id sid tid token lemma pos iajs-2428 1 1 microsoft microsoft PROPN iajs-2428 1 2 word word NOUN iajs-2428 1 3 72 72 NUM iajs-2428 1 4 - - SYM iajs-2428 1 5 80 80 NUM iajs-2428 1 6     SPACE iajs-2428 1 7 72 72 NUM iajs-2428 1 8     SPACE iajs-2428 1 9 ibn ibn PROPN iajs-2428 1 10 al al PROPN iajs-2428 1 11 - - PUNCT iajs-2428 1 12 haitham haitham PROPN iajs-2428 1 13 jour jour X iajs-2428 1 14 . . PROPN iajs-2428 2 1 for for ADP iajs-2428 2 2 pure pure ADJ iajs-2428 2 3 & & CCONJ iajs-2428 2 4 appl appl PROPN iajs-2428 2 5 . . PUNCT iajs-2428 3 1 sci sci PROPN iajs-2428 3 2 . . PROPN iajs-2428 4 1 33 33 NUM iajs-2428 4 2 ( ( PUNCT iajs-2428 4 3 2 2 NUM iajs-2428 4 4 ) ) PUNCT iajs-2428 4 5 2020 2020 NUM iajs-2428 4 6             SPACE iajs-2428 4 7 𝛌 𝛌 PROPN iajs-2428 4 8 – – PUNCT iajs-2428 4 9 𝐀𝐥𝐠𝐞𝐛𝐫𝐚 𝐀𝐥𝐠𝐞𝐛𝐫𝐚 PROPN iajs-2428 4 10 with with ADP iajs-2428 4 11 some some PRON iajs-2428 4 12 of of ADP iajs-2428 4 13 their their PRON iajs-2428 4 14 properties property NOUN iajs-2428 4 15 hassan hassan PROPN iajs-2428 4 16 hussien hussien PROPN iajs-2428 4 17 ebrahim ebrahim PROPN iajs-2428 4 18 rusul rusul PROPN iajs-2428 4 19 abd abd PROPN iajs-2428 4 20 al_salam al_salam PROPN iajs-2428 4 21 ali ali PROPN iajs-2428 4 22 article article PROPN iajs-2428 4 23 history history NOUN iajs-2428 4 24 : : PUNCT iajs-2428 4 25 received receive VERB iajs-2428 4 26 23 23 NUM iajs-2428 4 27 june june PROPN iajs-2428 4 28 2019 2019 NUM iajs-2428 4 29 , , PUNCT iajs-2428 4 30 accepted accept VERB iajs-2428 4 31 19 19 NUM iajs-2428 4 32 august august PROPN iajs-2428 4 33 2019 2019 NUM iajs-2428 4 34 , , PUNCT iajs-2428 4 35 published publish VERB iajs-2428 4 36 in in ADP iajs-2428 4 37 april april PROPN iajs-2428 4 38 2020 2020 NUM iajs-2428 4 39 . . PUNCT iajs-2428 5 1 abstract abstract ADV iajs-2428 5 2 the the DET iajs-2428 5 3 objective objective NOUN iajs-2428 5 4 of of ADP iajs-2428 5 5 this this DET iajs-2428 5 6 paper paper NOUN iajs-2428 5 7 is be AUX iajs-2428 5 8 , , PUNCT iajs-2428 5 9 firstly firstly ADV iajs-2428 5 10 , , PUNCT iajs-2428 5 11 we we PRON iajs-2428 5 12 study study VERB iajs-2428 5 13 a a DET iajs-2428 5 14 new new ADJ iajs-2428 5 15 concept concept NOUN iajs-2428 5 16 noted note VERB iajs-2428 5 17 by by ADP iajs-2428 5 18 λ λ PROPN iajs-2428 5 19 – – PUNCT iajs-2428 5 20 algebra algebra NOUN iajs-2428 5 21 and and CCONJ iajs-2428 5 22 discuss discuss VERB iajs-2428 5 23 the the DET iajs-2428 5 24 properties property NOUN iajs-2428 5 25 of of ADP iajs-2428 5 26 this this DET iajs-2428 5 27 concept concept NOUN iajs-2428 5 28 . . PUNCT iajs-2428 6 1 secondly secondly ADV iajs-2428 6 2 , , PUNCT iajs-2428 6 3 we we PRON iajs-2428 6 4 introduce introduce VERB iajs-2428 6 5 a a DET iajs-2428 6 6 new new ADJ iajs-2428 6 7 concept concept NOUN iajs-2428 6 8 related relate VERB iajs-2428 6 9 to to ADP iajs-2428 6 10 the the DET iajs-2428 6 11 λ λ NOUN iajs-2428 6 12 – – PUNCT iajs-2428 6 13 algebra algebra NOUN iajs-2428 6 14 such such ADJ iajs-2428 6 15 as as ADP iajs-2428 6 16 smallest small ADJ iajs-2428 6 17 λ λ NOUN iajs-2428 6 18 – – PUNCT iajs-2428 6 19 algebra algebra NOUN iajs-2428 6 20 . . PUNCT iajs-2428 7 1 thirdly thirdly ADV iajs-2428 7 2 , , PUNCT iajs-2428 7 3 we we PRON iajs-2428 7 4 introduce introduce VERB iajs-2428 7 5 the the DET iajs-2428 7 6 notion notion NOUN iajs-2428 7 7 of of ADP iajs-2428 7 8 the the DET iajs-2428 7 9 restriction restriction NOUN iajs-2428 7 10 of of ADP iajs-2428 7 11 λ λ PROPN iajs-2428 7 12 – – PUNCT iajs-2428 7 13 algebra algebra NOUN iajs-2428 7 14 on on ADP iajs-2428 7 15 a a DET iajs-2428 7 16 nonempty nonempty NOUN iajs-2428 7 17 subset subset VERB iajs-2428 7 18 𝔇 𝔇 NOUN iajs-2428 7 19 of of ADP iajs-2428 7 20 𝔓 𝔓 PROPN iajs-2428 7 21 and and CCONJ iajs-2428 7 22 investigate investigate VERB iajs-2428 7 23 some some PRON iajs-2428 7 24 of of ADP iajs-2428 7 25 its its PRON iajs-2428 7 26 basic basic ADJ iajs-2428 7 27 properties property NOUN iajs-2428 7 28 . . PUNCT iajs-2428 8 1 furthermore furthermore ADV iajs-2428 8 2 , , PUNCT iajs-2428 8 3 we we PRON iajs-2428 8 4 present present VERB iajs-2428 8 5 the the DET iajs-2428 8 6 relationships