3659	139	3	possess	possess	VERB
ejpam-3659	139	4	certain	certain	ADJ
ejpam-3659	139	5	properties	property	NOUN
ejpam-3659	139	6	almost	almost	ADV
ejpam-3659	139	7	everywhere	everywhere	ADV
ejpam-3659	139	8	,	,	PUNCT
ejpam-3659	139	9	then	then	ADV
ejpam-3659	139	10	some	some	DET
ejpam-3659	139	11	properties	property	NOUN
ejpam-3659	139	12	of	of	ADP
ejpam-3659	139	13	the	the	DET
ejpam-3659	139	14	henstock	henstock	NOUN
ejpam-3659	139	15	integral	integral	ADJ
ejpam-3659	139	16	are	be	AUX
ejpam-3659	139	17	preserved	preserve	VERB
ejpam-3659	139	18	.	.	PUNCT
ejpam-3659	140	1	in	in	ADP
ejpam-3659	140	2	particular	particular	ADJ
ejpam-3659	140	3	,	,	PUNCT
ejpam-3659	140	4	if	if	SCONJ
ejpam-3659	140	5	two	two	NUM
ejpam-3659	140	6	functions	function	NOUN
ejpam-3659	140	7	are	be	AUX
ejpam-3659	140	8	equal	equal	ADJ
ejpam-3659	140	9	a.e	a.e	PROPN
ejpam-3659	140	10	.	.	PROPN
ejpam-3659	141	1	and	and	CCONJ
ejpam-3659	141	2	one	one	NUM
ejpam-3659	141	3	of	of	ADP
ejpam-3659	141	4	the	the	DET
ejpam-3659	141	5	functions	function	NOUN
ejpam-3659	141	6	is	be	AUX
ejpam-3659	141	7	mcshane	mcshane	NOUN
ejpam-3659	141	8	integrable	integrable	ADJ
ejpam-3659	141	9	,	,	PUNCT
ejpam-3659	141	10	then	then	ADV
ejpam-3659	141	11	the	the	DET
ejpam-3659	141	12	other	other	ADJ
ejpam-3659	141	13	function	function	NOUN
ejpam-3659	141	14	is	be	AUX
ejpam-3659	141	15	also	also	ADV
ejpam-3659	141	16	mcshane	mcshane	PROPN
ejpam-3659	141	17	integrable	integrable	ADJ
ejpam-3659	141	18	and	and	CCONJ
ejpam-3659	141	19	their	their	PRON
ejpam-3659	141	20	integral	integral	ADJ
ejpam-3659	141	21	values	value	NOUN
ejpam-3659	141	22	coincide	coincide	NOUN
ejpam-3659	141	23	.	.	PUNCT
ejpam-3659	142	1	this	this	PRON
ejpam-3659	142	2	is	be	AUX
ejpam-3659	142	3	precisely	precisely	ADV
ejpam-3659	142	4	stated	state	VERB
ejpam-3659	142	5	in	in	ADP
ejpam-3659	142	6	theorem	theorem	NOUN
ejpam-3659	142	7	3	3	NUM
ejpam-3659	142	8	.	.	PUNCT
ejpam-3659	143	1	the	the	DET
ejpam-3659	143	2	proof	proof	NOUN
ejpam-3659	143	3	of	of	ADP
ejpam-3659	143	4	this	this	DET
ejpam-3659	143	5	result	result	NOUN
ejpam-3659	143	6	is	be	AUX
ejpam-3659	143	7	standard	standard	ADJ
ejpam-3659	143	8	and	and	CCONJ
ejpam-3659	143	9	one	one	PRON
ejpam-3659	143	10	may	may	AUX
ejpam-3659	143	11	follow	follow	VERB
ejpam-3659	143	12	the	the	DET
ejpam-3659	143	13	proof	proof	NOUN
ejpam-3659	143	14	of	of	ADP
ejpam-3659	143	15	theorem	theorem	ADJ
ejpam-3659	143	16	9.5	9.5	NUM
ejpam-3659	143	17	in	in	ADP
ejpam-3659	143	18	[	[	X
ejpam-3659	143	19	5	5	NUM
ejpam-3659	143	20	]	]	PUNCT
ejpam-3659	143	21	or	or	CCONJ
ejpam-3659	143	22	theorem	theorem	VERB
ejpam-3659	143	23	10	10	NUM
ejpam-3659	143	24	in	in	ADP
ejpam-3659	143	25	[	[	X
ejpam-3659	143	26	11	11	NUM
ejpam-3659	143	27	]	]	PUNCT
ejpam-3659	143	28	.	.	PUNCT
ejpam-3659	144	1	f.	f.	PROPN
ejpam-3659	144	2	sumalpong	sumalpong	PROPN
ejpam-3659	144	3	jr	jr	PROPN
ejpam-3659	144	4	.	.	PROPN
ejpam-3659	144	5	,	,	PUNCT
ejpam-3659	144	6	j.	j.	PROPN
ejpam-3659	144	7	benitez	benitez	PROPN
ejpam-3659	144	8	/	/	PUNCT
ejpam-3659	144	9	eur	eur	PROPN
ejpam-3659	144	10	.	.	PUNCT
ejpam-3659	145	1	j.	j.	PROPN
ejpam-3659	145	2	pure	pure	PROPN
ejpam-3659	145	3	appl	appl	PROPN
ejpam-3659	145	4	.	.	PROPN
ejpam-3659	145	5	math	math	PROPN
ejpam-3659	145	6	,	,	PUNCT
ejpam-3659	145	7	13	13	NUM
ejpam-3659	145	8	(	(	PUNCT
ejpam-3659	145	9	2	2	NUM
ejpam-3659	145	10	)	)	PUNCT
ejpam-3659	145	11	(	(	PUNCT
ejpam-3659	145	12	2020	2020	NUM
ejpam-3659	145	13	)	)	PUNCT
ejpam-3659	145	14	,	,	PUNCT
ejpam-3659	145	15	303	303	NUM
ejpam-3659	145	16	-	-	SYM
ejpam-3659	145	17	313	313	NUM
ejpam-3659	145	18	308	308	NUM
ejpam-3659	145	19	theorem	theorem	NOUN
ejpam-3659	145	20	3	3	X
ejpam-3659	145	21	.	.	PUNCT
ejpam-3659	146	1	let	let	VERB
ejpam-3659	146	2	f	f	NOUN
ejpam-3659	146	3	:	:	PUNCT
ejpam-3659	147	1	[	[	X
ejpam-3659	147	2	a	a	X
ejpam-3659	147	3	,	,	PUNCT
ejpam-3659	147	4	b	b	NOUN
ejpam-3659	147	5	]	]	X
ejpam-3659	147	6	→	→	PUNCT
ejpam-3659	147	7	r	r	AUX
ejpam-3659	147	8	be	be	PROPN
ejpam-3659	147	9	mcshane	mcshane	NOUN
ejpam-3659	147	10	integrable	integrable	ADJ
ejpam-3659	147	11	on	on	ADP
ejpam-3659	147	12	[	[	X
ejpam-3659	147	13	a	a	X
ejpam-3659	147	14	,	,	PUNCT
ejpam-3659	147	15	b	b	NOUN
ejpam-3659	147	16	]	]	X
ejpam-3659	147	17	.	.	PUNCT
ejpam-3659	148	1	if	if	SCONJ
ejpam-3659	148	2	g	g	PROPN
ejpam-3659	148	3	=	=	SYM
ejpam-3659	148	4	f	f	PROPN
ejpam-3659	148	5	a.e	a.e	PROPN
ejpam-3659	148	6	.	.	PROPN
ejpam-3659	149	1	on	on	ADP
ejpam-3659	149	2	[	[	X
ejpam-3659	149	3	a	a	X
ejpam-3659	149	4	,	,	PUNCT
ejpam-3659	149	5	b	b	NOUN
ejpam-3659	149	6	]	]	X
ejpam-3659	149	7	,	,	PUNCT
ejpam-3659	149	8	then	then	ADV
ejpam-3659	149	9	g	g	PROPN
ejpam-3659	149	10	is	be	AUX
ejpam-3659	149	11	mcshane	mcshane	PROPN
ejpam-3659	149	12	integrable	integrable	ADJ
ejpam-3659	149	13	on	on	ADP
ejpam-3659	149	14	[	[	X
ejpam-3659	149	15	a	a	X
ejpam-3659	149	16	,	,	PUNCT
ejpam-3659	149	17	b	b	NOUN
ejpam-3659	149	18	]	]	X
ejpam-3659	149	19	,	,	PUNCT
ejpam-3659	149	20	and∫	and∫	PROPN
ejpam-3659	149	21	b	b	X
ejpam-3659	149	22	a	a	DET
ejpam-3659	149	23	g(x	g(x	NOUN
ejpam-3659	149	24	)	)	PUNCT
ejpam-3659	149	25	dx	dx	PROPN
ejpam-3659	150	1	=	=	SYM
ejpam-3659	150	2	∫	∫	PROPN
ejpam-3659	150	3	b	b	PROPN
ejpam-3659	150	4	a	a	DET
ejpam-3659	150	5	f(x	f(x	PROPN
ejpam-3659	150	6	)	)	PUNCT
ejpam-3659	150	7	dx	dx	PROPN
ejpam-3659	150	8	.	.	PUNCT
ejpam-3659	151	1	in	in	ADP
ejpam-3659	151	2	view	view	NOUN
ejpam-3659	151	3	of	of	ADP
ejpam-3659	151	4	theorem	theorem	NOUN
ejpam-3659	151	5	3	3	NUM
ejpam-3659	151	6	,	,	PUNCT
ejpam-3659	151	7	the	the	DET
ejpam-3659	151	8	condition	condition	NOUN
ejpam-3659	151	9	“	"	PUNCT
ejpam-3659	151	10	fn(x	fn(x	X
ejpam-3659	151	11	)	)	PUNCT
ejpam-3659	151	12	→	→	SYM
ejpam-3659	151	13	f(x	f(x	PROPN
ejpam-3659	151	14	)	)	PUNCT
ejpam-3659	151	15	on	on	ADP
ejpam-3659	151	16	[	[	X
ejpam-3659	151	17	a	a	X
ejpam-3659	151	18	,	,	PUNCT
ejpam-3659	151	19	b	b	NOUN
ejpam-3659	151	20	]	]	X
ejpam-3659	151	21	”	"	PUNCT
ejpam-3659	151	22	in	in	ADP
ejpam-3659	151	23	all	all	DET
ejpam-3659	151	24	convergence	convergence	NOUN
ejpam-3659	151	25	theorems	theorem	NOUN
ejpam-3659	151	26	for	for	ADP
ejpam-3659	151	27	the	the	DET
ejpam-3659	151	28	mcshane	mcshane	PROPN
ejpam-3659	151	29	integral	integral	ADJ
ejpam-3659	151	30	(	(	PUNCT
ejpam-3659	151	31	see	see	VERB
ejpam-3659	151	32	[	[	X
ejpam-3659	151	33	5	5	NUM
ejpam-3659	151	34	]	]	PUNCT
ejpam-3659	151	35	)	)	PUNCT
ejpam-3659	151	36	can	can	AUX
ejpam-3659	151	37	now	now	ADV
ejpam-3659	151	38	be	be	AUX
ejpam-3659	151	39	replaced	replace	VERB
ejpam-3659	151	40	by	by	ADP
ejpam-3659	151	41	“	"	PUNCT
ejpam-3659	151	42	fn(x	fn(x	X
ejpam-3659	151	43	)	)	PUNCT
ejpam-3659	152	1	→	→	SYM
ejpam-3659	152	2	f(x	f(x	PROPN
ejpam-3659	152	3	)	)	PUNCT
ejpam-3659	152	4	a.e	a.e	PROPN
ejpam-3659	152	5	.	.	PROPN
ejpam-3659	153	1	on	on	ADP
ejpam-3659	153	2	[	[	X
ejpam-3659	153	3	a	a	PRON
ejpam-3659	153	4	,	,	PUNCT
ejpam-3659	153	5	b	b	NOUN
ejpam-3659	153	6	]	]	X
ejpam-3659	153	7	”	"	PUNCT
ejpam-3659	153	8	.	.	PUNCT
ejpam-3659	154	1	theorem	theorem	VERB
ejpam-3659	154	2	4	4	NUM
ejpam-3659	154	3	.	.	PUNCT
ejpam-3659	155	1	[	[	X
ejpam-3659	155	2	4	4	X
ejpam-3659	155	3	]	]	X
ejpam-3659	155	4	if	if	SCONJ
ejpam-3659	155	5	f	f	PROPN
ejpam-3659	155	6	:	:	PUNCT
ejpam-3659	156	1	[	[	X
ejpam-3659	156	2	a	a	X
ejpam-3659	156	3	,	,	PUNCT
ejpam-3659	156	4	b	b	NOUN
ejpam-3659	156	5	]	]	X
ejpam-3659	156	6	→	→	PUNCT
ejpam-3659	156	7	r	r	NOUN
ejpam-3659	156	8	is	be	AUX
ejpam-3659	156	9	mcshane	mcshane	NOUN
ejpam-3659	156	10	integrable	integrable	ADJ
ejpam-3659	156	11	on	on	ADP
ejpam-3659	156	12	[	[	X
ejpam-3659	156	13	a	a	X
ejpam-3659	156	14	,	,	PUNCT
ejpam-3659	156	15	b	b	NOUN
ejpam-3659	156	16	]	]	X
ejpam-3659	156	17	,	,	PUNCT
ejpam-3659	156	18	then	then	ADV
ejpam-3659	156	19	there	there	PRON
ejpam-3659	156	20	exists	exist	VERB
ejpam-3659	156	21	a	a	DET
ejpam-3659	156	22	sequence	sequence	NOUN
ejpam-3659	156	23	{	{	PUNCT
ejpam-3659	156	24	ϕn}∞n=1	ϕn}∞n=1	NUM
ejpam-3659	156	25	of	of	ADP
ejpam-3659	156	26	step	step	NOUN
ejpam-3659	156	27	functions	function	NOUN
ejpam-3659	156	28	such	such	ADJ
ejpam-3659	156	29	that	that	PRON
ejpam-3659	156	30	ϕn	ϕn	PRON
ejpam-3659	156	31	→	→	SYM
ejpam-3659	156	32	f	f	PROPN
ejpam-3659	156	33	a.e	a.e	PROPN
ejpam-3659	156	34	.	.	PROPN
ejpam-3659	157	1	on	on	ADP
ejpam-3659	157	2	[	[	X
ejpam-3659	157	3	a	a	X
ejpam-3659	157	4	,	,	PUNCT
ejpam-3659	157	5	b	b	NOUN
ejpam-3659	157	6	]	]	PUNCT
ejpam-3659	157	7	and	and	CCONJ
ejpam-3659	157	8	lim	lim	PROPN
ejpam-3659	157	9	n→∞	n→∞	NUM
ejpam-3659	158	1	∫	∫	PROPN
ejpam-3659	158	2	b	b	PROPN
ejpam-3659	158	3	a	a	PRON
ejpam-3659	158	4	|ϕn	|ϕn	NUM
ejpam-3659	158	5	−	−	PROPN
ejpam-3659	158	6	f	f	NOUN
ejpam-3659	158	7	|	|	NOUN
ejpam-3659	158	8	=	=	NOUN
ejpam-3659	158	9	0	0	X
ejpam-3659	158	10	.	.	PUNCT
ejpam-3659	159	1	in	in	ADP
ejpam-3659	159	2	the	the	DET
ejpam-3659	159	3	following	following	NOUN
ejpam-3659	159	4	,	,	PUNCT
ejpam-3659	159	5	if	if	SCONJ
ejpam-3659	159	6	f	f	X
ejpam-3659	159	7	:	:	PUNCT
ejpam-3659	159	8	[	[	X
ejpam-3659	159	9	a	a	X
ejpam-3659	159	10	,	,	PUNCT
ejpam-3659	159	11	b]→	b]→	ADJ
ejpam-3659	159	12	r	r	NOUN
ejpam-3659	159	13	is	be	AUX
ejpam-3659	159	14	a	a	DET
ejpam-3659	159	15	function	function	NOUN
ejpam-3659	159	16	and	and	CCONJ
ejpam-3659	159	17	c	c	NOUN
ejpam-3659	159	18	∈	∈	PROPN
ejpam-3659	159	19	r	r	NOUN
ejpam-3659	159	20	,	,	PUNCT
ejpam-3659	159	21	then	then	ADV
ejpam-3659	159	22	we	we	PRON
ejpam-3659	159	23	denote	denote	VERB
ejpam-3659	159	24	e(f	e(f	PROPN
ejpam-3659	159	25	<	<	X
ejpam-3659	159	26	c	c	X
ejpam-3659	159	27	)	)	PUNCT
ejpam-3659	159	28	=	=	PRON
ejpam-3659	160	1	{	{	PUNCT
ejpam-3659	160	2	x	x	PUNCT
ejpam-3659	160	3	∈	∈	PROPN
ejpam-3659	160	4	[	[	X
ejpam-3659	160	5	a	a	X
ejpam-3659	160	6	,	,	PUNCT
ejpam-3659	160	7	b	b	NOUN
ejpam-3659	160	8	]	]	X
ejpam-3659	160	9	:	:	PUNCT
ejpam-3659	160	10	f(x	f(x	PROPN
ejpam-3659	160	11	)	)	PUNCT
ejpam-3659	161	1	<	<	X
ejpam-3659	161	2	c	c	X
ejpam-3659	161	3	}	}	PUNCT
ejpam-3659	161	4	.	.	PUNCT