relationship NOUN iajs-2428 8 7 between between ADP iajs-2428 8 8 α α PROPN iajs-2428 8 9 – – PUNCT iajs-2428 8 10 σ σ NOUN iajs-2428 8 11 – – PUNCT iajs-2428 8 12 field field NOUN iajs-2428 8 13 , , PUNCT iajs-2428 8 14 monotone monotone ADJ iajs-2428 8 15 class class NOUN iajs-2428 8 16 , , PUNCT iajs-2428 8 17 β β X iajs-2428 8 18 – – PUNCT iajs-2428 8 19 σ σ NOUN iajs-2428 8 20 – – PUNCT iajs-2428 8 21 field field NOUN iajs-2428 8 22 and and CCONJ iajs-2428 8 23 λ λ NOUN iajs-2428 8 24 – – PUNCT iajs-2428 8 25 algebra algebra NOUN iajs-2428 8 26 . . PUNCT iajs-2428 9 1 finally finally ADV iajs-2428 9 2 , , PUNCT iajs-2428 9 3 we we PRON iajs-2428 9 4 introduce introduce VERB iajs-2428 9 5 the the DET iajs-2428 9 6 concept concept NOUN iajs-2428 9 7 of of ADP iajs-2428 9 8 measure measure NOUN iajs-2428 9 9 relative relative ADJ iajs-2428 9 10 to to ADP iajs-2428 9 11 the the DET iajs-2428 9 12 λ λ NOUN iajs-2428 9 13 – – PUNCT iajs-2428 9 14 algebra algebra NOUN iajs-2428 9 15 and and CCONJ iajs-2428 9 16 prove prove VERB iajs-2428 9 17 that that SCONJ iajs-2428 9 18 every every DET iajs-2428 9 19 measure measure NOUN iajs-2428 9 20 relative relative ADJ iajs-2428 9 21 to to ADP iajs-2428 9 22 the the DET iajs-2428 9 23 λ λ NOUN iajs-2428 9 24 – – PUNCT iajs-2428 9 25 algebra algebra NOUN iajs-2428 9 26 is be AUX iajs-2428 9 27 complete complete ADJ iajs-2428 9 28 . . PUNCT iajs-2428 10 1 keywords keyword NOUN iajs-2428 10 2 : : PUNCT iajs-2428 10 3 σ σ X iajs-2428 10 4 – – PUNCT iajs-2428 10 5 field field NOUN iajs-2428 10 6 , , PUNCT iajs-2428 10 7 increasing increase VERB iajs-2428 10 8 sequence sequence NOUN iajs-2428 10 9 , , PUNCT iajs-2428 10 10 α α X iajs-2428 10 11 – – PUNCT iajs-2428 10 12 σ σ NOUN iajs-2428 10 13 – – PUNCT iajs-2428 10 14 field field NOUN iajs-2428 10 15 , , PUNCT iajs-2428 10 16 monotone monotone ADJ iajs-2428 10 17 class class NOUN iajs-2428 10 18 , , PUNCT iajs-2428 10 19 β β X iajs-2428 10 20 – – PUNCT iajs-2428 10 21 σ σ NOUN iajs-2428 10 22 – – PUNCT iajs-2428 10 23 field field NOUN iajs-2428 10 24 . . PUNCT iajs-2428 11 1 1 1 X iajs-2428 11 2 . . X iajs-2428 11 3 introduction introduction NOUN iajs-2428 11 4 about about ADV iajs-2428 11 5 forty forty NUM iajs-2428 11 6 seven seven NUM iajs-2428 11 7 year year NOUN iajs-2428 11 8 ago ago ADV iajs-2428 11 9 , , PUNCT iajs-2428 11 10 robert robert PROPN iajs-2428 12 1 [ [ X iajs-2428 12 2 1 1 NUM iajs-2428 12 3 ] ] PUNCT iajs-2428 12 4 . . PUNCT iajs-2428 13 1 studied study VERB iajs-2428 13 2 the the DET iajs-2428 13 3 concept concept NOUN iajs-2428 13 4 of of ADP iajs-2428 13 5 σ σ PROPN iajs-2428 13 6 – – PUNCT iajs-2428 13 7 field field NOUN iajs-2428 13 8 , , PUNCT iajs-2428 13 9 where where SCONJ iajs-2428 13 10 a a DET iajs-2428 13 11 collection collection NOUN iajs-2428 13 12 𝒦 𝒦 PROPN iajs-2428 13 13 is be AUX iajs-2428 13 14 called call VERB iajs-2428 13 15 σ σ NOUN iajs-2428 13 16 – – PUNCT iajs-2428 13 17 field field NOUN iajs-2428 13 18 of of ADP iajs-2428 13 19 a a DET iajs-2428 13 20 set set NOUN iajs-2428 13 21 𝔓 𝔓 PROPN iajs-2428 13 22 if if SCONJ iajs-2428 13 23 𝔓ϵ𝒦 𝔓ϵ𝒦 PROPN iajs-2428 13 24 and and CCONJ iajs-2428 13 25 𝒦 𝒦 PROPN iajs-2428 13 26 is be AUX iajs-2428 13 27 closed close VERB iajs-2428 13 28 under under ADP iajs-2428 13 29 complementation complementation NOUN iajs-2428 13 30 and and CCONJ iajs-2428 13 31 countable countable ADJ iajs-2428 13 32 union union NOUN iajs-2428 13 33 . . PUNCT iajs-2428 14 1 many many ADJ iajs-2428 14 2 authors author NOUN iajs-2428 14 3 studied study VERB iajs-2428 14 4 the the DET iajs-2428 14 5 concept concept NOUN iajs-2428 14 6 of of ADP iajs-2428 14 7 σ σ PROPN iajs-2428 14 8 – – PUNCT iajs-2428 14 9 field field NOUN iajs-2428 14 10 , , PUNCT iajs-2428 14 11 for for ADP iajs-2428 14 12 example example NOUN iajs-2428 14 13 see see VERB iajs-2428 14 14 [ [ X iajs-2428 14 15 2 2 NUM iajs-2428 14 16 - - SYM iajs-2428 14 17 4 4 NUM iajs-2428 14 18 ] ] PUNCT iajs-2428 14 19 . . PUNCT iajs-2428 15 1 and and CCONJ iajs-2428 15 2 [ [ X iajs-2428 15 3 5 5 NUM iajs-2428 15 4 ] ] PUNCT iajs-2428 15 5 . . PUNCT iajs-2428 16 1 the the DET iajs-2428 16 2 notion notion NOUN iajs-2428 16 3 of of ADP iajs-2428 16 4 increasing increase VERB iajs-2428 16 5 sequence sequence NOUN iajs-2428 16 6 and and CCONJ iajs-2428 16 7 decreasing decrease VERB iajs-2428 16 8 sequence sequence NOUN iajs-2428 16 9 studied study VERB iajs-2428 16 10 by by ADP iajs-2428 16 11 robert robert PROPN iajs-2428 16 12 , , PUNCT iajs-2428 16 13 where where SCONJ iajs-2428 16 14 d d NOUN iajs-2428 16 15 , , PUNCT iajs-2428 16 16 d d X iajs-2428 16 17 , , PUNCT iajs-2428 16 18 … … PUNCT iajs-2428 16 19 are be AUX iajs-2428 16 20 subsets subset NOUN iajs-2428 16 21 of of ADP iajs-2428 16 22 a a DET iajs-2428 16 23 set set ADJ iajs-2428 16 24 𝔓 𝔓 NOUN iajs-2428 16 25 , , PUNCT iajs-2428 16 26 if if SCONJ iajs-2428 16 27 d d PROPN iajs-2428 16 28 ⊂ ⊂ PROPN iajs-2428 16 29 d d X iajs-2428 16 30 ⊂ ⊂ PROPN iajs-2428 16 31 ⋯ ⋯ PROPN iajs-2428 16 32 and and CCONJ iajs-2428 16 33 ⋃ ⋃ PROPN iajs-2428 16 34 d d X iajs-2428 16 35 d. d. PROPN iajs-2428 16 36 then then ADV iajs-2428 16 37 we we PRON iajs-2428 16 38 say say VERB iajs-2428 16 39 that that SCONJ iajs-2428 16 40 d d NOUN iajs-2428 16 41 increase increase NOUN iajs-2428 16 42 tod tod NOUN iajs-2428 16 43 ; ; PUNCT iajs-2428 16 44 we we PRON iajs-2428 16 45 write write VERB iajs-2428 16 46 d d PROPN iajs-2428 16 47 ↑ ↑ PROPN iajs-2428 16 48 d. d. PROPN iajs-2428 16 49 if if SCONJ iajs-2428 16 50 d d PROPN iajs-2428 16 51 ⊃ ⊃ PROPN iajs-2428 16 52 d d X iajs-2428 16 53 ⊃ ⊃ X iajs-2428 16 54 ⋯ ⋯ PROPN iajs-2428 16 55 and and CCONJ iajs-2428 16 56 ⋂ ⋂ PROPN iajs-2428 16 57 d d PROPN iajs-2428 16 58 d d PROPN iajs-2428 16 59 , , PUNCT iajs-2428 16 60 we we PRON iajs-2428 16 61 say say VERB iajs-2428 16 62 that that SCONJ iajs-2428 16 63 d d PROPN iajs-2428 16 64 decrease decrease NOUN iajs-2428 16 65 tod tod NOUN iajs-2428 16 66 ; ; PUNCT iajs-2428 16 67 we we PRON iajs-2428 16 68 write write VERB iajs-2428 16 69 d d PROPN iajs-2428 16 70 ↓ ↓ PROPN iajs-2428 17 1 d d PROPN iajs-2428 18 1 [ [ X iajs-2428 18 2 1 1 NUM iajs-2428 18 3 ] ] PUNCT iajs-2428 18 4 . . PUNCT iajs-2428 19 1 zhenyuan zhenyuan PROPN iajs-2428 19 2 and and CCONJ iajs-2428 19 3 george george PROPN iajs-2428 19 4 in in ADP iajs-2428 19 5 2009 2009 NUM iajs-2428 19 6 studied study VERB iajs-2428 19 7 the the DET iajs-2428 19 8 concept concept NOUN iajs-2428 19 9 of of ADP iajs-2428 19 10 monotone monotone ADJ iajs-2428 19 11 class class NOUN iajs-2428 19 12 which which PRON iajs-2428 19 13 represents represent VERB iajs-2428 19 14 the the DET iajs-2428 19 15 generalization generalization NOUN iajs-2428 19 16 of of ADP iajs-2428 19 17 σ σ PROPN iajs-2428 19 18 – – PUNCT iajs-2428 19 19 field field NOUN iajs-2428 19 20 , , PUNCT iajs-2428 19 21 where where SCONJ iajs-2428 19 22 a a DET iajs-2428 19 23 collection collection NOUN iajs-2428 19 24 𝒦 𝒦 PROPN iajs-2428 19 25 of of ADP iajs-2428 19 26 subsets subset NOUN iajs-2428 19 27 of of ADP iajs-2428 19 28 a a DET iajs-2428 19 29 nonempty nonempty ADV iajs-2428 19 30 set set VERB iajs-2428 19 31 𝔓 𝔓 PROPN iajs-2428 19 32 is be AUX iajs-2428 19 33 said say VERB iajs-2428 19 34 to to PART iajs-2428 19 35 be be AUX iajs-2428 19 36 monotone monotone ADJ iajs-2428 19 37 class class NOUN iajs-2428 19 38 iff iff NOUN iajs-2428 19 39 whenever whenever SCONJ iajs-2428 19 40 d d PROPN iajs-2428 19 41 , , PUNCT iajs-2428 19 42 d d PROPN iajs-2428 19 43 , , PUNCT iajs-2428 19 44 … … PUNCT iajs-2428 19 45 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 19 46 such such ADJ iajs-2428 19 47 that that SCONJ iajs-2428 19 48 d d PROPN iajs-2428 19 49 ↑ ↑ PROPN iajs-2428 19 50 d d PROPN iajs-2428 19 51 , , PUNCT iajs-2428 19 52 then then ADV iajs-2428 19 53 dϵ dϵ VERB iajs-2428 19 54 𝒦 𝒦 PROPN iajs-2428 19 55 and and CCONJ iajs-2428 20 1 if if SCONJ iajs-2428 20 2 d d PROPN iajs-2428 20 3 ↓ ↓ PROPN iajs-2428 20 4 d d PROPN iajs-2428 20 5 , , PUNCT iajs-2428 20 6 then then ADV iajs-2428 20 7 dϵ dϵ VERB iajs-2428 20 8 𝒦 𝒦 PROPN iajs-2428 20 9 [ [ X iajs-2428 20 10 6 6 NUM iajs-2428 20 11 ] ] PUNCT iajs-2428 20 12 . . PUNCT iajs-2428 21 1 in in ADP iajs-2428 21 2 2019 2019 NUM iajs-2428 21 3 , , PUNCT iajs-2428 21 4 ibrahim ibrahim PROPN iajs-2428 21 5 and and