ejpam-3659	162	1	theorem	theorem	ADJ
ejpam-3659	162	2	5	5	NUM
ejpam-3659	162	3	.	.	PUNCT
ejpam-3659	163	1	[	[	X
ejpam-3659	163	2	2	2	X
ejpam-3659	163	3	]	]	PUNCT
ejpam-3659	163	4	let	let	VERB
ejpam-3659	163	5	c	c	PRON
ejpam-3659	163	6	be	be	AUX
ejpam-3659	163	7	any	any	DET
ejpam-3659	163	8	real	real	ADJ
ejpam-3659	163	9	number	number	NOUN
ejpam-3659	163	10	.	.	PUNCT
ejpam-3659	164	1	if	if	SCONJ
ejpam-3659	164	2	f	f	X
ejpam-3659	164	3	:	:	PUNCT
ejpam-3659	165	1	[	[	X
ejpam-3659	165	2	a	a	X
ejpam-3659	165	3	,	,	PUNCT
ejpam-3659	165	4	b	b	NOUN
ejpam-3659	165	5	]	]	X
ejpam-3659	165	6	→	→	PUNCT
ejpam-3659	165	7	r	r	NOUN
ejpam-3659	165	8	is	be	AUX
ejpam-3659	165	9	mcshane	mcshane	NOUN
ejpam-3659	165	10	integrable	integrable	ADJ
ejpam-3659	165	11	on	on	ADP
ejpam-3659	165	12	[	[	X
ejpam-3659	165	13	a	a	X
ejpam-3659	165	14	,	,	PUNCT
ejpam-3659	165	15	b	b	NOUN
ejpam-3659	165	16	]	]	X
ejpam-3659	165	17	,	,	PUNCT
ejpam-3659	165	18	then	then	ADV
ejpam-3659	165	19	the	the	DET
ejpam-3659	165	20	characteristic	characteristic	ADJ
ejpam-3659	165	21	function	function	NOUN
ejpam-3659	165	22	χe(f	χe(f	PROPN
ejpam-3659	165	23	<	<	X
ejpam-3659	165	24	c	c	X
ejpam-3659	165	25	)	)	PUNCT
ejpam-3659	165	26	is	be	AUX
ejpam-3659	165	27	mcshane	mcshane	NOUN
ejpam-3659	165	28	integrable	integrable	ADJ
ejpam-3659	165	29	on	on	ADP
ejpam-3659	165	30	[	[	X
ejpam-3659	165	31	a	a	X
ejpam-3659	165	32	,	,	PUNCT
ejpam-3659	165	33	b	b	NOUN
ejpam-3659	165	34	]	]	X
ejpam-3659	165	35	.	.	PUNCT
ejpam-3659	166	1	since	since	SCONJ
ejpam-3659	166	2	e(f	e(f	PROPN
ejpam-3659	166	3	≥	≥	X
ejpam-3659	166	4	c	c	NOUN
ejpam-3659	166	5	)	)	PUNCT
ejpam-3659	166	6	=	=	PRON
ejpam-3659	166	7	{	{	PUNCT
ejpam-3659	166	8	x	x	PUNCT
ejpam-3659	166	9	∈	∈	PROPN
ejpam-3659	167	1	[	[	X
ejpam-3659	167	2	a	a	X
ejpam-3659	167	3	,	,	PUNCT
ejpam-3659	167	4	b	b	NOUN
ejpam-3659	167	5	]	]	X
ejpam-3659	167	6	:	:	PUNCT
ejpam-3659	167	7	f(x	f(x	PROPN
ejpam-3659	167	8	)	)	PUNCT
ejpam-3659	167	9	≥	≥	NOUN
ejpam-3659	167	10	c	c	NOUN
ejpam-3659	167	11	}	}	PUNCT
ejpam-3659	167	12	=	=	PUNCT
ejpam-3659	168	1	[	[	X
ejpam-3659	168	2	a	a	X
ejpam-3659	168	3	,	,	PUNCT
ejpam-3659	168	4	b	b	NOUN
ejpam-3659	168	5	]	]	X
ejpam-3659	168	6	r	r	NOUN
ejpam-3659	168	7	{	{	PUNCT
ejpam-3659	168	8	x	x	SYM
ejpam-3659	168	9	∈	∈	PROPN
ejpam-3659	168	10	[	[	X
ejpam-3659	168	11	a	a	X
ejpam-3659	168	12	,	,	PUNCT
ejpam-3659	168	13	b	b	NOUN
ejpam-3659	168	14	]	]	X
ejpam-3659	168	15	:	:	PUNCT
ejpam-3659	168	16	f(x	f(x	PROPN
ejpam-3659	168	17	)	)	PUNCT
ejpam-3659	168	18	<	<	X
ejpam-3659	169	1	c	c	X
ejpam-3659	169	2	}	}	PUNCT
ejpam-3659	169	3	=	=	PUNCT
ejpam-3659	170	1	[	[	X
ejpam-3659	170	2	a	a	X
ejpam-3659	170	3	,	,	PUNCT
ejpam-3659	170	4	b	b	NOUN
ejpam-3659	170	5	]	]	X
ejpam-3659	170	6	r	r	X
ejpam-3659	170	7	e(f	e(f	PROPN
ejpam-3659	170	8	<	<	X
ejpam-3659	170	9	c	c	PROPN
ejpam-3659	170	10	)	)	PUNCT
ejpam-3659	170	11	,	,	PUNCT
ejpam-3659	170	12	χe(f≥c	χe(f≥c	PROPN
ejpam-3659	170	13	)	)	PUNCT
ejpam-3659	170	14	is	be	AUX
ejpam-3659	170	15	also	also	ADV
ejpam-3659	170	16	mcshane	mcshane	PROPN
ejpam-3659	170	17	integrable	integrable	ADJ
ejpam-3659	170	18	.	.	PUNCT
ejpam-3659	171	1	similarly	similarly	ADV
ejpam-3659	171	2	,	,	PUNCT
ejpam-3659	171	3	χe(f≤c	χe(f≤c	PROPN
ejpam-3659	171	4	)	)	PUNCT
ejpam-3659	171	5	is	be	AUX
ejpam-3659	171	6	mcshane	mcshane	NOUN
ejpam-3659	171	7	integrable	integrable	ADJ
ejpam-3659	171	8	.	.	PUNCT
ejpam-3659	172	1	furthermore	furthermore	ADV
ejpam-3659	172	2	,	,	PUNCT
ejpam-3659	172	3	if	if	SCONJ
ejpam-3659	172	4	a	a	DET
ejpam-3659	172	5	set	set	NOUN
ejpam-3659	172	6	a	a	PRON
ejpam-3659	172	7	is	be	AUX
ejpam-3659	172	8	a	a	DET
ejpam-3659	172	9	countable	countable	ADJ
ejpam-3659	172	10	union	union	NOUN
ejpam-3659	172	11	of	of	ADP
ejpam-3659	172	12	sets	set	NOUN
ejpam-3659	172	13	of	of	ADP
ejpam-3659	172	14	the	the	DET
ejpam-3659	172	15	form	form	NOUN
ejpam-3659	172	16	:	:	PUNCT
ejpam-3659	172	17	e(f	e(f	PROPN
ejpam-3659	172	18	<	<	X
ejpam-3659	172	19	c	c	PROPN
ejpam-3659	172	20	)	)	PUNCT
ejpam-3659	172	21	,	,	PUNCT
ejpam-3659	172	22	e(f	e(f	PROPN
ejpam-3659	172	23	≥	≥	X
ejpam-3659	172	24	c	c	NOUN
ejpam-3659	172	25	)	)	PUNCT
ejpam-3659	172	26	,	,	PUNCT
ejpam-3659	172	27	e(f	e(f	PROPN
ejpam-3659	172	28	>	>	X
ejpam-3659	172	29	c	c	PROPN
ejpam-3659	172	30	)	)	PUNCT
ejpam-3659	172	31	,	,	PUNCT
ejpam-3659	172	32	or	or	CCONJ
ejpam-3659	172	33	e(f	e(f	PROPN
ejpam-3659	172	34	≤	≤	PROPN
ejpam-3659	172	35	c	c	PROPN
ejpam-3659	172	36	)	)	PUNCT
ejpam-3659	172	37	then	then	ADV
ejpam-3659	172	38	χa	χa	PROPN
ejpam-3659	172	39	is	be	AUX
ejpam-3659	172	40	mcshane	mcshane	PROPN
ejpam-3659	172	41	integrable	integrable	ADJ
ejpam-3659	172	42	on	on	ADP
ejpam-3659	172	43	[	[	X
ejpam-3659	172	44	a	a	X
ejpam-3659	172	45	,	,	PUNCT
ejpam-3659	172	46	b	b	NOUN
ejpam-3659	172	47	]	]	X
ejpam-3659	172	48	.	.	PUNCT
ejpam-3659	173	1	hence	hence	ADV
ejpam-3659	173	2	,	,	PUNCT
ejpam-3659	173	3	by	by	ADP
ejpam-3659	173	4	lemma	lemma	PROPN
ejpam-3659	173	5	3	3	NUM
ejpam-3659	173	6	and	and	CCONJ
ejpam-3659	173	7	theorem	theorem	VERB
ejpam-3659	173	8	5	5	NUM
ejpam-3659	173	9	,	,	PUNCT
ejpam-3659	173	10	we	we	PRON
ejpam-3659	173	11	have	have	VERB
ejpam-3659	173	12	the	the	DET
ejpam-3659	173	13	following	follow	VERB
ejpam-3659	173	14	result	result	NOUN
ejpam-3659	173	15	:	:	PUNCT
ejpam-3659	174	1	corollary	corollary	ADJ
ejpam-3659	174	2	1	1	X
ejpam-3659	174	3	.	.	PUNCT
ejpam-3659	175	1	let	let	VERB
ejpam-3659	175	2	c	c	PRON
ejpam-3659	175	3	be	be	AUX
ejpam-3659	175	4	any	any	DET
ejpam-3659	175	5	real	real	ADJ
ejpam-3659	175	6	number	number	NOUN
ejpam-3659	175	7	.	.	PUNCT
ejpam-3659	176	1	if	if	SCONJ
ejpam-3659	176	2	f	f	X
ejpam-3659	176	3	:	:	PUNCT
ejpam-3659	177	1	[	[	X
ejpam-3659	177	2	a	a	X
ejpam-3659	177	3	,	,	PUNCT
ejpam-3659	177	4	b	b	NOUN
ejpam-3659	177	5	]	]	X
ejpam-3659	177	6	→	→	PUNCT
ejpam-3659	177	7	r	r	NOUN
ejpam-3659	177	8	is	be	AUX
ejpam-3659	177	9	mcshane	mcshane	NOUN
ejpam-3659	177	10	integrable	integrable	ADJ
ejpam-3659	177	11	on	on	ADP
ejpam-3659	177	12	[	[	X
ejpam-3659	177	13	a	a	X
ejpam-3659	177	14	,	,	PUNCT
ejpam-3659	177	15	b	b	NOUN
ejpam-3659	177	16	]	]	X
ejpam-3659	177	17	,	,	PUNCT
ejpam-3659	177	18	then	then	ADV
ejpam-3659	177	19	the	the	DET
ejpam-3659	177	20	sets	set	NOUN
ejpam-3659	177	21	e(f	e(f	PROPN
ejpam-3659	177	22	<	<	X
ejpam-3659	177	23	c	c	PROPN
ejpam-3659	177	24	)	)	PUNCT
ejpam-3659	177	25	,	,	PUNCT
ejpam-3659	177	26	e(f	e(f	PROPN
ejpam-3659	177	27	≥	≥	X
ejpam-3659	177	28	c	c	NOUN
ejpam-3659	177	29	)	)	PUNCT
ejpam-3659	177	30	,	,	PUNCT
ejpam-3659	177	31	e(f	e(f	PROPN
ejpam-3659	177	32	>	>	X
ejpam-3659	177	33	c	c	PROPN
ejpam-3659	177	34	)	)	PUNCT
ejpam-3659	177	35	and	and	CCONJ
ejpam-3659	177	36	e(f	e(f	PROPN
ejpam-3659	177	37	≤	≤	PROPN
ejpam-3659	177	38	c	c	X
ejpam-3659	177	39	)	)	PUNCT
ejpam-3659	177	40	are	be	AUX
ejpam-3659	177	41	integrable	integrable	ADJ
ejpam-3659	177	42	sets	set	NOUN
ejpam-3659	177	43	.	.	PUNCT
ejpam-3659	178	1	below	below	ADV
ejpam-3659	178	2	is	be	AUX
ejpam-3659	178	3	a	a	DET
ejpam-3659	178	4	version	version	NOUN
ejpam-3659	178	5	of	of	ADP
ejpam-3659	178	6	the	the	DET
ejpam-3659	178	7	monotone	monotone	ADJ
ejpam-3659	178	8	convergence	convergence	NOUN
ejpam-3659	178	9	theorem	theorem	NOUN
ejpam-3659	178	10	(	(	PUNCT
ejpam-3659	178	11	mct	mct	PROPN
ejpam-3659	178	12	)	)	PUNCT
ejpam-3659	178	13	for	for	ADP
ejpam-3659	178	14	mcshane	mcshane	NOUN
ejpam-3659	178	15	integrals	integral	NOUN
ejpam-3659	178	16	.	.	PUNCT
ejpam-3659	179	1	theorem	theorem	NOUN
ejpam-3659	179	2	6	6	NUM
ejpam-3659	179	3	(	(	PUNCT
ejpam-3659	179	4	monotone	monotone	ADJ
ejpam-3659	179	5	convergence	convergence	NOUN
ejpam-3659	179	6	theorem	theorem	VERB
ejpam-3659	179	7	)	)	PUNCT
ejpam-3659	179	8	.	.	PUNCT
ejpam-3659	180	1	[	[	X
ejpam-3659	180	2	1	1	NUM
ejpam-3659	180	3	,	,	PUNCT
ejpam-3659	180	4	5	5	NUM
ejpam-3659	180	5	]	]	PUNCT
ejpam-3659	180	6	let	let	VERB
ejpam-3659	180	7	{	{	PUNCT
ejpam-3659	180	8	fn}∞n=1	fn}∞n=1	X
ejpam-3659	180	9	be	be	AUX
ejpam-3659	180	10	an	an	DET
ejpam-3659	180	11	increasing	increase	VERB
ejpam-3659	180	12	sequence	sequence	NOUN
ejpam-3659	180	13	of	of	ADP
ejpam-3659	180	14	mcshane	mcshane	PROPN
ejpam-3659	180	15	integrable	integrable	ADJ
ejpam-3659	180	16	functions	function	NOUN
ejpam-3659	180	17	on	on	ADP
ejpam-3659	180	18	[	[	X
ejpam-3659	180	19	a	a	DET
ejpam-3659	180	20	,	,	PUNCT
ejpam-3659	180	21	b	b	NOUN
ejpam-3659	180	22	]	]	X
ejpam-3659	180	23	such	such	ADJ
ejpam-3659	180	24	that	that	SCONJ
ejpam-3659	180	25	lim	lim	PROPN
ejpam-3659	180	26	n→∞	n→∞	X
ejpam-3659	180	27	fn(x	fn(x	NOUN
ejpam-3659	180	28	)	)	PUNCT
ejpam-3659	180	29	=	=	SYM
ejpam-3659	180	30	f(x	f(x	PROPN
ejpam-3659	180	31	)	)	PUNCT
ejpam-3659	180	32	,	,	PUNCT
ejpam-3659	180	33	for	for	ADP
ejpam-3659	180	34	each	each	DET
ejpam-3659	180	35	x	x	SYM
ejpam-3659	180	36	∈	∈	PROPN
ejpam-3659	180	37	[	[	X
ejpam-3659	180	38	a	a	X
ejpam-3659	180	39	,	,	PUNCT
ejpam-3659	180	40	b	b	NOUN
ejpam-3659	180	41	]	]	PUNCT
ejpam-3659	180	42	.	.	PUNCT
ejpam-3659	181	1	f.	f.	PROPN
ejpam-3659	181	2	sumalpong	sumalpong	PROPN
ejpam-3659	181	3	jr	jr	PROPN
ejpam-3659	181	4	.	.	PROPN
ejpam-3659	181	5	,	,	PUNCT
ejpam-3659	181	6	j.	j.	PROPN
ejpam-3659	181	7	benitez	benitez	PROPN
ejpam-3659	181	8	/	/	PUNCT
ejpam-3659	181	9	eur	eur	PROPN
ejpam-3659	181	10	.	.	PUNCT
ejpam-3659	182	1	j.	j.	PROPN
ejpam-3659	182	2	pure	pure	PROPN
ejpam-3659	182	3	appl	appl	PROPN
ejpam-3659	182	4	.	.	PROPN
ejpam-3659	182	5	math	math	PROPN
ejpam-3659	182	6	,	,	PUNCT
ejpam-3659	182	7	13	13	NUM
ejpam-3659	182	8	(	(	PUNCT
ejpam-3659	182	9	2	2	NUM
ejpam-3659	182	10	)	)	PUNCT
ejpam-3659	182	11	(	(	PUNCT
ejpam-3659	182	12	2020	2020	NUM
ejpam-3659	182	13	)	)	PUNCT
ejpam-3659	182	14	,	,	PUNCT
ejpam-3659	182	15	303	303	NUM
ejpam-3659	182	16	-	-	SYM
ejpam-3659	182	17	313	313	NUM
ejpam-3659	182	18	309	309	NUM
ejpam-3659	182	19	if	if	SCONJ
ejpam-3659	182	20	sup	sup	NOUN
ejpam-3659	182	21	{	{	PUNCT
ejpam-3659	182	22	∫	∫	PROPN
ejpam-3659	182	23	b	b	PROPN
ejpam-3659	183	1	a	a	DET
ejpam-3659	183	2	fn	fn	NOUN
ejpam-3659	183	3	:	:	PUNCT