CCONJ iajs-2428 21 6 hassan hassan PROPN iajs-2428 21 7 introduced introduce VERB iajs-2428 21 8 some some DET iajs-2428 21 9 concepts concept NOUN iajs-2428 21 10 such such ADJ iajs-2428 21 11 as as ADP iajs-2428 21 12 α α PROPN iajs-2428 21 13 – – PUNCT iajs-2428 21 14 σ σ NOUN iajs-2428 21 15 – – PUNCT iajs-2428 21 16 field field NOUN iajs-2428 21 17 and and CCONJ iajs-2428 21 18 β β X iajs-2428 21 19 – – PUNCT iajs-2428 21 20 σ σ NOUN iajs-2428 21 21 – – PUNCT iajs-2428 21 22 field field NOUN iajs-2428 21 23 which which PRON iajs-2428 21 24 represent represent VERB iajs-2428 21 25 the the DET iajs-2428 21 26 generalizations generalization NOUN iajs-2428 21 27 of of ADP iajs-2428 21 28 σ σ PROPN iajs-2428 21 29 – – PUNCT iajs-2428 21 30 field field NOUN iajs-2428 21 31 , , PUNCT iajs-2428 21 32 where where SCONJ iajs-2428 21 33 a a DET iajs-2428 21 34 collection collection NOUN iajs-2428 21 35 𝒦 𝒦 PROPN iajs-2428 21 36 is be AUX iajs-2428 21 37 said say VERB iajs-2428 21 38 to to PART iajs-2428 21 39 be be AUX iajs-2428 21 40 α α NUM iajs-2428 21 41 – – PUNCT iajs-2428 21 42 σ σ NOUN iajs-2428 21 43 – – PUNCT iajs-2428 21 44 field field NOUN iajs-2428 21 45 iff iff PROPN iajs-2428 21 46 φ φ NOUN iajs-2428 21 47 , , PUNCT iajs-2428 21 48 𝔓ϵ 𝔓ϵ NOUN iajs-2428 21 49 𝒦 𝒦 PROPN iajs-2428 21 50 and and CCONJ iajs-2428 21 51 𝒦 𝒦 PROPN iajs-2428 21 52 is be AUX iajs-2428 21 53 closed close VERB iajs-2428 21 54 under under ADP iajs-2428 21 55 countable countable ADJ iajs-2428 21 56 union union NOUN iajs-2428 22 1 [ [ X iajs-2428 22 2 7 7 NUM iajs-2428 22 3 ] ] PUNCT iajs-2428 22 4 . . PUNCT iajs-2428 23 1 and and CCONJ iajs-2428 23 2 a a DET iajs-2428 23 3 collection collection NOUN iajs-2428 23 4 𝒦 𝒦 PROPN iajs-2428 23 5 is be AUX iajs-2428 23 6 said say VERB iajs-2428 23 7 to to PART iajs-2428 23 8 be be AUX iajs-2428 23 9 β β X iajs-2428 23 10 – – PUNCT iajs-2428 23 11 σ σ NOUN iajs-2428 23 12 – – PUNCT iajs-2428 23 13 field field NOUN iajs-2428 23 14 if if SCONJ iajs-2428 23 15 φ φ PROPN iajs-2428 23 16 , , PUNCT iajs-2428 23 17 𝔓ϵ 𝔓ϵ NOUN iajs-2428 23 18 𝒦 𝒦 PROPN iajs-2428 23 19 and and CCONJ iajs-2428 23 20 𝒦 𝒦 PROPN iajs-2428 23 21 is be AUX iajs-2428 23 22 closed close VERB iajs-2428 23 23 under under ADP iajs-2428 23 24 countable countable ADJ iajs-2428 23 25 intersection intersection NOUN iajs-2428 23 26 [ [ X iajs-2428 23 27 7 7 NUM iajs-2428 23 28 ] ] PUNCT iajs-2428 23 29 . . PUNCT iajs-2428 24 1 ibrahim ibrahim PROPN iajs-2428 24 2 and and CCONJ iajs-2428 24 3 hassan hassan PROPN iajs-2428 24 4 in in ADP iajs-2428 24 5 2019 2019 NUM iajs-2428 24 6 also also ADV iajs-2428 24 7 introduced introduce VERB iajs-2428 24 8 the the DET iajs-2428 24 9 concept concept NOUN iajs-2428 24 10 of of ADP iajs-2428 24 11 δ δ PROPN iajs-2428 24 12 – – PUNCT iajs-2428 24 13 field field NOUN iajs-2428 24 14 as as ADP iajs-2428 24 15 a a DET iajs-2428 24 16 stronger strong ADJ iajs-2428 24 17 form form NOUN iajs-2428 24 18 of of ADP iajs-2428 24 19 these these DET iajs-2428 24 20 concepts concept NOUN iajs-2428 24 21 , , PUNCT iajs-2428 24 22 where where SCONJ iajs-2428 24 23 a a DET iajs-2428 24 24 collection collection NOUN iajs-2428 24 25 𝒦 𝒦 PROPN iajs-2428 24 26 is be AUX iajs-2428 24 27 said say VERB iajs-2428 24 28 to to ADP iajs-2428 24 29 δ δ PROPN iajs-2428 24 30 – – PUNCT iajs-2428 24 31 field field NOUN iajs-2428 24 32 iff iff NOUN iajs-2428 24 33 φϵ φϵ VERB iajs-2428 24 34 𝒦 𝒦 PROPN iajs-2428 24 35 and and CCONJ iajs-2428 24 36 if if SCONJ iajs-2428 24 37 φ φ PROPN iajs-2428 24 38 aϵ𝒦 aϵ𝒦 VERB iajs-2428 24 39 anda anda PROPN iajs-2428 24 40 ⊂ ⊂ PROPN iajs-2428 24 41 b b PROPN iajs-2428 25 1 ⊆ ⊆ NUM iajs-2428 25 2 𝔓 𝔓 PROPN iajs-2428 25 3 , , PUNCT iajs-2428 25 4 then then ADV iajs-2428 25 5 bϵ𝒦and bϵ𝒦and NOUN iajs-2428 25 6 𝒦 𝒦 PROPN iajs-2428 25 7 is be AUX iajs-2428 25 8 closed close VERB iajs-2428 25 9 under under ADP iajs-2428 25 10 countable countable ADJ iajs-2428 25 11 intersection intersection NOUN iajs-2428 25 12 [ [ X iajs-2428 25 13 8 8 NUM iajs-2428 25 14 ] ] PUNCT iajs-2428 25 15 . . PUNCT iajs-2428 26 1 the the DET iajs-2428 26 2 concept concept NOUN iajs-2428 26 3 of of ADP iajs-2428 26 