ejpam-3659	183	4	n	n	CCONJ
ejpam-3659	183	5	∈	∈	PROPN
ejpam-3659	183	6	n	n	CCONJ
ejpam-3659	183	7	}	}	PUNCT
ejpam-3659	183	8	<	<	X
ejpam-3659	183	9	∞	∞	PROPN
ejpam-3659	183	10	,	,	PUNCT
ejpam-3659	183	11	then	then	ADV
ejpam-3659	183	12	f	f	PROPN
ejpam-3659	183	13	is	be	AUX
ejpam-3659	183	14	mcshane	mcshane	PROPN
ejpam-3659	183	15	integrable	integrable	ADJ
ejpam-3659	183	16	on	on	ADP
ejpam-3659	183	17	[	[	X
ejpam-3659	183	18	a	a	DET
ejpam-3659	183	19	,	,	PUNCT
ejpam-3659	183	20	b	b	NOUN
ejpam-3659	183	21	]	]	PUNCT
ejpam-3659	183	22	and	and	CCONJ
ejpam-3659	183	23	lim	lim	PROPN
ejpam-3659	183	24	n→∞	n→∞	NUM
ejpam-3659	184	1	∫	∫	PROPN
ejpam-3659	184	2	b	b	PROPN
ejpam-3659	184	3	a	a	DET
ejpam-3659	184	4	fn	fn	NOUN
ejpam-3659	184	5	=	=	SYM
ejpam-3659	184	6	∫	∫	PROPN
ejpam-3659	184	7	b	b	PROPN
ejpam-3659	185	1	a	a	DET
ejpam-3659	185	2	f.	f.	PROPN
ejpam-3659	185	3	recall	recall	NOUN
ejpam-3659	185	4	that	that	SCONJ
ejpam-3659	185	5	if	if	SCONJ
ejpam-3659	185	6	f	f	X
ejpam-3659	185	7	:	:	PUNCT
ejpam-3659	186	1	[	[	X
ejpam-3659	186	2	a	a	X
ejpam-3659	186	3	,	,	PUNCT
ejpam-3659	186	4	b]→	b]→	ADJ
ejpam-3659	186	5	r	r	NOUN
ejpam-3659	186	6	is	be	AUX
ejpam-3659	186	7	mcshane	mcshane	NOUN
ejpam-3659	186	8	integrable	integrable	ADJ
ejpam-3659	186	9	on	on	ADP
ejpam-3659	186	10	[	[	X
ejpam-3659	186	11	a	a	X
ejpam-3659	186	12	,	,	PUNCT
ejpam-3659	186	13	b	b	NOUN
ejpam-3659	186	14	]	]	X
ejpam-3659	186	15	,	,	PUNCT
ejpam-3659	186	16	then	then	ADV
ejpam-3659	186	17	f	f	PROPN
ejpam-3659	186	18	is	be	AUX
ejpam-3659	186	19	mcshane	mcshane	PROPN
ejpam-3659	186	20	integrable	integrable	ADJ
ejpam-3659	186	21	on	on	ADP
ejpam-3659	186	22	every	every	DET
ejpam-3659	186	23	sub	sub	NOUN
ejpam-3659	186	24	-	-	NOUN
ejpam-3659	186	25	interval	interval	NOUN
ejpam-3659	186	26	[	[	X
ejpam-3659	186	27	c	c	X
ejpam-3659	186	28	,	,	PUNCT
ejpam-3659	186	29	d	d	X
ejpam-3659	186	30	]	]	X
ejpam-3659	186	31	of	of	ADP
ejpam-3659	186	32	[	[	X
ejpam-3659	186	33	a	a	X
ejpam-3659	186	34	,	,	PUNCT
ejpam-3659	186	35	b	b	NOUN
ejpam-3659	186	36	]	]	X
ejpam-3659	186	37	.	.	PUNCT
ejpam-3659	187	1	however	however	ADV
ejpam-3659	187	2	,	,	PUNCT
ejpam-3659	187	3	mcshane	mcshane	PROPN
ejpam-3659	187	4	integrability	integrability	NOUN
ejpam-3659	187	5	on	on	ADP
ejpam-3659	187	6	[	[	X
ejpam-3659	187	7	a	a	DET
ejpam-3659	187	8	,	,	PUNCT
ejpam-3659	187	9	b	b	NOUN
ejpam-3659	187	10	]	]	X
ejpam-3659	187	11	does	do	AUX
ejpam-3659	187	12	not	not	PART
ejpam-3659	187	13	imply	imply	VERB
ejpam-3659	187	14	mcshane	mcshane	PROPN
ejpam-3659	187	15	integrability	integrability	NOUN
ejpam-3659	187	16	on	on	ADP
ejpam-3659	187	17	any	any	DET
ejpam-3659	187	18	e	e	NOUN
ejpam-3659	187	19	⊆	⊆	NUM
ejpam-3659	187	20	[	[	X
ejpam-3659	187	21	a	a	X
ejpam-3659	187	22	,	,	PUNCT
ejpam-3659	187	23	b	b	NOUN
ejpam-3659	187	24	]	]	X
ejpam-3659	187	25	.	.	PUNCT
ejpam-3659	188	1	one	one	NUM
ejpam-3659	188	2	necessary	necessary	ADJ
ejpam-3659	188	3	condition	condition	NOUN
ejpam-3659	188	4	is	be	AUX
ejpam-3659	188	5	that	that	SCONJ
ejpam-3659	188	6	a	a	DET
ejpam-3659	188	7	subset	subset	NOUN
ejpam-3659	188	8	e	e	NOUN
ejpam-3659	188	9	⊆	⊆	NUM
ejpam-3659	188	10	[	[	X
ejpam-3659	188	11	a	a	X
ejpam-3659	188	12	,	,	PUNCT
ejpam-3659	188	13	b	b	NOUN
ejpam-3659	188	14	]	]	X
ejpam-3659	188	15	must	must	AUX
ejpam-3659	188	16	be	be	AUX
ejpam-3659	188	17	an	an	DET
ejpam-3659	188	18	integrable	integrable	ADJ
ejpam-3659	188	19	set	set	NOUN
ejpam-3659	188	20	.	.	PUNCT
ejpam-3659	189	1	theorem	theorem	VERB
ejpam-3659	189	2	7	7	NUM
ejpam-3659	189	3	.	.	PUNCT
ejpam-3659	190	1	[	[	X
ejpam-3659	190	2	3	3	X
ejpam-3659	190	3	]	]	PUNCT
ejpam-3659	190	4	if	if	SCONJ
ejpam-3659	190	5	f	f	X
ejpam-3659	190	6	:	:	PUNCT
ejpam-3659	191	1	[	[	X
ejpam-3659	191	2	a	a	X
ejpam-3659	191	3	,	,	PUNCT
ejpam-3659	191	4	b	b	NOUN
ejpam-3659	191	5	]	]	X
ejpam-3659	191	6	→	→	PUNCT
ejpam-3659	191	7	r	r	NOUN
ejpam-3659	191	8	is	be	AUX
ejpam-3659	191	9	mcshane	mcshane	NOUN
ejpam-3659	191	10	integrable	integrable	ADJ
ejpam-3659	191	11	on	on	ADP
ejpam-3659	191	12	[	[	X
ejpam-3659	191	13	a	a	X
ejpam-3659	191	14	,	,	PUNCT
ejpam-3659	191	15	b	b	NOUN
ejpam-3659	191	16	]	]	X
ejpam-3659	191	17	,	,	PUNCT
ejpam-3659	191	18	then	then	ADV
ejpam-3659	191	19	f	f	PROPN
ejpam-3659	191	20	is	be	AUX
ejpam-3659	191	21	mcshane	mcshane	PROPN
ejpam-3659	191	22	integrable	integrable	ADJ
ejpam-3659	191	23	on	on	ADP
ejpam-3659	191	24	every	every	DET
ejpam-3659	191	25	integrable	integrable	ADJ
ejpam-3659	191	26	subset	subset	NOUN
ejpam-3659	191	27	e	e	NOUN
ejpam-3659	191	28	of	of	ADP
ejpam-3659	191	29	[	[	X
ejpam-3659	191	30	a	a	X
ejpam-3659	191	31	,	,	PUNCT
ejpam-3659	191	32	b	b	NOUN
ejpam-3659	191	33	]	]	X
ejpam-3659	191	34	.	.	PUNCT
ejpam-3659	192	1	3	3	X
ejpam-3659	192	2	.	.	X
ejpam-3659	192	3	results	result	NOUN
ejpam-3659	192	4	in	in	ADP
ejpam-3659	192	5	what	what	PRON
ejpam-3659	192	6	follows	follow	VERB
ejpam-3659	192	7	,	,	PUNCT
ejpam-3659	192	8	we	we	PRON
ejpam-3659	192	9	denote	denote	VERB
ejpam-3659	192	10	the	the	DET
ejpam-3659	192	11	family	family	NOUN
ejpam-3659	192	12	of	of	ADP
ejpam-3659	192	13	all	all	DET
ejpam-3659	192	14	sub	sub	NOUN
ejpam-3659	192	15	-	-	NOUN
ejpam-3659	192	16	intervals	interval	NOUN
ejpam-3659	192	17	of	of	ADP
ejpam-3659	192	18	[	[	X
ejpam-3659	192	19	a	a	X
ejpam-3659	192	20	,	,	PUNCT
ejpam-3659	192	21	b	b	NOUN
ejpam-3659	192	22	]	]	PUNCT
ejpam-3659	192	23	by	by	ADP
ejpam-3659	192	24	i([a	i([a	PROPN
ejpam-3659	192	25	,	,	PUNCT
ejpam-3659	192	26	b	b	NOUN
ejpam-3659	192	27	]	]	X
ejpam-3659	192	28	)	)	PUNCT
ejpam-3659	192	29	and	and	CCONJ
ejpam-3659	192	30	δ	δ	PROPN
ejpam-3659	192	31	is	be	AUX
ejpam-3659	192	32	a	a	DET
ejpam-3659	192	33	gauge	gauge	NOUN
ejpam-3659	192	34	on	on	ADP
ejpam-3659	192	35	[	[	X
ejpam-3659	192	36	a	a	X
ejpam-3659	192	37	,	,	PUNCT
ejpam-3659	192	38	b	b	NOUN
ejpam-3659	192	39	]	]	PUNCT
ejpam-3659	192	40	.	.	PUNCT
ejpam-3659	193	1	definition	definition	NOUN
ejpam-3659	193	2	5	5	NUM
ejpam-3659	193	3	.	.	PUNCT
ejpam-3659	194	1	[	[	X
ejpam-3659	194	2	8	8	NUM
ejpam-3659	194	3	]	]	PUNCT
ejpam-3659	194	4	let	let	VERB
ejpam-3659	194	5	f	f	PROPN
ejpam-3659	194	6	:	:	PUNCT
ejpam-3659	194	7	i([a	i([a	PROPN
ejpam-3659	194	8	,	,	PUNCT
ejpam-3659	194	9	b	b	NOUN
ejpam-3659	194	10	]	]	X
ejpam-3659	194	11	)	)	PUNCT
ejpam-3659	194	12	→	→	SYM
ejpam-3659	194	13	r.	r.	NOUN
ejpam-3659	194	14	for	for	ADP
ejpam-3659	194	15	any	any	DET
ejpam-3659	194	16	subset	subset	NOUN
ejpam-3659	194	17	x	x	X
ejpam-3659	194	18	of	of	ADP
ejpam-3659	194	19	[	[	X
ejpam-3659	194	20	a	a	X
ejpam-3659	194	21	,	,	PUNCT
ejpam-3659	194	22	b	b	NOUN
ejpam-3659	194	23	]	]	X
ejpam-3659	194	24	,	,	PUNCT
ejpam-3659	194	25	the	the	DET
ejpam-3659	194	26	mcshane	mcshane	PROPN
ejpam-3659	194	27	δvariation	δvariation	NOUN
ejpam-3659	194	28	of	of	ADP
ejpam-3659	194	29	f	f	PROPN
ejpam-3659	194	30	on	on	ADP
ejpam-3659	194	31	x	x	PROPN
ejpam-3659	194	32	is	be	AUX
ejpam-3659	194	33	given	give	VERB
ejpam-3659	194	34	by	by	ADP
ejpam-3659	194	35	v	v	PROPN
ejpam-3659	194	36	(	(	PUNCT
ejpam-3659	194	37	f	f	PROPN
ejpam-3659	194	38	,	,	PUNCT
ejpam-3659	194	39	x	x	NOUN
ejpam-3659	194	40	,	,	PUNCT
ejpam-3659	194	41	δ	δ	PROPN
ejpam-3659	194	42	)	)	PUNCT
ejpam-3659	194	43	=	=	SYM
ejpam-3659	194	44	sup	sup	NOUN
ejpam-3659	194	45	p∈p([a	p∈p([a	PROPN
ejpam-3659	194	46	,	,	PUNCT
ejpam-3659	194	47	b	b	NOUN
ejpam-3659	194	48	]	]	X
ejpam-3659	194	49	)	)	PUNCT
ejpam-3659	194	50	(	(	PUNCT
ejpam-3659	194	51	p	p	NOUN
ejpam-3659	194	52	)	)	PUNCT
ejpam-3659	194	53	∑	∑	PROPN
ejpam-3659	194	54	|f	|f	PROPN
ejpam-3659	194	55	(	(	PUNCT
ejpam-3659	194	56	u	u	NOUN
ejpam-3659	194	57	,	,	PUNCT
ejpam-3659	194	58	v)|	v)|	NOUN
ejpam-3659	194	59	,	,	PUNCT
ejpam-3659	194	60	where	where	SCONJ
ejpam-3659	194	61	p([a	p([a	NOUN
ejpam-3659	194	62	,	,	PUNCT
ejpam-3659	194	63	b	b	NOUN
ejpam-3659	194	64	]	]	X
ejpam-3659	194	65	)	)	PUNCT
ejpam-3659	194	66	is	be	AUX
ejpam-3659	194	67	the	the	DET
ejpam-3659	194	68	collection	collection	NOUN
ejpam-3659	194	69	of	of	ADP
ejpam-3659	194	70	all	all	DET
ejpam-3659	194	71	mcshane	mcshane	PROPN
ejpam-3659	194	72	δ	δ	PROPN
ejpam-3659	194	73	-	-	PUNCT
ejpam-3659	194	74	fine	fine	ADJ
ejpam-3659	194	75	partial	partial	ADJ
ejpam-3659	194	76	division	division	NOUN
ejpam-3659	194	77	p	p	NOUN
ejpam-3659	194	78	=	=	PUNCT
ejpam-3659	194	79	{	{	PUNCT
ejpam-3659	194	80	(	(	PUNCT
ejpam-3659	194	81	[	[	X
ejpam-3659	194	82	u	u	NOUN
ejpam-3659	194	83	,	,	PUNCT
ejpam-3659	194	84	v	v	ADP
ejpam-3659	194	85	]	]	X
ejpam-3659	194	86	,	,	PUNCT
ejpam-3659	194	87	ξ	ξ	X
ejpam-3659	194	88	)	)	PUNCT
ejpam-3659	194	89	}	}	PUNCT
ejpam-3659	194	90	of	of	ADP
ejpam-3659	194	91	[	[	X
ejpam-3659	194	92	a	a	X
ejpam-3659	194	93	,	,	PUNCT
ejpam-3659	194	94	b	b	NOUN
ejpam-3659	194	95	]	]	X
ejpam-3659	194	96	with	with	ADP
ejpam-3659	194	97	ξ	ξ	PROPN
ejpam-3659	194	98	∈	∈	PROPN
ejpam-3659	194	99	x.	x.	NOUN
ejpam-3659	195	1	the	the	DET
ejpam-3659	195	2	mcshane	mcshane	PROPN
ejpam-3659	195	3	variational	variational	ADJ
ejpam-3659	195	4	measure	measure	NOUN
ejpam-3659	195	5	of	of	ADP
ejpam-3659	195	6	f	f	PROPN
ejpam-3659	195	7	on	on	ADP
ejpam-3659	195	8	x	x	PROPN
ejpam-3659	195	9	is	be	AUX
ejpam-3659	195	10	given	give	VERB
ejpam-3659	195	11	by	by	ADP
ejpam-3659	195	12	vmf	vmf	PROPN
ejpam-3659	195	13	(	(	PUNCT
ejpam-3659	195	14	x	x	NOUN
ejpam-3659	195	15	)	)	PUNCT
ejpam-3659	195	16	=	=	SYM
ejpam-3659	195	17	inf	inf	NOUN
ejpam-3659	195	18	{	{	PUNCT
ejpam-3659	195	19	v	v	NOUN
ejpam-3659	195	20	(	(	PUNCT
ejpam-3659	195	21	f	f	PROPN
ejpam-3659	195	22	,	,	PUNCT
ejpam-3659	195	23	x	x	NOUN
ejpam-3659	195	24	,	,	PUNCT
ejpam-3659	195	25	δ	δ	PROPN
ejpam-3659	195	26	)	)	PUNCT
ejpam-3659	195	27	:	:	PUNCT
ejpam-3659	196	1	δ	δ	PROPN
ejpam-3659	196	2	is	be	AUX
ejpam-3659	196	3	a	a	DET
ejpam-3659	196	4	gauge	gauge	NOUN
ejpam-3659	196	5	on	on	ADP
ejpam-3659	196	6	x	x	SYM
ejpam-3659	196	7	}	}	PUNCT
ejpam-3659	196	8	.	.	PUNCT
ejpam-3659	197	1	if	if	SCONJ
ejpam-3659	197	2	f	f	X
ejpam-3659	197	3	:	:	PUNCT
ejpam-3659	198	1	[	[	X
ejpam-3659	198	2	a	a	X
ejpam-3659	198	3	,	,	PUNCT
ejpam-3659	198	4	b]→	b]→	ADJ
ejpam-3659	198	5	r	r	NOUN
ejpam-3659	198	6	is	be	AUX
ejpam-3659	198	7	mcshane	mcshane	NOUN
ejpam-3659	198	8	integrable	integrable	ADJ
ejpam-3659	198	9	on	on	ADP
ejpam-3659	198	10	[	[	X
ejpam-3659	198	11	a	a	DET
ejpam-3659	198	12	,	,	PUNCT
ejpam-3659	198	13	b	b	NOUN
ejpam-3659	198	14	]	]	X
ejpam-3659	198	15	with	with	ADP
ejpam-3659	198	16	primitive	primitive	ADJ
ejpam-3659	198	17	f	f	NOUN
ejpam-3659	198	18	,	,	PUNCT
ejpam-3659	198	19	then	then	ADV
ejpam-3659	198	20	we	we	PRON
ejpam-3659	198	21	write	write	VERB
ejpam-3659	198	22	f	f	PROPN
ejpam-3659	198	23	(	(	PUNCT
ejpam-3659	198	24	u	u	NOUN
ejpam-3659	198	25	,	,	PUNCT
ejpam-3659	198	26	v	v	NOUN
ejpam-3659	198	27	)	)	PUNCT
ejpam-3659	199	1	=	=	SYM
ejpam-3659	199	2	f	f	PROPN
ejpam-3659	199	3	(	(	PUNCT
ejpam-3659	199	4	v)−	v)−	PROPN
ejpam-3659	199	5	f	f	X
ejpam-3659	199	6	(	(	PUNCT
ejpam-3659	199	7	u	u	NOUN
ejpam-3659	199	8	)	)	PUNCT
ejpam-3659	199	9	,	,	PUNCT
ejpam-3659	199	10	for	for	ADP
ejpam-3659	199	11	any	any	DET
ejpam-3659	199	12	u	u	NOUN
ejpam-3659	199	13	≤	≤	X
ejpam-3659	199	14	v	v	NOUN
ejpam-3659	199	15	in	in	ADP
ejpam-3659	199	16	[	[	X
ejpam-3659	199	17	a	a	PRON
ejpam-3659	199	18	,	,	PUNCT
ejpam-3659	199	19	b	b	NOUN
ejpam-3659	199	20	]	]	PUNCT
ejpam-3659	199	21	.	.	PUNCT
ejpam-3659	200	1	lemma	lemma	PROPN
ejpam-3659	200	2	4	4	X
ejpam-3659	200	3	.	.	PUNCT
ejpam-3659	201	1	let	let	VERB
ejpam-3659	201	2	f	f	NOUN
ejpam-3659	201	3	:	:	PUNCT
ejpam-3659	202	1	[	[	X
ejpam-3659	202	2	a	a	X
ejpam-3659	202	3	,	,	PUNCT
ejpam-3659	202	4	b	b	NOUN
ejpam-3659	202	5	]	]	X
ejpam-3659	202	6	→	→	PUNCT
ejpam-3659	202	7	r	r	AUX
ejpam-3659	202	8	be	be	PROPN
ejpam-3659	202	9	mcshane	mcshane	NOUN
ejpam-3659	202	10	integrable	integrable	ADJ
ejpam-3659	202	11	on	on	ADP
ejpam-3659	202	12	[	[	X
ejpam-3659	202	13	a	a	DET
ejpam-3659	202	14	,	,	PUNCT
ejpam-3659	202	15	b	b	NOUN
ejpam-3659	202	16	]	]	X
ejpam-3659	202	17	with	with	ADP
ejpam-3659	202	18	primitive	primitive	ADJ
ejpam-3659	202	19	f	f	PROPN
ejpam-3659	202	20	and	and	CCONJ
ejpam-3659	202	21	x	x	SYM
ejpam-3659	202	22	⊆	⊆	NUM
ejpam-3659	202	23	[	[	X
ejpam-3659	202	24	a	a	X
ejpam-3659	202	25	,	,	PUNCT
ejpam-3659	202	26	b	b	NOUN
ejpam-3659	202	27	]	]	X
ejpam-3659	202	28	.	.	PUNCT
ejpam-3659	203	1	if	if	SCONJ
ejpam-3659	203	2	f	f	PROPN
ejpam-3659	203	3	is	be	AUX
ejpam-3659	203	4	mcshane	mcshane	PROPN
ejpam-3659	203	5	integrable	integrable	ADJ
ejpam-3659	203	6	on	on	ADP
ejpam-3659	203	7	x	x	PRON
ejpam-3659	203	8	,	,	PUNCT
ejpam-3659	203	9	then	then	ADV
ejpam-3659	203	10	vmf	vmf	PROPN
ejpam-3659	203	11	(	(	PUNCT
ejpam-3659	203	12	x	x	X
ejpam-3659	203	13	)	)	PUNCT
ejpam-3659	203	14	<	<	X
ejpam-3659	203	15	∞	∞	NUM
ejpam-3659	203	16	and∫	and∫	PROPN
ejpam-3659	203	17	b	b	PROPN
ejpam-3659	203	18	a	a	PRON
ejpam-3659	203	19	|f	|f	PROPN
ejpam-3659	203	20	·	·	PUNCT
ejpam-3659	203	21	χx	χx	PROPN
ejpam-3659	204	1	|	|	ADV
ejpam-3659	204	2	=	=	SYM
ejpam-3659	204	3	vmf	vmf	PROPN
ejpam-3659	204	4	(	(	PUNCT
ejpam-3659	204	5	x	x	NOUN
ejpam-3659	204	6	)	)	PUNCT
ejpam-3659	204	7	.	.	PUNCT
ejpam-3659	205	1	proof	proof	NOUN
ejpam-3659	205	2	.	.	PUNCT
ejpam-3659	206	1	let	let	VERB
ejpam-3659	206	2	ε	ε	PROPN
ejpam-3659	206	3	>	>	X
ejpam-3659	206	4	0	0	PROPN
ejpam-3659	206	5	.	.	PUNCT
ejpam-3659	207	1	by	by	ADP
ejpam-3659	207	2	henstock	henstock	PROPN
ejpam-3659	207	3	lemma	lemma	PROPN
ejpam-3659	207	4	,	,	PUNCT
ejpam-3659	207	5	there	there	PRON
ejpam-3659	207	6	exists	exist	VERB
ejpam-3659	207	7	a	a	DET
ejpam-3659	207	8	gauge	gauge	NOUN
ejpam-3659	207	9	δ1	δ1	NOUN
ejpam-3659	207	10	on	on	ADP
ejpam-3659	207	11	[	[	X
ejpam-3659	207	12	a	a	DET
ejpam-3659	207	13	,	,	PUNCT
ejpam-3659	207	14	b	b	NOUN
ejpam-3659	207	15	]	]	X
ejpam-3659	207	16	such	such	ADJ
ejpam-3659	207	17	that	that	SCONJ
ejpam-3659	207	18	(	(	PUNCT
ejpam-3659	207	19	p	p	NOUN
ejpam-3659	207	20	)	)	PUNCT
ejpam-3659	207	21	∑	∑	PROPN
ejpam-3659	207	22	|f	|f	PROPN
ejpam-3659	207	23	(	(	PUNCT
ejpam-3659	207	24	u	u	PROPN
ejpam-3659	207	25	,	,	PUNCT
ejpam-3659	207	26	v)−	v)−	PROPN
ejpam-3659	207	27	f(ξ)(v	f(ξ)(v	PUNCT
ejpam-3659	207	28	−	−	PROPN
ejpam-3659	207	29	u)|	u)|	NOUN
ejpam-3659	207	30	<	<	X
ejpam-3659	207	31	ε	ε	PROPN
ejpam-3659	207	32	3	3	NUM
ejpam-3659	207	33	whenever	whenever	SCONJ
ejpam-3659	207	34	p	p	NOUN
ejpam-3659	207	35	=	=	X
ejpam-3659	207	36	{	{	PUNCT
ejpam-3659	207	37	(	(	PUNCT
ejpam-3659	207	38	[	[	X
ejpam-3659	207	39	u	u	NOUN
ejpam-3659	207	40	,	,	PUNCT
ejpam-3659	207	41	v	v	ADP
ejpam-3659	207	42	]	]	X
ejpam-3659	207	43	,	,	PUNCT
ejpam-3659	207	44	ξ	ξ	X
ejpam-3659	207	45	)	)	PUNCT
ejpam-3659	207	46	}	}	PUNCT
ejpam-3659	207	47	is	be	AUX
ejpam-3659	207	48	a	a	DET
ejpam-3659	207	49	mcshane	mcshane	PROPN
ejpam-3659	207	50	δ1	δ1	NOUN
ejpam-3659	207	51	-	-	PUNCT
ejpam-3659	207	52	fine	fine	ADJ
ejpam-3659	207	53	partial	partial	ADJ
ejpam-3659	207	54	division	division	NOUN
ejpam-3659	207	55	of	of	ADP
ejpam-3659	207	56	[	[	X
ejpam-3659	207	57	a	a	X
ejpam-3659	207	58	,	,	PUNCT
ejpam-3659	207	59	b	b	NOUN
ejpam-3659	207	60	]	]	PUNCT
ejpam-3659	207	61	.	.	PUNCT
ejpam-3659	208	1	f.	f.	PROPN
ejpam-3659	208	2	sumalpong	sumalpong	PROPN
ejpam-3659	208	3	jr	jr	PROPN
ejpam-3659	208	4	.	.	PROPN
ejpam-3659	208	5	,	,	PUNCT
ejpam-3659	208	6	j.	j.	PROPN
ejpam-3659	208	7	benitez	benitez	PROPN
ejpam-3659	208	8	/	/	PUNCT
ejpam-3659	208	9	eur	eur	PROPN
ejpam-3659	208	10	.	.	PUNCT
ejpam-3659	209	1	j.	j.	PROPN
ejpam-3659	209	2	pure	pure	PROPN
ejpam-3659	209	3	appl	appl	PROPN
ejpam-3659	209	4	.	.	PROPN
ejpam-3659	209	5	math	math	PROPN
ejpam-3659	209	6	,	,	PUNCT
ejpam-3659	209	7	13	13	NUM
ejpam-3659	209	8	(	(	PUNCT
ejpam-3659	209	9	2	2	NUM
ejpam-3659	209	10	)	)	PUNCT
ejpam-3659	209	11	(	(	PUNCT
ejpam-3659	209	12	2020	2020	NUM
ejpam-3659	209	13	)	)	PUNCT
ejpam-3659	209	14	,	,	PUNCT
ejpam-3659	209	15	303	303	NUM
ejpam-3659	209	16	-	-	SYM
ejpam-3659	209	17	313	313	NUM
ejpam-3659	209	18	310	310	NUM
ejpam-3659	209	19	note	note	NOUN
ejpam-3659	209	20	that	that	SCONJ
ejpam-3659	209	21	if	if	SCONJ
ejpam-3659	209	22	f	f	PROPN
ejpam-3659	209	23	is	be	AUX
ejpam-3659	209	24	mcshane	mcshane	PROPN
ejpam-3659	209	25	integrable	integrable	ADJ
ejpam-3659	209	26	on	on	ADP
ejpam-3659	209	27	x	x	X
ejpam-3659	209	28	⊆	⊆	NUM
ejpam-3659	209	29	[	[	X
ejpam-3659	209	30	a	a	X
ejpam-3659	209	31	,	,	PUNCT
ejpam-3659	209	32	b	b	NOUN
ejpam-3659	209	33	]	]	X
ejpam-3659	209	34	,	,	PUNCT
ejpam-3659	209	35	then	then	ADV
ejpam-3659	209	36	f	f	PROPN
ejpam-3659	209	37	·	·	PUNCT
ejpam-3659	209	38	χx	χx	PROPN
ejpam-3659	209	39	is	be	AUX
ejpam-3659	209	40	mcshane	mcshane	PROPN
ejpam-3659	209	41	integrable	integrable	ADJ
ejpam-3659	209	42	on	on	ADP
ejpam-3659	209	43	[	[	X
ejpam-3659	209	44	a	a	X
ejpam-3659	209	45	,	,	PUNCT
ejpam-3659	209	46	b	b	NOUN
ejpam-3659	209	47	]	]	PUNCT
ejpam-3659	209	48	.	.	PUNCT
ejpam-3659	210	1	by	by	ADP
ejpam-3659	210	2	theorem	theorem	NOUN
ejpam-3659	210	3	1	1	NUM
ejpam-3659	210	4	,	,	PUNCT
ejpam-3659	210	5	|f	|f	PROPN
ejpam-3659	210	6	·	·	PUNCT
ejpam-3659	210	7	χx	χx	INTJ
ejpam-3659	211	1	|	|	ADV
ejpam-3659	211	2	is	be	AUX
ejpam-3659	211	3	mcshane	mcshane	NOUN
ejpam-3659	211	4	integrable	integrable	ADJ
ejpam-3659	211	5	on	on	ADP
ejpam-3659	211	6	[	[	X
ejpam-3659	211	7	a	a	X
ejpam-3659	211	8	,	,	PUNCT
ejpam-3659	211	9	b	b	NOUN
ejpam-3659	211	10	]	]	X
ejpam-3659	211	11	.	.	PUNCT
ejpam-3659	212	1	again	again	ADV
ejpam-3659	212	2	by	by	ADP
ejpam-3659	212	3	henstock	henstock	PROPN
ejpam-3659	212	4	lemma	lemma	PROPN
ejpam-3659	212	5	,	,	PUNCT
ejpam-3659	212	6	there	there	PRON
ejpam-3659	212	7	exists	exist	VERB
ejpam-3659	212	8	a	a	DET
ejpam-3659	212	9	gauge	gauge	NOUN
ejpam-3659	212	10	δ2	δ2	VERB
ejpam-3659	212	11	on	on	ADP
ejpam-3659	212	12	[	[	X
ejpam-3659	212	13	a	a	DET
ejpam-3659	212	14	,	,	PUNCT
ejpam-3659	212	15	b	b	NOUN
ejpam-3659	212	16	]	]	X
ejpam-3659	212	17	such	such	ADJ
ejpam-3659	212	18	that	that	SCONJ
ejpam-3659	212	19	(	(	PUNCT
ejpam-3659	212	20	p	p	NOUN
ejpam-3659	212	21	)	)	PUNCT
ejpam-3659	212	22	∑∣∣∣|(f	∑∣∣∣|(f	NUM
ejpam-3659	212	23	·	·	PUNCT
ejpam-3659	212	24	χx)(ξ)|(v	χx)(ξ)|(v	NOUN
ejpam-3659	212	25	−	−	PROPN
ejpam-3659	212	26	u)−	u)−	PROPN
ejpam-3659	212	27	∫	∫	PROPN
ejpam-3659	212	28	v	v	NUM
ejpam-3659	212	29	u	u	PROPN
ejpam-3659	212	30	|f	|f	PROPN
ejpam-3659	212	31	·	·	PUNCT
ejpam-3659	212	32	χx	χx	NOUN
ejpam-3659	212	33	|	|	ADV
ejpam-3659	212	34	∣∣∣	∣∣∣	ADJ
ejpam-3659	212	35	<	<	X
ejpam-3659	212	36	ε	ε	PROPN
ejpam-3659	212	37	3	3	NUM
ejpam-3659	212	38	whenever	whenever	SCONJ
ejpam-3659	212	39	p	p	NOUN
ejpam-3659	212	40	=	=	X
ejpam-3659	212	41	{	{	PUNCT
ejpam-3659	212	42	(	(	PUNCT
ejpam-3659	212	43	[	[	X
ejpam-3659	212	44	u	u	NOUN
ejpam-3659	212	45	,	,	PUNCT
ejpam-3659	212	46	v	v	ADP
ejpam-3659	212	47	]	]	X
ejpam-3659	212	48	,	,	PUNCT
ejpam-3659	212	49	ξ	ξ	X
ejpam-3659	212	50	)	)	PUNCT
ejpam-3659	212	51	}	}	PUNCT
ejpam-3659	212	52	is	be	AUX
ejpam-3659	212	53	a	a	DET
ejpam-3659	212	54	mcshane	mcshane	PROPN
ejpam-3659	212	55	δ2	δ2	VERB
ejpam-3659	212	56	-	-	PUNCT
ejpam-3659	212	57	fine	fine	ADJ
ejpam-3659	212	58	partial	partial	ADJ
ejpam-3659	212	59	division	division	NOUN
ejpam-3659	212	60	of	of	ADP
ejpam-3659	212	61	[	[	X
ejpam-3659	212	62	a	a	X
ejpam-3659	212	63	,	,	PUNCT
ejpam-3659	212	64	b	b	NOUN
ejpam-3659	212	65	]	]	PUNCT
ejpam-3659	212	66	.	.	PUNCT
ejpam-3659	213	1	let	let	VERB
ejpam-3659	213	2	δ3	δ3	PROPN
ejpam-3659	213	3	=	=	SYM
ejpam-3659	213	4	min{δ1	min{δ1	NOUN
ejpam-3659	213	5	,	,	PUNCT
ejpam-3659	213	6	δ2	δ2	ADJ
ejpam-3659	213	7	}	}	PUNCT
ejpam-3659	213	8	and	and	CCONJ
ejpam-3659	213	9	p	p	NOUN
ejpam-3659	213	10	=	=	PUNCT
ejpam-3659	213	11	{	{	PUNCT
ejpam-3659	213	12	(	(	PUNCT
ejpam-3659	213	13	[	[	X
ejpam-3659	213	14	u	u	NOUN
ejpam-3659	213	15	,	,	PUNCT
ejpam-3659	213	16	v	v	ADP
ejpam-3659	213	17	]	]	X
ejpam-3659	213	18	,	,	PUNCT
ejpam-3659	213	19	ξ	ξ	X
ejpam-3659	213	20	)	)	PUNCT
ejpam-3659	213	21	}	}	PUNCT
ejpam-3659	213	22	be	be	AUX
ejpam-3659	213	23	a	a	DET
ejpam-3659	213	24	mcshane	mcshane	PROPN
ejpam-3659	213	25	δ3	δ3	PROPN
ejpam-3659	213	26	-	-	PUNCT
ejpam-3659	213	27	fine	fine	ADJ