4 complete complete ADJ iajs-2428 26 5 measure measure NOUN iajs-2428 26 6 on on ADP iajs-2428 26 7 department department NOUN iajs-2428 26 8 of of ADP iajs-2428 26 9 mathematic mathematic PROPN iajs-2428 26 10 /college /college PROPN iajs-2428 26 11 of of ADP iajs-2428 26 12 computer computer NOUN iajs-2428 26 13 science science NOUN iajs-2428 26 14 and and CCONJ iajs-2428 26 15 mathematics mathematic NOUN iajs-2428 26 16 /tikrit /tikrit VERB iajs-2428 26 17 university/ university/ NUM iajs-2428 26 18 tikrit tikrit PROPN iajs-2428 26 19 / / SYM iajs-2428 26 20 iraq iraq PROPN iajs-2428 26 21 . . PUNCT iajs-2428 27 1 hassan1962pl@tu.edu.iq hassan1962pl@tu.edu.iq PROPN iajs-2428 27 2     SPACE iajs-2428 27 3 ibn ibn PROPN iajs-2428 27 4 al al PROPN iajs-2428 27 5 haitham haitham PROPN iajs-2428 27 6 journal journal PROPN iajs-2428 27 7 for for ADP iajs-2428 27 8 pure pure ADJ iajs-2428 27 9 and and CCONJ iajs-2428 27 10 applied apply VERB iajs-2428 27 11 science science NOUN iajs-2428 27 12 journal journal PROPN iajs-2428 27 13 homepage homepage NOUN iajs-2428 27 14 : : PUNCT iajs-2428 27 15 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2428 27 16 doi doi NOUN iajs-2428 27 17 : : PUNCT iajs-2428 27 18 10.30526/33.2.2428 10.30526/33.2.2428 NUM iajs-2428 27 19 68@gmail.comrusulsalam6 68@gmail.comrusulsalam6 NOUN iajs-2428 27 20     SPACE iajs-2428 27 21 73 73 NUM iajs-2428 28 1 ibn ibn PROPN iajs-2428 28 2 al al PROPN iajs-2428 28 3 - - PUNCT iajs-2428 28 4 haitham haitham PROPN iajs-2428 28 5 jour jour X iajs-2428 28 6 . . PROPN iajs-2428 28 7 for for ADP iajs-2428 28 8 pure pure ADJ iajs-2428 28 9 & & CCONJ iajs-2428 28 10 appl appl PROPN iajs-2428 28 11 . . PUNCT iajs-2428 29 1 sci sci PROPN iajs-2428 29 2 . . PROPN iajs-2428 30 1 33 33 NUM iajs-2428 30 2 ( ( PUNCT iajs-2428 30 3 2 2 NUM iajs-2428 30 4 ) ) PUNCT iajs-2428 30 5 2020 2020 NUM iajs-2428 30 6 σ σ PROPN iajs-2428 30 7 – – PUNCT iajs-2428 30 8 field field NOUN iajs-2428 30 9 was be AUX iajs-2428 30 10 studied study VERB iajs-2428 30 11 by by ADP iajs-2428 30 12 robert robert PROPN iajs-2428 30 13 in in ADP iajs-2428 30 14 1972 1972 NUM iajs-2428 30 15 , , PUNCT iajs-2428 30 16 but but CCONJ iajs-2428 30 17 not not PART iajs-2428 30 18 necessarily necessarily ADV iajs-2428 30 19 that that SCONJ iajs-2428 30 20 every every DET iajs-2428 30 21 measure measure NOUN iajs-2428 30 22 defined define VERB iajs-2428 30 23 on on ADP iajs-2428 30 24 σ σ PROPN iajs-2428 30 25 – – PUNCT iajs-2428 30 26 field field NOUN iajs-2428 30 27 is be AUX iajs-2428 30 28 complete complete ADJ iajs-2428 30 29 . . PUNCT iajs-2428 31 1 in in ADP iajs-2428 31 2 this this DET iajs-2428 31 3 work work NOUN iajs-2428 31 4 , , PUNCT iajs-2428 31 5 we we PRON iajs-2428 31 6 prove prove VERB iajs-2428 31 7 that that SCONJ iajs-2428 31 8 every every DET iajs-2428 31 9 measure measure NOUN iajs-2428 31 10 defined define VERB iajs-2428 31 11 on on ADP iajs-2428 31 12 λ λ PROPN iajs-2428 31 13 – – PUNCT iajs-2428 31 14 algebra algebra NOUN iajs-2428 31 15 is be AUX iajs-2428 31 16 complete complete ADJ iajs-2428 31 17 . . PUNCT iajs-2428 32 1 the the DET iajs-2428 32 2 main main ADJ iajs-2428 32 3 aim aim NOUN iajs-2428 32 4 of of ADP iajs-2428 32 5 this this DET iajs-2428 32 6 paper paper NOUN iajs-2428 32 7 is be AUX iajs-2428 32 8 to to PART iajs-2428 32 9 introduce introduce VERB iajs-2428 32 10 and and CCONJ iajs-2428 32 11 study study VERB iajs-2428 32 12 new new ADJ iajs-2428 32 13 concept concept NOUN iajs-2428 32 14 such such ADJ iajs-2428 32 15 as as ADP iajs-2428 32 16 λ λ PROPN iajs-2428 32 17 – – PUNCT iajs-2428 32 18 algebra algebra NOUN iajs-2428 32 19 as as ADP iajs-2428 32 20 a a DET iajs-2428 32 21 stronger strong ADJ iajs-2428 32 22 from from ADP iajs-2428 32 23 of of ADP iajs-2428 32 24 α α PRON iajs-2428 32 25 – – PUNCT iajs-2428 32 26 σ σ NOUN iajs-2428 32 27 – – PUNCT iajs-2428 32 28 field field NOUN iajs-2428 32 29 and and CCONJ iajs-2428 32 30 monotone monotone ADJ iajs-2428 32 31 class class NOUN iajs-2428 32 32 . . PUNCT iajs-2428 33 1 and and CCONJ iajs-2428 33 2 we we PRON iajs-2428 33 3 give give VERB iajs-2428 33 4 basic basic ADJ iajs-2428 33 