ejpam-3659	213	28	partial	partial	ADJ
ejpam-3659	213	29	division	division	NOUN
ejpam-3659	213	30	of	of	ADP
ejpam-3659	213	31	[	[	X
ejpam-3659	213	32	a	a	X
ejpam-3659	213	33	,	,	PUNCT
ejpam-3659	213	34	b	b	NOUN
ejpam-3659	213	35	]	]	X
ejpam-3659	213	36	such	such	ADJ
ejpam-3659	213	37	that	that	SCONJ
ejpam-3659	213	38	ξ	ξ	PROPN
ejpam-3659	213	39	∈	∈	PROPN
ejpam-3659	213	40	x.	x.	NOUN
ejpam-3659	213	41	then	then	ADV
ejpam-3659	213	42	(	(	PUNCT
ejpam-3659	213	43	p	p	NOUN
ejpam-3659	213	44	)	)	PUNCT
ejpam-3659	213	45	∑	∑	PROPN
ejpam-3659	213	46	|f	|f	PROPN
ejpam-3659	213	47	(	(	PUNCT
ejpam-3659	213	48	u	u	NOUN
ejpam-3659	213	49	,	,	PUNCT
ejpam-3659	213	50	v)|	v)|	ADJ
ejpam-3659	213	51	≤	≤	X
ejpam-3659	213	52	(	(	PUNCT
ejpam-3659	213	53	p	p	NOUN
ejpam-3659	213	54	)	)	PUNCT
ejpam-3659	213	55	∑	∑	PROPN
ejpam-3659	213	56	|f	|f	PROPN
ejpam-3659	213	57	(	(	PUNCT
ejpam-3659	213	58	u	u	PROPN
ejpam-3659	213	59	,	,	PUNCT
ejpam-3659	213	60	v)−	v)−	PROPN
ejpam-3659	213	61	f(ξ)(v	f(ξ)(v	PUNCT
ejpam-3659	213	62	−	−	PROPN
ejpam-3659	213	63	u)|+	u)|+	ADJ
ejpam-3659	213	64	(	(	PUNCT
ejpam-3659	213	65	p	p	NOUN
ejpam-3659	213	66	)	)	PUNCT
ejpam-3659	213	67	∑	∑	PROPN
ejpam-3659	213	68	|f(ξ)(v	|f(ξ)(v	NOUN
ejpam-3659	213	69	−	−	PROPN
ejpam-3659	214	1	u)|	u)|	NOUN
ejpam-3659	214	2	<	<	X
ejpam-3659	214	3	ε	ε	PROPN
ejpam-3659	214	4	3	3	NUM
ejpam-3659	214	5	+	+	CCONJ
ejpam-3659	214	6	(	(	PUNCT
ejpam-3659	214	7	p	p	NOUN
ejpam-3659	214	8	)	)	PUNCT
ejpam-3659	214	9	∑	∑	PROPN
ejpam-3659	214	10	|f(ξ)(v	|f(ξ)(v	NOUN
ejpam-3659	214	11	−	−	PROPN
ejpam-3659	215	1	u)|	u)|	NOUN
ejpam-3659	215	2	≤	≤	NOUN
ejpam-3659	215	3	ε	ε	PROPN
ejpam-3659	215	4	3	3	NUM
ejpam-3659	215	5	+	+	CCONJ
ejpam-3659	215	6	(	(	PUNCT
ejpam-3659	215	7	p	p	NOUN
ejpam-3659	215	8	)	)	PUNCT
ejpam-3659	215	9	∑∣∣∣|f(ξ	∑∣∣∣|f(ξ	INTJ
ejpam-3659	215	10	)	)	PUNCT
ejpam-3659	215	11	·	·	PUNCT
ejpam-3659	216	1	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	216	2	−	−	PROPN
ejpam-3659	216	3	u)−	u)−	PROPN
ejpam-3659	216	4	∫	∫	PROPN
ejpam-3659	216	5	v	v	NUM
ejpam-3659	216	6	u	u	PROPN
ejpam-3659	216	7	|f	|f	PROPN
ejpam-3659	216	8	·	·	PUNCT
ejpam-3659	216	9	χx	χx	NOUN
ejpam-3659	216	10	|	|	ADV
ejpam-3659	216	11	∣∣∣	∣∣∣	ADJ
ejpam-3659	216	12	+	+	PROPN
ejpam-3659	216	13	(	(	PUNCT
ejpam-3659	216	14	p	p	NOUN
ejpam-3659	216	15	)	)	PUNCT
ejpam-3659	216	16	∑∫	∑∫	PROPN
ejpam-3659	216	17	v	v	NUM
ejpam-3659	216	18	u	u	NOUN
ejpam-3659	216	19	|f	|f	PROPN
ejpam-3659	216	20	·	·	PUNCT
ejpam-3659	216	21	χx	χx	INTJ
ejpam-3659	217	1	|	|	ADV
ejpam-3659	217	2	<	<	X
ejpam-3659	217	3	2ε	2ε	NOUN
ejpam-3659	217	4	3	3	NUM
ejpam-3659	217	5	+	+	CCONJ
ejpam-3659	217	6	(	(	PUNCT
ejpam-3659	217	7	p	p	NOUN
ejpam-3659	217	8	)	)	PUNCT
ejpam-3659	217	9	∑∫	∑∫	PROPN
ejpam-3659	217	10	v	v	NUM
ejpam-3659	217	11	u	u	NOUN
ejpam-3659	217	12	|f	|f	PROPN
ejpam-3659	217	13	·	·	PUNCT
ejpam-3659	217	14	χx	χx	PROPN
ejpam-3659	218	1	|	|	ADV
ejpam-3659	218	2	≤	≤	NUM
ejpam-3659	218	3	2ε	2ε	NOUN
ejpam-3659	218	4	3	3	NUM
ejpam-3659	218	5	+	+	CCONJ
ejpam-3659	218	6	∫	∫	PROPN
ejpam-3659	218	7	b	b	PROPN
ejpam-3659	218	8	a	a	PRON
ejpam-3659	218	9	|f	|f	PROPN
ejpam-3659	218	10	·	·	PUNCT
ejpam-3659	218	11	χx	χx	NOUN
ejpam-3659	219	1	|	|	ADV
ejpam-3659	219	2	.	.	PUNCT
ejpam-3659	220	1	this	this	PRON
ejpam-3659	220	2	implies	imply	VERB
ejpam-3659	220	3	that	that	SCONJ
ejpam-3659	220	4	vmf	vmf	PROPN
ejpam-3659	220	5	(	(	PUNCT
ejpam-3659	220	6	x	x	NOUN
ejpam-3659	220	7	)	)	PUNCT
ejpam-3659	220	8	≤	≤	NOUN
ejpam-3659	220	9	v	v	NOUN
ejpam-3659	220	10	(	(	PUNCT
ejpam-3659	220	11	f	f	PROPN
ejpam-3659	220	12	,	,	PUNCT
ejpam-3659	220	13	x	x	NOUN
ejpam-3659	220	14	,	,	PUNCT
ejpam-3659	220	15	δ3	δ3	PROPN
ejpam-3659	220	16	)	)	PUNCT
ejpam-3659	220	17	≤	≤	NOUN
ejpam-3659	220	18	2ε	2ε	NOUN
ejpam-3659	220	19	3	3	NUM
ejpam-3659	221	1	+	+	CCONJ
ejpam-3659	221	2	∫	∫	PROPN
ejpam-3659	222	1	b	b	PROPN
ejpam-3659	222	2	a	a	PRON
ejpam-3659	222	3	|f	|f	PROPN
ejpam-3659	222	4	·	·	PUNCT
ejpam-3659	222	5	χx	χx	NOUN
ejpam-3659	222	6	|	|	INTJ
ejpam-3659	222	7	.	.	PUNCT
ejpam-3659	223	1	since	since	SCONJ
ejpam-3659	223	2	ε	ε	PROPN
ejpam-3659	223	3	>	>	X
ejpam-3659	223	4	0	0	NUM
ejpam-3659	223	5	is	be	AUX
ejpam-3659	223	6	arbitrary	arbitrary	ADJ
ejpam-3659	223	7	,	,	PUNCT
ejpam-3659	223	8	we	we	PRON
ejpam-3659	223	9	have	have	VERB
ejpam-3659	223	10	vmf	vmf	NOUN
ejpam-3659	223	11	(	(	PUNCT
ejpam-3659	223	12	x	x	NOUN
ejpam-3659	223	13	)	)	PUNCT
ejpam-3659	223	14	≤	≤	NUM
ejpam-3659	223	15	∫	∫	PROPN
ejpam-3659	224	1	b	b	PROPN
ejpam-3659	224	2	a	a	PRON
ejpam-3659	224	3	|f	|f	PROPN
ejpam-3659	224	4	·	·	PUNCT
ejpam-3659	224	5	χx	χx	NOUN
ejpam-3659	224	6	|	|	INTJ
ejpam-3659	224	7	.	.	PUNCT
ejpam-3659	225	1	(	(	PUNCT
ejpam-3659	225	2	1	1	X
ejpam-3659	225	3	)	)	PUNCT
ejpam-3659	225	4	by	by	ADP
ejpam-3659	225	5	definition	definition	NOUN
ejpam-3659	225	6	of	of	ADP
ejpam-3659	225	7	vmf	vmf	PROPN
ejpam-3659	225	8	(	(	PUNCT
ejpam-3659	225	9	x	x	NOUN
ejpam-3659	225	10	)	)	PUNCT
ejpam-3659	225	11	,	,	PUNCT
ejpam-3659	225	12	there	there	PRON
ejpam-3659	225	13	exists	exist	VERB
ejpam-3659	225	14	δ4(ξ	δ4(ξ	PROPN
ejpam-3659	225	15	)	)	PUNCT
ejpam-3659	225	16	>	>	X
ejpam-3659	225	17	0	0	NUM
ejpam-3659	226	1	such	such	ADJ
ejpam-3659	226	2	that	that	DET
ejpam-3659	226	3	v	v	NOUN
ejpam-3659	226	4	(	(	PUNCT
ejpam-3659	226	5	f	f	PROPN
ejpam-3659	226	6	,	,	PUNCT
ejpam-3659	226	7	x	x	NOUN
ejpam-3659	226	8	,	,	PUNCT
ejpam-3659	226	9	δ4	δ4	NOUN
ejpam-3659	226	10	)	)	PUNCT
ejpam-3659	226	11	≤	≤	NOUN
ejpam-3659	226	12	vmf	vmf	NOUN
ejpam-3659	226	13	(	(	PUNCT
ejpam-3659	226	14	x	x	NOUN
ejpam-3659	226	15	)	)	PUNCT
ejpam-3659	227	1	+	+	CCONJ
ejpam-3659	227	2	ε	ε	PROPN
ejpam-3659	227	3	3	3	NUM
ejpam-3659	227	4	.	.	PUNCT
ejpam-3659	227	5	let	let	VERB
ejpam-3659	227	6	δ0(ξ	δ0(ξ	PRON
ejpam-3659	227	7	)	)	PUNCT
ejpam-3659	227	8	=	=	SYM
ejpam-3659	227	9	min{δ3(ξ	min{δ3(ξ	PROPN
ejpam-3659	227	10	)	)	PUNCT
ejpam-3659	227	11	,	,	PUNCT
ejpam-3659	227	12	δ4(ξ	δ4(ξ	NOUN
ejpam-3659	227	13	)	)	PUNCT
ejpam-3659	227	14	}	}	PUNCT
ejpam-3659	227	15	,	,	PUNCT
ejpam-3659	227	16	for	for	ADP
ejpam-3659	227	17	all	all	DET
ejpam-3659	227	18	ξ	ξ	X
ejpam-3659	227	19	∈	∈	PROPN
ejpam-3659	227	20	[	[	X
ejpam-3659	227	21	a	a	X
ejpam-3659	227	22	,	,	PUNCT
ejpam-3659	227	23	b	b	NOUN
ejpam-3659	227	24	]	]	PUNCT
ejpam-3659	227	25	.	.	PUNCT
ejpam-3659	228	1	let	let	VERB
ejpam-3659	228	2	p	p	NOUN
ejpam-3659	228	3	=	=	PRON
ejpam-3659	228	4	{	{	PUNCT
ejpam-3659	228	5	(	(	PUNCT
ejpam-3659	228	6	[	[	X
ejpam-3659	228	7	u	u	NOUN
ejpam-3659	228	8	,	,	PUNCT
ejpam-3659	228	9	v	v	ADP
ejpam-3659	228	10	]	]	X
ejpam-3659	228	11	,	,	PUNCT
ejpam-3659	228	12	ξ	ξ	X
ejpam-3659	228	13	)	)	PUNCT
ejpam-3659	228	14	}	}	PUNCT
ejpam-3659	228	15	be	be	AUX
ejpam-3659	228	16	any	any	DET
ejpam-3659	228	17	mcshane	mcshane	NOUN
ejpam-3659	228	18	δ0	δ0	NOUN
ejpam-3659	228	19	-	-	PUNCT
ejpam-3659	228	20	fine	fine	ADJ
ejpam-3659	228	21	partial	partial	ADJ
ejpam-3659	228	22	division	division	NOUN
ejpam-3659	228	23	of	of	ADP
ejpam-3659	228	24	[	[	X
ejpam-3659	228	25	a	a	X
ejpam-3659	228	26	,	,	PUNCT
ejpam-3659	228	27	b	b	NOUN
ejpam-3659	228	28	]	]	X
ejpam-3659	228	29	with	with	ADP
ejpam-3659	228	30	ξ	ξ	PROPN
ejpam-3659	228	31	∈	∈	PROPN
ejpam-3659	228	32	x.	x.	NOUN
ejpam-3659	228	33	suppose	suppose	VERB
ejpam-3659	229	1	d	d	X
ejpam-3659	229	2	=	=	SYM
ejpam-3659	229	3	p	p	NOUN
ejpam-3659	229	4	∪	∪	NOUN
ejpam-3659	229	5	p	p	NOUN
ejpam-3659	229	6	′	′	NOUN
ejpam-3659	229	7	is	be	AUX
ejpam-3659	229	8	a	a	DET
ejpam-3659	229	9	mcshane	mcshane	NOUN
ejpam-3659	229	10	δ0	δ0	NOUN
ejpam-3659	229	11	-	-	PUNCT
ejpam-3659	229	12	fine	fine	ADJ
ejpam-3659	229	13	division	division	NOUN
ejpam-3659	229	14	of	of	ADP
ejpam-3659	229	15	[	[	X
ejpam-3659	229	16	a	a	X
ejpam-3659	229	17	,	,	PUNCT
ejpam-3659	229	18	b	b	NOUN
ejpam-3659	229	19	]	]	X
ejpam-3659	229	20	,	,	PUNCT
ejpam-3659	229	21	where	where	SCONJ
ejpam-3659	229	22	p	p	NOUN
ejpam-3659	230	1	′	′	NOUN
ejpam-3659	230	2	=	=	PUNCT
ejpam-3659	230	3	{	{	PUNCT
ejpam-3659	230	4	(	(	PUNCT
ejpam-3659	230	5	[	[	X
ejpam-3659	230	6	u	u	NOUN
ejpam-3659	230	7	,	,	PUNCT
ejpam-3659	230	8	v	v	ADP
ejpam-3659	230	9	]	]	X
ejpam-3659	230	10	,	,	PUNCT
ejpam-3659	230	11	ξ	ξ	X
ejpam-3659	230	12	)	)	PUNCT
ejpam-3659	230	13	}	}	PUNCT
ejpam-3659	230	14	such	such	ADJ
ejpam-3659	230	15	that	that	SCONJ
ejpam-3659	230	16	ξ	ξ	PROPN
ejpam-3659	230	17	/∈	/∈	PUNCT
ejpam-3659	230	18	x.	x.	NOUN
ejpam-3659	231	1	then	then	ADV
ejpam-3659	231	2	(	(	PUNCT
ejpam-3659	231	3	d	d	X
ejpam-3659	231	4	)	)	PUNCT
ejpam-3659	231	5	∑	∑	ADV
ejpam-3659	231	6	|f(ξ	|f(ξ	PROPN
ejpam-3659	231	7	)	)	PUNCT
ejpam-3659	231	8	·	·	PUNCT
ejpam-3659	231	9	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	231	10	−	−	PROPN
ejpam-3659	231	11	u	u	NOUN
ejpam-3659	231	12	)	)	PUNCT
ejpam-3659	231	13	=	=	SYM
ejpam-3659	231	14	(	(	PUNCT
ejpam-3659	231	15	p	p	NOUN
ejpam-3659	231	16	)	)	PUNCT
ejpam-3659	231	17	∑	∑	PROPN
ejpam-3659	231	18	|f(ξ	|f(ξ	PROPN
ejpam-3659	231	19	)	)	PUNCT
ejpam-3659	231	20	·	·	PUNCT
ejpam-3659	231	21	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	231	22	−	−	PROPN
ejpam-3659	231	23	u	u	NOUN
ejpam-3659	231	24	)	)	PUNCT
ejpam-3659	231	25	.	.	PUNCT
ejpam-3659	232	1	note	note	VERB
ejpam-3659	232	2	that	that	SCONJ
ejpam-3659	232	3	d	d	NOUN
ejpam-3659	232	4	is	be	AUX
ejpam-3659	232	5	also	also	ADV
ejpam-3659	232	6	a	a	DET
ejpam-3659	232	7	mcshane	mcshane	NOUN
ejpam-3659	232	8	δ0	δ0	NOUN
ejpam-3659	232	9	-	-	PUNCT
ejpam-3659	232	10	fine	fine	ADJ
ejpam-3659	232	11	partial	partial	ADJ
ejpam-3659	232	12	division	division	NOUN
ejpam-3659	232	13	of	of	ADP
ejpam-3659	232	14	[	[	X
ejpam-3659	232	15	a	a	X
ejpam-3659	232	16	,	,	PUNCT
ejpam-3659	232	17	b	b	NOUN
ejpam-3659	232	18	]	]	X
ejpam-3659	232	19	.	.	PUNCT
ejpam-3659	233	1	thus,∫	thus,∫	PROPN
ejpam-3659	233	2	b	b	PROPN
ejpam-3659	233	3	a	a	PRON
ejpam-3659	233	4	|f	|f	PROPN
ejpam-3659	233	5	·	·	PUNCT
ejpam-3659	233	6	χx	χx	PROPN
ejpam-3659	234	1	|	|	ADV
ejpam-3659	234	2	≤	≤	X
ejpam-3659	234	3	(	(	PUNCT
ejpam-3659	234	4	d	d	NOUN
ejpam-3659	234	5	)	)	PUNCT
ejpam-3659	234	6	∑∣∣∣	∑∣∣∣	PROPN
ejpam-3659	235	1	∫	∫	PROPN
ejpam-3659	235	2	v	v	NUM
ejpam-3659	235	3	u	u	PROPN
ejpam-3659	235	4	|f	|f	PROPN
ejpam-3659	235	5	·	·	PUNCT
ejpam-3659	235	6	χx	χx	INTJ
ejpam-3659	236	1	|	|	ADV
ejpam-3659	236	2	−	−	PROPN
ejpam-3659	236	3	|f(ξ	|f(ξ	PROPN
ejpam-3659	236	4	)	)	PUNCT
ejpam-3659	236	5	·	·	PUNCT
ejpam-3659	237	1	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	237	2	−	−	PROPN
ejpam-3659	237	3	u	u	NOUN
ejpam-3659	237	4	)	)	PUNCT
ejpam-3659	237	5	∣∣∣+	∣∣∣+	PROPN
ejpam-3659	237	6	(	(	PUNCT
ejpam-3659	237	7	d	d	NOUN
ejpam-3659	237	8	)	)	PUNCT
ejpam-3659	237	9	∑	∑	ADV
ejpam-3659	237	10	|f(ξ	|f(ξ	PROPN
ejpam-3659	237	11	)	)	PUNCT
ejpam-3659	237	12	·	·	PUNCT
ejpam-3659	237	13	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	237	14	−	−	PROPN
ejpam-3659	237	15	u	u	NOUN
ejpam-3659	237	16	)	)	PUNCT
ejpam-3659	237	17	f.	f.	PROPN
ejpam-3659	237	18	sumalpong	sumalpong	PROPN
ejpam-3659	237	19	jr	jr	PROPN
ejpam-3659	237	20	.	.	PROPN
ejpam-3659	237	21	,	,	PUNCT
ejpam-3659	237	22	j.	j.	PROPN
ejpam-3659	237	23	benitez	benitez	PROPN
ejpam-3659	237	24	/	/	PUNCT
ejpam-3659	237	25	eur	eur	PROPN
ejpam-3659	237	26	.	.	PUNCT
ejpam-3659	238	1	j.	j.	PROPN
ejpam-3659	238	2	pure	pure	PROPN
ejpam-3659	238	3	appl	appl	PROPN
ejpam-3659	238	4	.	.	PROPN
ejpam-3659	238	5	math	math	PROPN
ejpam-3659	238	6	,	,	PUNCT
ejpam-3659	238	7	13	13	NUM
ejpam-3659	238	8	(	(	PUNCT
ejpam-3659	238	9	2	2	NUM
ejpam-3659	238	10	)	)	PUNCT
ejpam-3659	238	11	(	(	PUNCT
ejpam-3659	238	12	2020	2020	NUM
ejpam-3659	238	13	)	)	PUNCT
ejpam-3659	238	14	,	,	PUNCT
ejpam-3659	238	15	303	303	NUM
ejpam-3659	238	16	-	-	SYM
ejpam-3659	238	17	313	313	NUM
ejpam-3659	238	18	311	311	NUM
ejpam-3659	238	19	≤	≤	NOUN
ejpam-3659	238	20	(	(	PUNCT
ejpam-3659	238	21	d	d	NOUN
ejpam-3659	238	22	)	)	PUNCT
ejpam-3659	238	23	∑∣∣∣	∑∣∣∣	PROPN
ejpam-3659	239	1	∫	∫	PROPN
ejpam-3659	239	2	v	v	NUM
ejpam-3659	239	3	u	u	PROPN
ejpam-3659	239	4	|f	|f	PROPN
ejpam-3659	239	5	·	·	PUNCT
ejpam-3659	239	6	χx	χx	INTJ
ejpam-3659	240	1	|	|	ADV
ejpam-3659	240	2	−	−	PROPN
ejpam-3659	240	3	|f(ξ	|f(ξ	PROPN
ejpam-3659	240	4	)	)	PUNCT
ejpam-3659	240	5	·	·	PUNCT
ejpam-3659	241	1	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	241	2	−	−	PROPN
ejpam-3659	241	3	u	u	NOUN
ejpam-3659	241	4	)	)	PUNCT
ejpam-3659	241	5	∣∣∣+	∣∣∣+	PROPN
ejpam-3659	241	6	(	(	PUNCT
ejpam-3659	241	7	p	p	NOUN
ejpam-3659	241	8	)	)	PUNCT
ejpam-3659	241	9	∑	∑	PROPN
ejpam-3659	241	10	|f(ξ	|f(ξ	PROPN
ejpam-3659	241	11	)	)	PUNCT
ejpam-3659	241	12	·	·	PUNCT
ejpam-3659	241	13	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	241	14	−	−	PROPN
ejpam-3659	241	15	u	u	NOUN
ejpam-3659	241	16	)	)	PUNCT
ejpam-3659	241	17	<	<	X
ejpam-3659	241	18	ε	ε	PROPN
ejpam-3659	241	19	3	3	NUM
ejpam-3659	241	20	+	+	CCONJ
ejpam-3659	241	21	(	(	PUNCT
ejpam-3659	241	22	p	p	NOUN
ejpam-3659	241	23	)	)	PUNCT
ejpam-3659	241	24	∑	∑	PROPN
ejpam-3659	241	25	|f(ξ	|f(ξ	PROPN
ejpam-3659	241	26	)	)	PUNCT
ejpam-3659	241	27	·	·	PUNCT
ejpam-3659	241	28	χx(ξ)|(v	χx(ξ)|(v	PROPN
ejpam-3659	241	29	−	−	PROPN
ejpam-3659	241	30	u	u	NOUN
ejpam-3659	241	31	)	)	PUNCT
ejpam-3659	241	32	=	=	SYM
ejpam-3659	241	33	ε	ε	PROPN
ejpam-3659	241	34	3	3	NUM
ejpam-3659	241	35	+	+	CCONJ
ejpam-3659	241	36	(	(	PUNCT
ejpam-3659	241	37	p	p	NOUN
ejpam-3659	241	38	)	)	PUNCT
ejpam-3659	241	39	∑	∑	PUNCT
ejpam-3659	241	40	|f(ξ)|(v	|f(ξ)|(v	PROPN
ejpam-3659	241	41	−	−	PROPN
ejpam-3659	241	42	u	u	NOUN
ejpam-3659	241	43	)	)	PUNCT
ejpam-3659	241	44	≤	≤	PUNCT
ejpam-3659	241	45	ε	ε	PROPN
ejpam-3659	241	46	3	3	NUM
ejpam-3659	241	47	+	+	CCONJ
ejpam-3659	241	48	(	(	PUNCT
ejpam-3659	241	49	p	p	NOUN
ejpam-3659	241	50	)	)	PUNCT
ejpam-3659	241	51	∑∣∣f(ξ)(v	∑∣∣f(ξ)(v	PUNCT
ejpam-3659	241	52	−	−	PROPN
ejpam-3659	241	53	u)−	u)−	PROPN
ejpam-3659	241	54	f	f	PROPN
ejpam-3659	241	55	(	(	PUNCT
ejpam-3659	241	56	u	u	NOUN
ejpam-3659	241	57	,	,	PUNCT
ejpam-3659	241	58	v	v	NOUN
ejpam-3659	241	59	)	)	PUNCT
ejpam-3659	241	60	∣∣+	∣∣+	NOUN
ejpam-3659	241	61	(	(	PUNCT
ejpam-3659	241	62	p	p	NOUN
ejpam-3659	241	63	)	)	PUNCT
ejpam-3659	241	64	∑∣∣f	∑∣∣f	NOUN
ejpam-3659	241	65	(	(	PUNCT
ejpam-3659	241	66	u	u	NOUN
ejpam-3659	241	67	,	,	PUNCT
ejpam-3659	241	68	v	v	NOUN
ejpam-3659	241	69	)	)	PUNCT
ejpam-3659	241	70	∣∣	∣∣	X
ejpam-3659	241	71	<	<	X
ejpam-3659	241	72	2ε	2ε	NOUN
ejpam-3659	241	73	3	3	NUM
ejpam-3659	241	74	+	+	CCONJ
ejpam-3659	241	75	(	(	PUNCT
ejpam-3659	241	76	p	p	NOUN
ejpam-3659	241	77	)	)	PUNCT
ejpam-3659	241	78	∑∣∣f	∑∣∣f	NOUN
ejpam-3659	241	79	(	(	PUNCT
ejpam-3659	241	80	u	u	NOUN
ejpam-3659	241	81	,	,	PUNCT
ejpam-3659	241	82	v	v	NOUN
ejpam-3659	241	83	)	)	PUNCT
ejpam-3659	241	84	∣∣	∣∣	X
ejpam-3659	241	85	≤	≤	NUM
ejpam-3659	241	86	2ε	2ε	NOUN
ejpam-3659	241	87	3	3	NUM
ejpam-3659	241	88	+	+	NUM
ejpam-3659	241	89	v	v	PROPN
ejpam-3659	241	90	(	(	PUNCT
ejpam-3659	241	91	f	f	PROPN
ejpam-3659	241	92	,	,	PUNCT
ejpam-3659	241	93	x	x	NOUN
ejpam-3659	241	94	,	,	PUNCT
ejpam-3659	241	95	δ0	δ0	NOUN
ejpam-3659	241	96	)	)	PUNCT
ejpam-3659	241	97	.	.	PUNCT
ejpam-3659	242	1	hence	hence	ADV
ejpam-3659	242	2	,	,	PUNCT
ejpam-3659	242	3	∫	∫	PROPN
ejpam-3659	242	4	b	b	PROPN
ejpam-3659	242	5	a	a	PRON
ejpam-3659	242	6	|f	|f	PROPN
ejpam-3659	242	7	·	·	PUNCT
ejpam-3659	242	8	χx	χx	PROPN
ejpam-3659	242	9	|	|	ADV
ejpam-3659	242	10	≤	≤	NUM
ejpam-3659	242	11	ε+	ε+	X
ejpam-3659	242	12	vmf	vmf	PROPN
ejpam-3659	242	13	(	(	PUNCT
ejpam-3659	242	14	x	x	NOUN
ejpam-3659	242	15	)	)	PUNCT
ejpam-3659	242	16	.	.	PUNCT
ejpam-3659	243	1	again	again	ADV
ejpam-3659	243	2	,	,	PUNCT
ejpam-3659	243	3	since	since	SCONJ
ejpam-3659	243	4	ε	ε	PROPN
ejpam-3659	243	5	>	>	X
ejpam-3659	243	6	0	0	NUM
ejpam-3659	243	7	is	be	AUX
ejpam-3659	243	8	arbitrary	arbitrary	ADJ
ejpam-3659	243	9	,	,	PUNCT
ejpam-3659	243	10	we	we	PRON
ejpam-3659	243	11	have∫	have∫	VERB
ejpam-3659	243	12	b	b	ADP
ejpam-3659	243	13	a	a	PRON
ejpam-3659	243	14	|f	|f	PROPN
ejpam-3659	243	15	·	·	PUNCT
ejpam-3659	243	16	χx	χx	PROPN
ejpam-3659	244	1	|	|	ADV
ejpam-3659	244	2	≤	≤	PROPN
ejpam-3659	244	3	vmf	vmf	NOUN
ejpam-3659	244	4	(	(	PUNCT
ejpam-3659	244	5	x	x	NOUN
ejpam-3659	244	6	)	)	PUNCT
ejpam-3659	244	7	.	.	PUNCT
ejpam-3659	245	1	(	(	PUNCT
ejpam-3659	245	2	2	2	X
ejpam-3659	245	3	)	)	PUNCT
ejpam-3659	245	4	combining	combine	VERB
ejpam-3659	245	5	(	(	PUNCT
ejpam-3659	245	6	1	1	NUM
ejpam-3659	245	7	)	)	PUNCT
ejpam-3659	245	8	and	and	CCONJ
ejpam-3659	245	9	(	(	PUNCT
ejpam-3659	245	10	2	2	NUM
ejpam-3659	245	11	)	)	PUNCT
ejpam-3659	245	12	,	,	PUNCT
ejpam-3659	245	13	we	we	PRON
ejpam-3659	245	14	have	have	VERB
ejpam-3659	245	15	∫	∫	PROPN
ejpam-3659	245	16	b	b	PROPN
ejpam-3659	245	17	a	a	PRON
ejpam-3659	245	18	|f	|f	PROPN
ejpam-3659	245	19	·	·	PUNCT
ejpam-3659	245	20	χx	χx	PROPN
ejpam-3659	246	1	|	|	ADV
ejpam-3659	246	2	=	=	SYM
ejpam-3659	246	3	vmf	vmf	PROPN
ejpam-3659	246	4	(	(	PUNCT
ejpam-3659	246	5	x	x	NOUN
ejpam-3659	246	6	)	)	PUNCT
ejpam-3659	246	7	.	.	PUNCT
ejpam-3659	247	1	theorem	theorem	ADJ
ejpam-3659	247	2	8	8	NUM
ejpam-3659	247	3	.	.	PUNCT
ejpam-3659	248	1	let	let	VERB
ejpam-3659	248	2	f	f	NOUN
ejpam-3659	248	3	:	:	PUNCT
ejpam-3659	249	1	[	[	X
ejpam-3659	249	2	a	a	X
ejpam-3659	249	3	,	,	PUNCT
ejpam-3659	249	4	b	b	NOUN
ejpam-3659	249	5	]	]	X
ejpam-3659	249	6	→	→	PUNCT
ejpam-3659	249	7	r	r	AUX
ejpam-3659	249	8	be	be	PROPN
ejpam-3659	249	9	mcshane	mcshane	NOUN
ejpam-3659	249	10	integrable	integrable	ADJ
ejpam-3659	249	11	on	on	ADP
ejpam-3659	249	12	[	[	X
ejpam-3659	249	13	a	a	DET
ejpam-3659	249	14	,	,	PUNCT
ejpam-3659	249	15	b	b	NOUN
ejpam-3659	249	16	]	]	X
ejpam-3659	249	17	with	with	ADP
ejpam-3659	249	18	primitive	primitive	ADJ
ejpam-3659	249	19	f	f	PROPN
ejpam-3659	249	20	and	and	CCONJ
ejpam-3659	249	21	x	x	SYM
ejpam-3659	249	22	⊆	⊆	NUM
ejpam-3659	249	23	[	[	X
ejpam-3659	249	24	a	a	X
ejpam-3659	249	25	,	,	PUNCT
ejpam-3659	249	26	b	b	NOUN
ejpam-3659	249	27	]	]	X
ejpam-3659	249	28	.	.	PUNCT
ejpam-3659	250	1	then	then	ADV
ejpam-3659	250	2	f	f	PROPN
ejpam-3659	250	3	is	be	AUX
ejpam-3659	250	4	mcshane	mcshane	PROPN
ejpam-3659	250	5	integrable	integrable	ADJ
ejpam-3659	250	6	on	on	ADP
ejpam-3659	250	7	x	x	SYM
ejpam-3659	250	8	if	if	SCONJ
ejpam-3659	251	1	and	and	CCONJ
ejpam-3659	251	2	only	only	ADV
ejpam-3659	251	3	if	if	SCONJ
ejpam-3659	251	4	vmf	vmf	PROPN
ejpam-3659	251	5	(	(	PUNCT
ejpam-3659	251	6	x	x	X
ejpam-3659	251	7	)	)	PUNCT
ejpam-3659	251	8	<	<	X
ejpam-3659	251	9	∞.	∞.	PROPN
ejpam-3659	251	10	in	in	ADP
ejpam-3659	251	11	this	this	DET
ejpam-3659	251	12	case,∫	case,∫	NOUN
ejpam-3659	251	13	b	b	PROPN
ejpam-3659	251	14	a	a	PRON
ejpam-3659	251	15	|f	|f	PROPN
ejpam-3659	251	16	·	·	PUNCT
ejpam-3659	251	17	χx	χx	PROPN
ejpam-3659	252	1	|	|	ADV
ejpam-3659	252	2	=	=	SYM
ejpam-3659	252	3	vmf	vmf	PROPN
ejpam-3659	252	4	(	(	PUNCT
ejpam-3659	252	5	x	x	NOUN
ejpam-3659	252	6	)	)	PUNCT
ejpam-3659	252	7	.	.	PUNCT
ejpam-3659	253	1	proof	proof	NOUN
ejpam-3659	253	2	.	.	PUNCT
ejpam-3659	254	1	necessity	necessity	NOUN
ejpam-3659	254	2	follows	follow	VERB
ejpam-3659	254	3	from	from	ADP
ejpam-3659	254	4	lemma	lemma	PROPN
ejpam-3659	254	5	4	4	NUM
ejpam-3659	254	6	.	.	PUNCT
ejpam-3659	254	7	conversely	conversely	ADV
ejpam-3659	254	8	,	,	PUNCT
ejpam-3659	254	9	assume	assume	VERB
ejpam-3659	254	10	that	that	SCONJ
ejpam-3659	254	11	vmf	vmf	PROPN
ejpam-3659	254	12	(	(	PUNCT
ejpam-3659	254	13	x	x	X
ejpam-3659	254	14	)	)	PUNCT
ejpam-3659	255	1	<	<	X
ejpam-3659	255	2	∞.	∞.	PROPN
ejpam-3659	255	3	for	for	ADP
ejpam-3659	255	4	each	each	DET
ejpam-3659	255	5	n	n	PRON
ejpam-3659	255	6	∈	∈	PROPN
ejpam-3659	255	7	n	n	CCONJ
ejpam-3659	255	8	,	,	PUNCT
ejpam-3659	255	9	we	we	PRON
ejpam-3659	255	10	let	let	VERB
ejpam-3659	255	11	xn	xn	PUNCT
ejpam-3659	256	1	=	=	PUNCT
ejpam-3659	256	2	{	{	PUNCT
ejpam-3659	256	3	x	x	PUNCT
ejpam-3659	256	4	∈	∈	NOUN
ejpam-3659	256	5	x	x	X
ejpam-3659	256	6	:	:	PUNCT
ejpam-3659	256	7	|f(x)|	|f(x)|	NOUN
ejpam-3659	256	8	≤	≤	NOUN
ejpam-3659	256	9	n	n	CCONJ
ejpam-3659	256	10	}	}	PUNCT
ejpam-3659	256	11	.	.	PUNCT
ejpam-3659	257	1	then	then	ADV
ejpam-3659	257	2	each	each	DET
ejpam-3659	257	3	xn	xn	PROPN
ejpam-3659	257	4	is	be	AUX
ejpam-3659	257	5	an	an	DET