5 properties property NOUN iajs-2428 33 6 and and CCONJ iajs-2428 33 7 examples example NOUN iajs-2428 33 8 of of ADP iajs-2428 33 9 this this DET iajs-2428 33 10 concept concept NOUN iajs-2428 33 11 . . PUNCT iajs-2428 34 1 2 2 X iajs-2428 34 2 . . X iajs-2428 34 3 the the DET iajs-2428 34 4 main main ADJ iajs-2428 34 5 results result NOUN iajs-2428 34 6 : : PUNCT iajs-2428 34 7 let let VERB iajs-2428 34 8 p(𝔓 p(𝔓 NUM iajs-2428 34 9 ) ) PUNCT iajs-2428 34 10 denoted denote VERB iajs-2428 34 11 to to ADP iajs-2428 34 12 the the DET iajs-2428 34 13 power power NOUN iajs-2428 34 14 set set NOUN iajs-2428 34 15 of of ADP iajs-2428 34 16 a a DET iajs-2428 34 17 nonempty nonempty ADV iajs-2428 34 18 set set VERB iajs-2428 34 19 𝔓 𝔓 NOUN iajs-2428 34 20 and and CCONJ iajs-2428 34 21 we we PRON iajs-2428 34 22 start start VERB iajs-2428 34 23 this this DET iajs-2428 34 24 section section NOUN iajs-2428 34 25 by by ADP iajs-2428 34 26 the the DET iajs-2428 34 27 definition definition NOUN iajs-2428 34 28 of of ADP iajs-2428 34 29 λ λ PROPN iajs-2428 34 30 – – PUNCT iajs-2428 34 31 algebra algebra NOUN iajs-2428 34 32 . . PUNCT iajs-2428 35 1 definition definition NOUN iajs-2428 35 2 1 1 NUM iajs-2428 35 3 a a DET iajs-2428 35 4 nonempty nonempty ADJ iajs-2428 35 5 collection collection NOUN iajs-2428 35 6 𝒦 𝒦 PROPN iajs-2428 35 7 of of ADP iajs-2428 35 8 a a DET iajs-2428 35 9 set set ADJ iajs-2428 35 10 𝔓 𝔓 NOUN iajs-2428 35 11 , , PUNCT iajs-2428 35 12 𝒦 𝒦 PROPN iajs-2428 35 13 𝔓 𝔓 PROPN iajs-2428 35 14 is be AUX iajs-2428 35 15 called call VERB iajs-2428 35 16 λ λ NOUN iajs-2428 35 17 – – PUNCT iajs-2428 35 18 algebra algebra NOUN iajs-2428 35 19 or or CCONJ iajs-2428 35 20 ( ( PUNCT iajs-2428 35 21 λ λ NOUN iajs-2428 35 22 – – PUNCT iajs-2428 35 23 field field NOUN iajs-2428 35 24 of of ADP iajs-2428 35 25 a a DET iajs-2428 35 26 set set NOUN iajs-2428 35 27 𝔓 𝔓 NOUN iajs-2428 35 28 if if SCONJ iajs-2428 35 29 : : PUNCT iajs-2428 35 30 1𝔓ϵ𝒦. 1𝔓ϵ𝒦. NUM iajs-2428 35 31 2if 2if NOUN iajs-2428 35 32 dϵ𝒦 dϵ𝒦 VERB iajs-2428 35 33 and and CCONJ iajs-2428 35 34 e e X iajs-2428 35 35 ⊂ ⊂ PROPN iajs-2428 35 36 d d X iajs-2428 35 37 ⊂ ⊂ PROPN iajs-2428 35 38 𝔓 𝔓 PROPN iajs-2428 35 39 , , PUNCT iajs-2428 35 40 then then ADV iajs-2428 35 41 eϵ𝒦. eϵ𝒦. NOUN iajs-2428 36 1 3if 3if PROPN iajs-2428 37 1 d d X iajs-2428 37 2 , , PUNCT iajs-2428 37 3 d d PROPN iajs-2428 37 4 , , PUNCT iajs-2428 37 5 … … PUNCT iajs-2428 37 6 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 37 7 , , PUNCT iajs-2428 37 8 then then ADV iajs-2428 37 9 ⋃ ⋃ PROPN iajs-2428 37 10 d d NOUN iajs-2428 37 11 ϵ𝒦. ϵ𝒦. PROPN iajs-2428 37 12 definition definition NOUN iajs-2428 37 13 2 2 NUM iajs-2428 37 14 if if SCONJ iajs-2428 37 15 𝒦 𝒦 PROPN iajs-2428 37 16 is be AUX iajs-2428 37 17 a a DET iajs-2428 37 18 λ λ NOUN iajs-2428 37 19 – – PUNCT iajs-2428 37 20 algebra algebra NOUN iajs-2428 37 21 of of ADP iajs-2428 37 22 a a DET iajs-2428 37 23 set set ADJ iajs-2428 37 24 𝔓 𝔓 PROPN iajs-2428 37 25 .then .then VERB iajs-2428 37 26 a a DET iajs-2428 37 27 pair pair NOUN iajs-2428 37 28 ( ( PUNCT iajs-2428 37 29 𝔓 𝔓 NOUN iajs-2428 37 30 , , PUNCT iajs-2428 37 31 𝒦 𝒦 PROPN iajs-2428 37 32 ) ) PUNCT iajs-2428 37 33 is be AUX iajs-2428 37 34 called call VERB iajs-2428 37 35 measurable measurable ADJ iajs-2428 37 36 space space NOUN iajs-2428 37 37 relative relative ADJ iajs-2428 37 38 to to ADP iajs-2428 37 39 the the DET iajs-2428 37 40 λ λ NOUN iajs-2428 37 41 – – PUNCT iajs-2428 37 42 algebra algebra NOUN iajs-2428 37 43 𝒦 𝒦 PROPN iajs-2428 37 44 and and CCONJ iajs-2428 37 45 the the DET iajs-2428 37 46 elements element NOUN iajs-2428 37 47 of of ADP iajs-2428 37 48 𝒦 𝒦 PROPN iajs-2428 37 49 are be AUX iajs-2428 37 50 called call VERB iajs-2428 37 51 the the DET iajs-2428 37 52 measurable measurable ADJ iajs-2428 37 53 sets set NOUN iajs-2428 37 54 . . PUNCT iajs-2428 38 1 example example NOUN iajs-2428 38 2 3 3 NUM iajs-2428 38 3 let let VERB iajs-2428 38 4 𝔓 𝔓 PRON iajs-2428 38 5 = = NOUN iajs-2428 38 6 { { PUNCT iajs-2428 38 7 1,2,3,4 1,2,3,4 NUM iajs-2428 38 8 } } PUNCT iajs-2428 38 9 and and CCONJ iajs-2428 38 10 𝒦 𝒦 PROPN iajs-2428 38 11 { { PUNCT iajs-2428 38 12 φ,{1},{2},{4},{1,2},{1,4},{2,4},{1,2,4},𝔓 φ,{1},{2},{4},{1,2},{1,4},{2,4},{1,2,4},𝔓 PROPN iajs-2428 38 13 } } PUNCT iajs-2428 38 14 . . PUNCT iajs-2428 39 1 then then ADV iajs-2428 39 2 ( ( PUNCT iajs-2428 39 3 𝔓 𝔓 NOUN iajs-2428 39 4 , , PUNCT iajs-2428 39 5 𝒦 𝒦 PROPN iajs-2428 39 6 ) ) PUNCT iajs-2428 39 7 is be AUX iajs-2428 39 8 measurable measurable ADJ iajs-2428 39 9 space space NOUN iajs-2428 39 10 relative relative ADJ iajs-2428 39 11 to to ADP iajs-2428 39 12 the the DET iajs-2428 39 13 λ λ NOUN iajs-2428 39 14 – – PUNCT iajs-2428 39 15 algebra algebra NOUN iajs-2428 39 16 𝒦. 