ejpam-3659	257	6	integrable	integrable	ADJ
ejpam-3659	257	7	set	set	NOUN
ejpam-3659	257	8	,	,	PUNCT
ejpam-3659	257	9	xn	xn	PROPN
ejpam-3659	257	10	⊆	⊆	NUM
ejpam-3659	257	11	xn+1	xn+1	PROPN
ejpam-3659	257	12	for	for	ADP
ejpam-3659	257	13	all	all	DET
ejpam-3659	257	14	n	n	NOUN
ejpam-3659	257	15	and	and	CCONJ
ejpam-3659	257	16	x	x	X
ejpam-3659	257	17	=	=	SYM
ejpam-3659	257	18	∞⋃	∞⋃	PROPN
ejpam-3659	257	19	n=1	n=1	PROPN
ejpam-3659	257	20	xn	xn	PROPN
ejpam-3659	257	21	.	.	PUNCT
ejpam-3659	258	1	by	by	ADP
ejpam-3659	258	2	theorem	theorem	NOUN
ejpam-3659	258	3	7	7	NUM
ejpam-3659	258	4	,	,	PUNCT
ejpam-3659	258	5	f	f	PROPN
ejpam-3659	258	6	is	be	AUX
ejpam-3659	258	7	mcshane	mcshane	PROPN
ejpam-3659	258	8	integrable	integrable	ADJ
ejpam-3659	258	9	on	on	ADP
ejpam-3659	258	10	xn	xn	PROPN
ejpam-3659	258	11	for	for	ADP
ejpam-3659	258	12	each	each	DET
ejpam-3659	258	13	n.	n.	NOUN
ejpam-3659	258	14	by	by	ADP
ejpam-3659	258	15	lemma	lemma	PROPN
ejpam-3659	258	16	4	4	NUM
ejpam-3659	258	17	,	,	PUNCT
ejpam-3659	258	18	vmf	vmf	PROPN
ejpam-3659	258	19	(	(	PUNCT
ejpam-3659	258	20	xn	xn	PROPN
ejpam-3659	258	21	)	)	PUNCT
ejpam-3659	259	1	<	<	X
ejpam-3659	259	2	∞	∞	NUM
ejpam-3659	259	3	and	and	CCONJ
ejpam-3659	259	4	∫	∫	PROPN
ejpam-3659	259	5	b	b	PROPN
ejpam-3659	259	6	a	a	PRON
ejpam-3659	259	7	|f	|f	PROPN
ejpam-3659	259	8	·	·	PUNCT
ejpam-3659	259	9	χxn	χxn	ADJ
ejpam-3659	260	1	|	|	ADV
ejpam-3659	260	2	=	=	SYM
ejpam-3659	260	3	vmf	vmf	PROPN
ejpam-3659	260	4	(	(	PUNCT
ejpam-3659	260	5	xn	xn	PROPN
ejpam-3659	260	6	)	)	PUNCT
ejpam-3659	260	7	.	.	PUNCT
ejpam-3659	261	1	references	reference	NOUN
ejpam-3659	261	2	312	312	NUM
ejpam-3659	261	3	consider	consider	VERB
ejpam-3659	261	4	the	the	DET
ejpam-3659	261	5	sequence	sequence	NOUN
ejpam-3659	261	6	{	{	PUNCT
ejpam-3659	261	7	fn}∞n=1	fn}∞n=1	X
ejpam-3659	261	8	,	,	PUNCT
ejpam-3659	261	9	defined	define	VERB
ejpam-3659	261	10	by	by	ADP
ejpam-3659	261	11	fn	fn	PROPN
ejpam-3659	261	12	=	=	SYM
ejpam-3659	261	13	|f	|f	PROPN
ejpam-3659	261	14	·	·	PUNCT
ejpam-3659	261	15	χxn	χxn	PROPN
ejpam-3659	261	16	|	|	ADV
ejpam-3659	261	17	for	for	ADP
ejpam-3659	261	18	all	all	DET
ejpam-3659	261	19	n	n	PRON
ejpam-3659	261	20	∈	∈	PROPN
ejpam-3659	261	21	n.	n.	NOUN
ejpam-3659	261	22	since	since	SCONJ
ejpam-3659	261	23	xn	xn	PROPN
ejpam-3659	261	24	⊆	⊆	NUM
ejpam-3659	261	25	xn+1	xn+1	PROPN
ejpam-3659	261	26	for	for	ADP
ejpam-3659	261	27	each	each	DET
ejpam-3659	261	28	n	n	CCONJ
ejpam-3659	261	29	,	,	PUNCT
ejpam-3659	261	30	{	{	PUNCT
ejpam-3659	261	31	fn}∞n=1	fn}∞n=1	X
ejpam-3659	261	32	is	be	AUX
ejpam-3659	261	33	an	an	DET
ejpam-3659	261	34	increasing	increase	VERB
ejpam-3659	261	35	sequence	sequence	NOUN
ejpam-3659	261	36	of	of	ADP
ejpam-3659	261	37	mcshane	mcshane	PROPN
ejpam-3659	261	38	integrable	integrable	ADJ
ejpam-3659	261	39	functions	function	NOUN
ejpam-3659	261	40	on	on	ADP
ejpam-3659	261	41	[	[	X
ejpam-3659	261	42	a	a	X
ejpam-3659	261	43	,	,	PUNCT
ejpam-3659	261	44	b	b	NOUN
ejpam-3659	261	45	]	]	PUNCT
ejpam-3659	261	46	and	and	CCONJ
ejpam-3659	261	47	fn	fn	ADJ
ejpam-3659	261	48	→	→	SYM
ejpam-3659	261	49	|f	|f	PROPN
ejpam-3659	261	50	·	·	PUNCT
ejpam-3659	261	51	χx	χx	INTJ
ejpam-3659	261	52	|	|	ADV
ejpam-3659	261	53	pointwisely	pointwisely	ADV
ejpam-3659	261	54	on	on	ADP
ejpam-3659	261	55	[	[	X
ejpam-3659	261	56	a	a	X
ejpam-3659	261	57	,	,	PUNCT
ejpam-3659	261	58	b	b	NOUN
ejpam-3659	261	59	]	]	PUNCT
ejpam-3659	261	60	.	.	PUNCT
ejpam-3659	262	1	note	note	VERB
ejpam-3659	262	2	that	that	DET
ejpam-3659	262	3	sup	sup	NOUN
ejpam-3659	262	4	{	{	PUNCT
ejpam-3659	262	5	∫	∫	PROPN
ejpam-3659	262	6	b	b	PROPN
ejpam-3659	262	7	a	a	PRON
ejpam-3659	262	8	|f	|f	PROPN
ejpam-3659	262	9	·	·	PUNCT
ejpam-3659	262	10	χxn	χxn	ADJ
ejpam-3659	262	11	|	|	ADV
ejpam-3659	262	12	:	:	PUNCT
ejpam-3659	262	13	n	n	CCONJ
ejpam-3659	262	14	∈	∈	NOUN
ejpam-3659	262	15	n	n	NOUN
ejpam-3659	262	16	}	}	PUNCT
ejpam-3659	262	17	=	=	PUNCT
ejpam-3659	262	18	sup	sup	NOUN
ejpam-3659	262	19	{	{	PUNCT
ejpam-3659	262	20	vmf	vmf	PROPN
ejpam-3659	262	21	(	(	PUNCT
ejpam-3659	262	22	xn	xn	PROPN
ejpam-3659	262	23	)	)	PUNCT
ejpam-3659	262	24	:	:	PUNCT
ejpam-3659	262	25	n	n	X
ejpam-3659	262	26	∈	∈	PROPN
ejpam-3659	262	27	n	n	CCONJ
ejpam-3659	262	28	}	}	PUNCT
ejpam-3659	262	29	≤	≤	PROPN
ejpam-3659	262	30	vmf	vmf	NOUN
ejpam-3659	262	31	(	(	PUNCT
ejpam-3659	262	32	x	x	X
ejpam-3659	262	33	)	)	PUNCT
ejpam-3659	262	34	<	<	X
ejpam-3659	262	35	∞.	∞.	PROPN
ejpam-3659	262	36	by	by	ADP
ejpam-3659	262	37	theorem	theorem	NOUN
ejpam-3659	262	38	6	6	NUM
ejpam-3659	262	39	,	,	PUNCT
ejpam-3659	262	40	|f	|f	PROPN
ejpam-3659	262	41	·	·	PUNCT
ejpam-3659	263	1	χx	χx	INTJ
ejpam-3659	263	2	|	|	ADV
ejpam-3659	263	3	is	be	AUX
ejpam-3659	263	4	mcshane	mcshane	NOUN
ejpam-3659	263	5	integrable	integrable	ADJ
ejpam-3659	263	6	on	on	ADP
ejpam-3659	263	7	[	[	X
ejpam-3659	263	8	a	a	PRON
ejpam-3659	263	9	,	,	PUNCT
ejpam-3659	263	10	b	b	NOUN
ejpam-3659	263	11	]	]	X
ejpam-3659	263	12	and∫	and∫	PROPN
ejpam-3659	263	13	b	b	PROPN
ejpam-3659	263	14	a	a	PRON
ejpam-3659	263	15	|f	|f	PROPN
ejpam-3659	263	16	·	·	PUNCT
ejpam-3659	263	17	χx	χx	PROPN
ejpam-3659	264	1	|	|	ADV
ejpam-3659	264	2	=	=	SYM
ejpam-3659	264	3	lim	lim	PROPN
ejpam-3659	264	4	n→∞	n→∞	NUM
ejpam-3659	265	1	∫	∫	PROPN
ejpam-3659	265	2	b	b	PROPN
ejpam-3659	265	3	a	a	PRON
ejpam-3659	265	4	|f	|f	PROPN
ejpam-3659	265	5	·	·	PUNCT
ejpam-3659	265	6	χxn	χxn	ADJ
ejpam-3659	266	1	|	|	ADV
ejpam-3659	266	2	=	=	SYM
ejpam-3659	266	3	lim	lim	PROPN
ejpam-3659	266	4	n→∞	n→∞	NUM
ejpam-3659	266	5	vmf	vmf	PROPN
ejpam-3659	266	6	(	(	PUNCT
ejpam-3659	266	7	xn	xn	PROPN
ejpam-3659	266	8	)	)	PUNCT
ejpam-3659	267	1	=	=	SYM
ejpam-3659	267	2	vmf	vmf	PROPN
ejpam-3659	267	3	(	(	PUNCT
ejpam-3659	267	4	x	x	NOUN
ejpam-3659	267	5	)	)	PUNCT
ejpam-3659	267	6	.	.	PUNCT
ejpam-3659	268	1	4	4	X
ejpam-3659	268	2	.	.	X
ejpam-3659	268	3	conclusion	conclusion	NOUN
ejpam-3659	268	4	let	let	VERB
ejpam-3659	268	5	f	f	NOUN
ejpam-3659	268	6	:	:	PUNCT
ejpam-3659	269	1	[	[	X
ejpam-3659	269	2	a	a	X
ejpam-3659	269	3	,	,	PUNCT
ejpam-3659	269	4	b	b	NOUN
ejpam-3659	269	5	]	]	X
ejpam-3659	269	6	→	→	PUNCT
ejpam-3659	269	7	r	r	NOUN
ejpam-3659	269	8	to	to	PART
ejpam-3659	269	9	be	be	AUX
ejpam-3659	269	10	mcshane	mcshane	NOUN
ejpam-3659	269	11	(	(	PUNCT
ejpam-3659	269	12	lebesgue	lebesgue	PROPN
ejpam-3659	269	13	)	)	PUNCT
ejpam-3659	269	14	integrable	integrable	ADJ
ejpam-3659	269	15	on	on	ADP
ejpam-3659	269	16	[	[	X
ejpam-3659	269	17	a	a	X
ejpam-3659	269	18	,	,	PUNCT
ejpam-3659	269	19	b	b	NOUN
ejpam-3659	269	20	]	]	X
ejpam-3659	269	21	with	with	ADP
ejpam-3659	269	22	primitive	primitive	ADJ
ejpam-3659	269	23	f	f	NOUN
ejpam-3659	269	24	.	.	PUNCT
ejpam-3659	270	1	it	it	PRON
ejpam-3659	270	2	is	be	AUX
ejpam-3659	270	3	shown	show	VERB
ejpam-3659	270	4	in	in	ADP
ejpam-3659	270	5	this	this	DET
ejpam-3659	270	6	paper	paper	NOUN
ejpam-3659	270	7	that	that	PRON
ejpam-3659	270	8	for	for	SCONJ
ejpam-3659	270	9	f	f	PROPN
ejpam-3659	270	10	to	to	PART
ejpam-3659	270	11	be	be	AUX
ejpam-3659	270	12	mcshane	mcshane	PROPN
ejpam-3659	270	13	integrable	integrable	ADJ
ejpam-3659	270	14	on	on	ADP
ejpam-3659	270	15	x	x	X
ejpam-3659	270	16	⊆	⊆	NUM
ejpam-3659	270	17	[	[	X
ejpam-3659	270	18	a	a	X
ejpam-3659	270	19	,	,	PUNCT
ejpam-3659	270	20	b	b	NOUN
ejpam-3659	270	21	]	]	X
ejpam-3659	270	22	,	,	PUNCT
ejpam-3659	270	23	a	a	DET
ejpam-3659	270	24	necessary	necessary	ADJ
ejpam-3659	270	25	and	and	CCONJ
ejpam-3659	270	26	sufficient	sufficient	ADJ
ejpam-3659	270	27	condition	condition	NOUN
ejpam-3659	270	28	is	be	AUX
ejpam-3659	270	29	that	that	SCONJ
ejpam-3659	270	30	the	the	DET
ejpam-3659	270	31	primitive	primitive	NOUN
ejpam-3659	270	32	must	must	AUX
ejpam-3659	270	33	be	be	AUX
ejpam-3659	270	34	of	of	ADP
ejpam-3659	270	35	finite	finite	PROPN
ejpam-3659	270	36	mcshane	mcshane	PROPN
ejpam-3659	270	37	variational	variational	ADJ
ejpam-3659	270	38	measure	measure	NOUN
ejpam-3659	270	39	on	on	ADP
ejpam-3659	270	40	x	x	X
ejpam-3659	270	41	,	,	PUNCT
ejpam-3659	270	42	that	that	ADV
ejpam-3659	270	43	is	is	ADV
ejpam-3659	270	44	,	,	PUNCT
ejpam-3659	270	45	vmf	vmf	PROPN
ejpam-3659	270	46	(	(	PUNCT
ejpam-3659	270	47	x	x	X
ejpam-3659	270	48	)	)	PUNCT
ejpam-3659	270	49	<	<	AUX
ejpam-3659	270	50	∞.	∞.	PROPN
ejpam-3659	270	51	the	the	DET
ejpam-3659	270	52	authors	author	NOUN
ejpam-3659	270	53	recommend	recommend	VERB
ejpam-3659	270	54	that	that	SCONJ
ejpam-3659	270	55	interested	interested	ADJ
ejpam-3659	270	56	readers	reader	NOUN
ejpam-3659	270	57	may	may	AUX
ejpam-3659	270	58	investigate	investigate	VERB
ejpam-3659	270	59	this	this	DET
ejpam-3659	270	60	study	study	NOUN
ejpam-3659	270	61	for	for	ADP
ejpam-3659	270	62	non	non	ADJ
ejpam-3659	270	63	-	-	ADJ
ejpam-3659	270	64	absolute	absolute	ADJ
ejpam-3659	270	65	integrals	integral	NOUN
ejpam-3659	270	66	(	(	PUNCT
ejpam-3659	270	67	since	since	SCONJ
ejpam-3659	270	68	mcshane	mcshane	PROPN
ejpam-3659	270	69	integrals	integral	NOUN
ejpam-3659	270	70	are	be	AUX
ejpam-3659	270	71	absolute	absolute	ADJ
ejpam-3659	270	72	)	)	PUNCT
ejpam-3659	270	73	.	.	PUNCT
ejpam-3659	271	1	references	reference	NOUN
ejpam-3659	271	2	[	[	X
ejpam-3659	271	3	1	1	NUM
ejpam-3659	271	4	]	]	X
ejpam-3659	271	5	benitez	benitez	PROPN
ejpam-3659	271	6	,	,	PUNCT
ejpam-3659	271	7	j.v	j.v	PROPN
ejpam-3659	271	8	.	.	PROPN
ejpam-3659	271	9	,	,	PUNCT
ejpam-3659	271	10	equi	equi	NOUN
ejpam-3659	271	11	-	-	PUNCT
ejpam-3659	271	12	integrability	integrability	NOUN
ejpam-3659	271	13	ad	ad	NOUN
ejpam-3659	271	14	harnack	harnack	NOUN
ejpam-3659	271	15	extensions	extension	NOUN
ejpam-3659	271	16	for	for	ADP
ejpam-3659	271	17	mcshane	mcshane	PROPN
ejpam-3659	271	18	and	and	CCONJ
ejpam-3659	271	19	henstock	henstock	NOUN
ejpam-3659	271	20	integrals	integral	NOUN
ejpam-3659	271	21	,	,	PUNCT
ejpam-3659	271	22	ph.d	ph.d	PROPN
ejpam-3659	271	23	.	.	PUNCT
ejpam-3659	272	1	dissertation	dissertation	NOUN
ejpam-3659	272	2	,	,	PUNCT
ejpam-3659	272	3	msu	msu	PROPN
ejpam-3659	272	4	-	-	PUNCT
ejpam-3659	272	5	iligan	iligan	PROPN
ejpam-3659	272	6	institute	institute	PROPN
ejpam-3659	272	7	of	of	ADP
ejpam-3659	272	8	technology	technology	PROPN
ejpam-3659	272	9	,	,	PUNCT
ejpam-3659	272	10	2012	2012	NUM
ejpam-3659	272	11	.	.	PUNCT
ejpam-3659	273	1	[	[	X
ejpam-3659	273	2	2	2	NUM
ejpam-3659	273	3	]	]	X
ejpam-3659	273	4	benitez	benitez	PROPN
ejpam-3659	273	5	,	,	PUNCT
ejpam-3659	273	6	j.v	j.v	PROPN
ejpam-3659	273	7	.	.	PROPN
ejpam-3659	273	8	,	,	PUNCT
ejpam-3659	273	9	jamil	jamil	PROPN
ejpam-3659	273	10	,	,	PUNCT
ejpam-3659	273	11	f.p	f.p	PROPN
ejpam-3659	273	12	.	.	PROPN
ejpam-3659	273	13	and	and	CCONJ
ejpam-3659	273	14	chew	chew	VERB
ejpam-3659	273	15	,	,	PUNCT
ejpam-3659	273	16	t.s	t.s	PROPN
ejpam-3659	273	17	.	.	PROPN
ejpam-3659	273	18	,	,	PUNCT
ejpam-3659	273	19	mcshane	mcshane	PROPN
ejpam-3659	273	20	integrability	integrability	PROPN
ejpam-3659	273	21	and	and	CCONJ
ejpam-3659	273	22	egoroff	egoroff	NOUN
ejpam-3659	273	23	’s	’s	PART
ejpam-3659	273	24	theorem	theorem	ADJ
ejpam-3659	273	25	,	,	PUNCT
ejpam-3659	273	26	matimyás	matimyás	NOUN
ejpam-3659	273	27	matematika	matematika	NOUN
ejpam-3659	273	28	,	,	PUNCT
ejpam-3659	273	29	32(2	32(2	NUM
ejpam-3659	273	30	)	)	PUNCT
ejpam-3659	273	31	,	,	PUNCT
ejpam-3659	273	32	13	13	NUM
ejpam-3659	273	33	-	-	SYM
ejpam-3659	273	34	20	20	NUM
ejpam-3659	273	35	,	,	PUNCT
ejpam-3659	273	36	2009	2009	NUM
ejpam-3659	273	37	.	.	PUNCT
ejpam-3659	274	1	[	[	X
ejpam-3659	274	2	3	3	NUM
ejpam-3659	274	3	]	]	X
ejpam-3659	274	4	benitez	benitez	PROPN
ejpam-3659	274	5	,	,	PUNCT
ejpam-3659	274	6	j.v	j.v	PROPN
ejpam-3659	274	7	.	.	PROPN
ejpam-3659	274	8	,	,	PUNCT
ejpam-3659	274	9	integrable	integrable	ADJ
ejpam-3659	274	10	set	set	NOUN
ejpam-3659	274	11	and	and	CCONJ
ejpam-3659	274	12	measurable	measurable	ADJ
ejpam-3659	274	13	function	function	NOUN
ejpam-3659	274	14	,	,	PUNCT
ejpam-3659	274	15	mindanawan	mindanawan	PROPN
ejpam-3659	274	16	journal	journal	NOUN
ejpam-3659	274	17	of	of	ADP
ejpam-3659	274	18	mathematics	mathematic	NOUN
ejpam-3659	274	19	,	,	PUNCT
ejpam-3659	274	20	3(1	3(1	NUM
ejpam-3659	274	21	)	)	PUNCT
ejpam-3659	274	22	,	,	PUNCT
ejpam-3659	274	23	2012	2012	NUM
ejpam-3659	274	24	.	.	PUNCT
ejpam-3659	275	1	[	[	X
ejpam-3659	275	2	4	4	NUM
ejpam-3659	275	3	]	]	PUNCT
ejpam-3659	275	4	chew	chew	VERB
ejpam-3659	275	5	,	,	PUNCT
ejpam-3659	275	6	t.s	t.s	PROPN
ejpam-3659	275	7	.	.	PROPN
ejpam-3659	275	8	,	,	PUNCT
ejpam-3659	275	9	the	the	DET
ejpam-3659	275	10	riemann	riemann	NOUN
ejpam-3659	275	11	-	-	PUNCT
ejpam-3659	275	12	type	type	NOUN
ejpam-3659	275	13	integral	integral	ADJ
ejpam-3659	275	14	that	that	PRON
ejpam-3659	275	15	includes	include	VERB
ejpam-3659	275	16	lebesgue	lebesgue	NOUN
ejpam-3659	275	17	-	-	PUNCT
ejpam-3659	275	18	stieltjes	stieltjes	NOUN
ejpam-3659	275	19	and	and	CCONJ
ejpam-3659	275	20	stochastic	stochastic	ADJ
ejpam-3659	275	21	integrals	integral	NOUN
ejpam-3659	275	22	,	,	PUNCT
ejpam-3659	275	23	lecture	lecture	NOUN
ejpam-3659	275	24	note	note	NOUN
ejpam-3659	275	25	,	,	PUNCT
ejpam-3659	275	26	chulalongkorn	chulalongkorn	ADJ
ejpam-3659	275	27	university	university	NOUN
ejpam-3659	275	28	,	,	PUNCT
ejpam-3659	275	29	2004	2004	NUM
ejpam-3659	275	30	.	.	PUNCT
ejpam-3659	276	1	[	[	X
ejpam-3659	276	2	5	5	NUM
ejpam-3659	276	3	]	]	X
ejpam-3659	276	4	gordon	gordon	PROPN
ejpam-3659	276	5	,	,	PUNCT
ejpam-3659	276	6	r.	r.	PROPN
ejpam-3659	276	7	,	,	PUNCT
ejpam-3659	276	8	the	the	DET
ejpam-3659	276	9	integrals	integral	NOUN
ejpam-3659	276	10	of	of	ADP
ejpam-3659	276	11	lebesgue	lebesgue	NOUN
ejpam-3659	276	12	,	,	PUNCT
ejpam-3659	276	13	denjoy	denjoy	PROPN
ejpam-3659	276	14	,	,	PUNCT
ejpam-3659	276	15	perron	perron	PROPN
ejpam-3659	276	16	,	,	PUNCT
ejpam-3659	276	17	and	and	CCONJ
ejpam-3659	276	18	henstock	henstock	NOUN
ejpam-3659	276	19	,	,	PUNCT
ejpam-3659	276	20	graduate	graduate	NOUN
ejpam-3659	276	21	studies	study	NOUN
ejpam-3659	276	22	in	in	ADP
ejpam-3659	276	23	mathematics	mathematic	NOUN
ejpam-3659	276	24	,	,	PUNCT
ejpam-3659	276	25	4	4	NUM
ejpam-3659	276	26	,	,	PUNCT
ejpam-3659	276	27	amer	amer	PROPN
ejpam-3659	276	28	.	.	PROPN
ejpam-3659	276	29	math	math	PROPN
ejpam-3659	276	30	.	.	PUNCT
ejpam-3659	277	1	soc	soc	PROPN
ejpam-3659	277	2	.	.	PROPN
ejpam-3659	277	3	,	,	PUNCT
ejpam-3659	277	4	1994	1994	NUM
ejpam-3659	277	5	.	.	PUNCT
ejpam-3659	278	1	[	[	X
ejpam-3659	278	2	6	6	NUM
ejpam-3659	278	3	]	]	X
ejpam-3659	278	4	lee	lee	PROPN
ejpam-3659	278	5	,	,	PUNCT
ejpam-3659	278	6	p.y	p.y	PROPN
ejpam-3659	278	7	.	.	PROPN
ejpam-3659	278	8	,	,	PUNCT
ejpam-3659	278	9	lanzhou	lanzhou	PROPN
ejpam-3659	278	10	lectures	lecture	VERB
ejpam-3659	278	11	on	on	ADP
ejpam-3659	278	12	henstock	henstock	NOUN
ejpam-3659	278	13	integration	integration	NOUN
ejpam-3659	278	14	,	,	PUNCT
ejpam-3659	278	15	world	world	NOUN
ejpam-3659	278	16	scientific	scientific	NOUN
ejpam-3659	278	17	,	,	PUNCT
ejpam-3659	278	18	1989	1989	NUM
ejpam-3659	278	19	.	.	PUNCT
ejpam-3659	279	1	[	[	X
ejpam-3659	279	2	7	7	NUM
ejpam-3659	279	3	]	]	X
ejpam-3659	279	4	lee	lee	PROPN
ejpam-3659	279	5	,	,	PUNCT
ejpam-3659	279	6	p.y	p.y	PROPN
ejpam-3659	279	7	.	.	PROPN
ejpam-3659	279	8	and	and	CCONJ
ejpam-3659	279	9	yýborný	yýborný	PROPN
ejpam-3659	279	10	,	,	PUNCT
ejpam-3659	279	11	r.	r.	PROPN
ejpam-3659	279	12	,	,	PUNCT
ejpam-3659	279	13	the	the	DET
ejpam-3659	279	14	integral	integral	ADJ
ejpam-3659	279	15	:	:	PUNCT
ejpam-3659	279	16	an	an	DET
ejpam-3659	279	17	easy	easy	ADJ
ejpam-3659	279	18	approach	approach	NOUN
ejpam-3659	279	19	after	after	ADP
ejpam-3659	279	20	kurzweil	kurzweil	PROPN
ejpam-3659	279	21	and	and	CCONJ
ejpam-3659	279	22	henstock	henstock	PROPN
ejpam-3659	279	23	,	,	PUNCT
ejpam-3659	279	24	cambridge	cambridge	PROPN
ejpam-3659	279	25	university	university	PROPN
ejpam-3659	279	26	press	press	NOUN
ejpam-3659	279	27	,	,	PUNCT
ejpam-3659	279	28	2000	2000	NUM
ejpam-3659	279	29	.	.	PUNCT
ejpam-3659	280	1	[	[	X
ejpam-3659	280	2	8	8	NUM
ejpam-3659	280	3	]	]	X
ejpam-3659	280	4	lee	lee	PROPN
ejpam-3659	280	5	,	,	PUNCT
ejpam-3659	280	6	t.y	t.y	PROPN
ejpam-3659	280	7	.	.	PROPN
ejpam-3659	280	8	,	,	PUNCT
ejpam-3659	280	9	henstock	henstock	NOUN
ejpam-3659	280	10	-	-	PUNCT
ejpam-3659	280	11	kurzweil	kurzweil	NOUN
ejpam-3659	280	12	integration	integration	NOUN
ejpam-3659	280	13	on	on	ADP
ejpam-3659	280	14	euclidean	euclidean	ADJ
ejpam-3659	280	15	spaces	space	NOUN
ejpam-3659	280	16	,	,	PUNCT
ejpam-3659	280	17	world	world	NOUN
ejpam-3659	280	18	scientific	scientific	ADJ
ejpam-3659	280	19	,	,	PUNCT
ejpam-3659	280	20	volume	volume	NOUN
ejpam-3659	280	21	12	12	NUM
ejpam-3659	280	22	,	,	PUNCT
ejpam-3659	280	23	2011	2011	NUM
ejpam-3659	280	24	.	.	PUNCT
ejpam-3659	281	1	references	reference	NOUN
ejpam-3659	281	2	313	313	NUM
ejpam-3659	281	3	[	[	X
ejpam-3659	281	4	9	9	NUM
ejpam-3659	281	5	]	]	SYM
ejpam-3659	281	6	benitez	benitez	PROPN
ejpam-3659	281	7	,	,	PUNCT
ejpam-3659	281	8	j.v	j.v	PROPN
ejpam-3659	281	9	.	.	PROPN
ejpam-3659	281	10	and	and	CCONJ
ejpam-3659	281	11	quindala	quindala	PROPN
ejpam-3659	281	12	iii	iii	PROPN
ejpam-3659	281	13	,	,	PUNCT
ejpam-3659	281	14	k.m	k.m	PROPN
ejpam-3659	281	15	.	.	PROPN
ejpam-3659	281	16	,	,	PUNCT
ejpam-3659	281	17	on	on	ADP
ejpam-3659	281	18	mcshane	mcshane	PROPN
ejpam-3659	281	19	integrable	integrable	ADJ
ejpam-3659	281	20	sets	set	NOUN
ejpam-3659	281	21	and	and	CCONJ
ejpam-3659	281	22	lebesgue	lebesgue	NOUN
ejpam-3659	281	23	measure	measure	NOUN
ejpam-3659	281	24	,	,	PUNCT
ejpam-3659	281	25	international	international	ADJ
ejpam-3659	281	26	journal	journal	NOUN
ejpam-3659	281	27	of	of	ADP
ejpam-3659	281	28	mathematical	mathematical	ADJ
ejpam-3659	281	29	analysis	analysis	NOUN
ejpam-3659	281	30	,	,	PUNCT
ejpam-3659	281	31	9(3	9(3	NUM
ejpam-3659	281	32	)	)	PUNCT
ejpam-3659	281	33	,	,	PUNCT
ejpam-3659	281	34	127	127	NUM
ejpam-3659	281	35	-	-	SYM
ejpam-3659	281	36	139	139	NUM
ejpam-3659	281	37	,	,	PUNCT
ejpam-3659	281	38	2015	2015	NUM
ejpam-3659	281	39	.	.	PUNCT
ejpam-3659	282	1	[	[	X
ejpam-3659	282	2	10	10	NUM
ejpam-3659	282	3	]	]	X
ejpam-3659	282	4	swartz	swartz	PROPN
ejpam-3659	282	5	,	,	PUNCT
ejpam-3659	282	6	c.	c.	PROPN
ejpam-3659	282	7	,	,	PUNCT
ejpam-3659	282	8	introduction	introduction	NOUN
ejpam-3659	282	9	to	to	PART
ejpam-3659	282	10	gauge	gauge	VERB
ejpam-3659	282	11	integrals	integral	NOUN
ejpam-3659	282	12	,	,	PUNCT
ejpam-3659	282	13	world	world	NOUN
ejpam-3659	282	14	scientific	scientific	ADJ
ejpam-3659	282	15	,	,	PUNCT
ejpam-3659	282	16	2001	2001	NUM
ejpam-3659	282	17	.	.	PUNCT
ejpam-3659	283	1	[	[	X
ejpam-3659	283	2	11	11	NUM
ejpam-3659	283	3	]	]	PUNCT
ejpam-3659	283	4	wells	well	NOUN
ejpam-3659	283	5	,	,	PUNCT
ejpam-3659	283	6	j.	j.	PROPN
ejpam-3659	283	7	,	,	PUNCT
ejpam-3659	283	8	generalizations	generalization	NOUN
ejpam-3659	283	9	of	of	ADP
ejpam-3659	283	10	the	the	DET
ejpam-3659	283	11	riemann	riemann	PROPN
ejpam-3659	283	12	integral	integral	PROPN
ejpam-3659	283	13	:	:	PUNCT
ejpam-3659	283	14	an	an	DET
ejpam-3659	283	15	investigation	investigation	NOUN
ejpam-3659	283	16	of	of	ADP
ejpam-3659	283	17	the	the	DET
ejpam-3659	283	18	henstock	henstock	NOUN
ejpam-3659	283	19	integral	integral	ADJ
ejpam-3659	283	20	,	,	PUNCT
ejpam-3659	283	21	2011	2011	NUM
ejpam-3659	283	22	.	.	PUNCT
ejpam-3659	284	1	[	[	X
ejpam-3659	284	2	12	12	NUM
ejpam-3659	284	3	]	]	X
ejpam-3659	284	4	yang	yang	PROPN
ejpam-3659	284	5	,	,	PUNCT
ejpam-3659	284	6	c.h	c.h	PROPN
ejpam-3659	284	7	.	.	PROPN
ejpam-3659	284	8	,	,	PUNCT
ejpam-3659	284	9	measure	measure	NOUN
ejpam-3659	284	10	theory	theory	NOUN
ejpam-3659	284	11	and	and	CCONJ
ejpam-3659	284	12	the	the	DET
ejpam-3659	284	13	henstock	henstock	NOUN
ejpam-3659	284	14	integral	integral	ADJ
ejpam-3659	284	15	,	,	PUNCT
ejpam-3659	284	16	academic	academic	ADJ
ejpam-3659	284	17	exercise	exercise	NOUN
ejpam-3659	284	18	,	,	PUNCT
ejpam-3659	284	19	national	national	ADJ
ejpam-3659	284	20	university	university	PROPN
ejpam-3659	284	21	of	of	ADP
ejpam-3659	284	22	singapore	singapore	PROPN
ejpam-3659	284	23	,	,	PUNCT
ejpam-3659	284	24	singapore	singapore	PROPN
ejpam-3659	284	25	,	,	PUNCT
ejpam-3659	284	26	1996	1996	NUM
ejpam-3659	284	27	-	-	SYM
ejpam-3659	284	28	97	97	NUM
ejpam-3659	284	29	.	.	PUNCT