𝒦. NOUN iajs-2428 39 17 proposition proposition NOUN iajs-2428 39 18 4 4 NUM iajs-2428 39 19 for for ADP iajs-2428 39 20 any any DET iajs-2428 39 21 λ λ NOUN iajs-2428 39 22 – – PUNCT iajs-2428 39 23 algebra algebra NOUN iajs-2428 39 24 𝒦 𝒦 PROPN iajs-2428 39 25 of of ADP iajs-2428 39 26 a a DET iajs-2428 39 27 set𝔓 set𝔓 NOUN iajs-2428 39 28 , , PUNCT iajs-2428 39 29 the the DET iajs-2428 39 30 following follow VERB iajs-2428 39 31 hold hold NOUN iajs-2428 39 32 : : PUNCT iajs-2428 40 1 1φϵ𝒦 1φϵ𝒦 NUM iajs-2428 40 2 2if 2if ADJ iajs-2428 40 3 d d PROPN iajs-2428 40 4 , , PUNCT iajs-2428 40 5 d d PROPN iajs-2428 40 6 , , PUNCT iajs-2428 40 7 … … PUNCT iajs-2428 40 8 , , PUNCT iajs-2428 40 9 d d X iajs-2428 40 10 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 40 11 , , PUNCT iajs-2428 40 12 then then ADV iajs-2428 40 13 ⋃ ⋃ PROPN iajs-2428 40 14 d d PROPN iajs-2428 40 15 ϵ𝒦. ϵ𝒦. PROPN iajs-2428 40 16 3if 3if PROPN iajs-2428 41 1 d d NOUN iajs-2428 41 2 , , PUNCT iajs-2428 41 3 d d PROPN iajs-2428 41 4 , , PUNCT iajs-2428 41 5 … … PUNCT iajs-2428 41 6 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 41 7 , , PUNCT iajs-2428 41 8 then then ADV iajs-2428 41 9 ⋂ ⋂ PROPN iajs-2428 41 10 d d X iajs-2428 41 11 ϵ𝒦. ϵ𝒦. PROPN iajs-2428 41 12 4if 4if NOUN iajs-2428 42 1 d d NOUN iajs-2428 42 2 , , PUNCT iajs-2428 42 3 d d PROPN iajs-2428 42 4 , , PUNCT iajs-2428 42 5 … … PUNCT iajs-2428 42 6 , , PUNCT iajs-2428 43 1 d d X iajs-2428 43 2 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 43 3 , , PUNCT iajs-2428 43 4 then then ADV iajs-2428 43 5 ⋂ ⋂ PROPN iajs-2428 43 6 d d X iajs-2428 43 7 ϵ𝒦. ϵ𝒦. PROPN iajs-2428 43 8 proof proof NOUN iajs-2428 43 9 the the DET iajs-2428 43 10 proof proof NOUN iajs-2428 43 11 follows follow VERB iajs-2428 43 12 from from ADP iajs-2428 43 13 definition definition NOUN iajs-2428 43 14 of of ADP iajs-2428 43 15 λ λ PROPN iajs-2428 43 16 – – PUNCT iajs-2428 43 17 algebra algebra NOUN iajs-2428 43 18 . . PUNCT iajs-2428 44 1 lemma lemma PROPN iajs-2428 44 2 5 5 NUM iajs-2428 44 3 let let VERB iajs-2428 44 4 𝒦 𝒦 PROPN iajs-2428 44 5 ∈ ∈ PRON iajs-2428 44 6 be be AUX iajs-2428 44 7 a a DET iajs-2428 44 8 collection collection NOUN iajs-2428 44 9 of of ADP iajs-2428 44 10 λ λ PROPN iajs-2428 44 11 – – PUNCT iajs-2428 44 12 algebra algebra NOUN iajs-2428 44 13 on on ADP iajs-2428 44 14 𝔓. 𝔓. PROPN iajs-2428 44 15 then then ADV iajs-2428 44 16 ⋂ ⋂ PROPN iajs-2428 44 17 𝒦∈ 𝒦∈ PROPN iajs-2428 44 18 is be AUX iajs-2428 44 19 a a DET iajs-2428 44 20 λ λ NOUN iajs-2428 44 21 – – PUNCT iajs-2428 44 22 algebra algebra NOUN iajs-2428 44 23 on on ADP iajs-2428 44 24 𝔓. 𝔓. ADJ iajs-2428 44 25 proof proof NOUN iajs-2428 44 26 since since SCONJ iajs-2428 44 27 𝒦 𝒦 PROPN iajs-2428 44 28 is be AUX iajs-2428 44 29 λ λ NOUN iajs-2428 44 30 – – PUNCT iajs-2428 44 31 algebra algebra NOUN iajs-2428 44 32 ∀ ∀ PUNCT iajs-2428 45 1 α α X iajs-2428 45 2 ∈ ∈ PROPN iajs-2428 45 3 ι ι X iajs-2428 45 4 , , PUNCT iajs-2428 45 5 then then ADV iajs-2428 45 6 𝔓ϵ𝒦 𝔓ϵ𝒦 X iajs-2428 45 7 ∀ ∀ X iajs-2428 45 8 α α X iajs-2428 45 9 ∈ ∈ PROPN iajs-2428 45 10 ι ι NOUN iajs-2428 45 11 , , PUNCT iajs-2428 45 12 hence hence ADV iajs-2428 45 13 𝒦 𝒦 PROPN iajs-2428 45 14 φ φ PROPN iajs-2428 45 15 ∀α ∀α VERB iajs-2428 45 16 ∈ ∈ PROPN iajs-2428 45 17 ι ι X iajs-2428 46 1 and and CCONJ iajs-2428 46 2 ⋂ ⋂ PROPN iajs-2428 46 3 𝒦∈ 𝒦∈ PROPN iajs-2428 46 4 φ φ PROPN iajs-2428 46 5 , , PUNCT iajs-2428 46 6 therefore therefore ADV iajs-2428 46 7 𝔓 𝔓 PROPN iajs-2428 47 1 ϵ ϵ X iajs-2428 47 2 ⋂ ⋂ PROPN iajs-2428 48 1 𝒦∈ 𝒦∈ X iajs-2428 48 2 . . PUNCT iajs-2428 49 1 let let VERB iajs-2428 49 2 dϵ dϵ VERB iajs-2428 49 3 ⋂ ⋂ PROPN iajs-2428 50 1 𝒦∈ 𝒦∈ PROPN iajs-2428 50 2 and and CCONJ iajs-2428 50 3 e e X iajs-2428 50 4 ⊂ ⊂ PROPN iajs-2428 50 5 d d X iajs-2428 50 6 ⊂ ⊂ PROPN iajs-2428 50 7 𝔓 𝔓 PROPN iajs-2428 50 8 , , PUNCT iajs-2428 50 9 then then ADV iajs-2428 50 10 dϵ𝒦 dϵ𝒦 VERB iajs-2428 50 11 ∀α ∀α X iajs-2428 50 12 ∈ ∈ PROPN iajs-2428 50 13 ι ι NOUN iajs-2428 50 14 , , PUNCT iajs-2428 50 15 but but CCONJ iajs-2428 50 16 𝒦 𝒦 PROPN iajs-2428 50 17 is be AUX iajs-2428 50 18 λ λ NOUN iajs-2428 50 19 – – PUNCT iajs-2428 50 20 algebra algebra NOUN iajs-2428 50 21 ∀ ∀ NUM iajs-2428 50 22 α α PRON iajs-2428 50 23 ∈ ∈ X iajs-2428 50 24 ι ι X iajs-2428 50 25 and and CCONJ iajs-2428 50 26 e e PROPN iajs-2428 50 27 ⊂ ⊂ PROPN iajs-2428 50 28 d. d. PROPN iajs-2428 51 1 so so ADV iajs-2428 51 2 , , PUNCT iajs-2428 51 3 we we PRON iajs-2428 51 4 get get VERB iajs-2428 51 5 eϵ𝒦 eϵ𝒦 PROPN iajs-2428 51 6 ∀α ∀α NOUN iajs-2428 51 7 ∈ ∈ PROPN iajs-2428 51 8 ι ι NOUN iajs-2428 51 9 , , PUNCT iajs-2428 51 10 hence hence ADV iajs-2428 51 11 eϵ eϵ ADP iajs-2428 51 12 ⋂ ⋂ PROPN iajs-2428 52 1 𝒦∈ 𝒦∈ PROPN iajs-2428 52 2 . . PUNCT iajs-2428 53 1 let let VERB iajs-2428 53 2 d d NOUN iajs-2428 53 3 , , PUNCT iajs-2428 53 4 d d PROPN iajs-2428 53 5 , , PUNCT iajs-2428 53 6 … … PUNCT iajs-2428 53 7 ϵ ϵ X iajs-2428 53 8 ⋂ ⋂ PROPN iajs-2428 53 9 𝒦∈ 𝒦∈ PROPN iajs-2428 53 10 .then .then X iajs-2428 53 11 , , PUNCT iajs-2428 53 12 d d INTJ iajs-2428 53 13 , , PUNCT iajs-2428 53 14 d d PROPN iajs-2428 53 15 , , PUNCT iajs-2428 53 16 … … PUNCT iajs-2428 53 17 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 53 18 , , PUNCT iajs-2428 53 19 ∀α ∀α VERB iajs-2428 53 20 ∈ ∈ PROPN iajs-2428 53 21 ι ι NOUN iajs-2428 53 22 , , PUNCT iajs-2428 53 23 but but CCONJ iajs-2428 53 24 𝒦 𝒦 PROPN iajs-2428 53 25 is be AUX iajs-2428 53 26 λ λ NOUN iajs-2428 53 27 – – PUNCT iajs-2428 53 28 algebra algebra NOUN iajs-2428 53 29 ∀ ∀ PUNCT iajs-2428 54 1 α α X iajs-2428 54 2 ∈ ∈ PROPN iajs-2428 54 3 ι ι X iajs-2428 54 4 which which PRON iajs-2428 54 5 implies imply VERB iajs-2428 54 6 that that SCONJ iajs-2428 54 7 ⋃ ⋃ PROPN iajs-2428 54 8 d d PROPN iajs-2428 54 9 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 54 10 , , PUNCT iajs-2428 54 11 ∀α ∀α VERB iajs-2428 54 12 ∈ ∈ PROPN iajs-2428 54 13 ι ι NOUN iajs-2428 54 14 , , PUNCT iajs-2428 54 15 hence hence ADV iajs-2428 54 16 ⋃ ⋃ PUNCT iajs-2428 54 17 d d NOUN iajs-2428 54 18 ϵ ϵ X iajs-2428 54 19 ⋂ ⋂ PROPN iajs-2428 54 20 𝒦∈ 𝒦∈ PROPN iajs-2428 54 21 . . PUNCT iajs-2428 55 1 therefore therefore ADV iajs-2428 55 2 , , PUNCT iajs-2428 55 3 ⋂ ⋂ PROPN iajs-2428 55 4 𝒦∈ 𝒦∈ PROPN iajs-2428 55 5 is be AUX iajs-2428 55 6 a a DET iajs-2428 55 7 λ λ NOUN iajs-2428 55 8 – – PUNCT iajs-2428 55 9 algebra algebra NOUN iajs-2428 55 10 . . PUNCT iajs-2428 55 11     SPACE iajs-2428 56 1 74 74 NUM iajs-2428 56 2 ibn ibn PROPN iajs-2428 56 3 al al PROPN iajs-2428 56 4 - - PUNCT iajs-2428 56 5 haitham haitham PROPN iajs-2428 56 6 jour jour X iajs-2428 56 7 . . PROPN iajs-2428 56 8 for for ADP iajs-2428 56 9 pure pure ADJ iajs-2428 56 10 & & CCONJ iajs-2428 56 11 appl appl PROPN iajs-2428 56 12 . . PUNCT iajs-2428 57 1 sci sci PROPN iajs-2428 57 2 . . PROPN iajs-2428 58 1 33 33 NUM iajs-2428 58 2 ( ( PUNCT iajs-2428 58 3 2 2 NUM iajs-2428 58 4 ) ) PUNCT iajs-2428 58 5 2020 2020 NUM iajs-2428 58 6 definition definition NOUN iajs-2428 58 7 6 6 NUM iajs-2428 58 8 let let VERB iajs-2428 58 9 𝒥 𝒥 PRON iajs-2428 58 10 ⊆ ⊆ NUM iajs-2428 58 11 p p PROPN iajs-2428 58 12 𝔓 𝔓 PROPN iajs-2428 58 13 . . PUNCT iajs-2428 59 1 then then ADV iajs-2428 59 2 the the DET iajs-2428 59 3 intersection intersection NOUN iajs-2428 59 4 of of ADP iajs-2428 59 5 all all DET iajs-2428 59 6 λ λ PROPN iajs-2428 59 7 – – PUNCT iajs-2428 59 8 algebra algebra NOUN iajs-2428 59 9 of of ADP iajs-2428 59 10 𝔓 𝔓 PROPN iajs-2428 59 11 which which PRON iajs-2428 59 12 includes include VERB iajs-2428 59 13 𝒥 𝒥 PRON iajs-2428 59 14 is be AUX iajs-2428 59 15 called call VERB iajs-2428 59 16 the the DET iajs-2428 59 17 λ λ PROPN iajs-2428 59 18 – – PUNCT iajs-2428 59 19 algebra algebra NOUN iajs-2428 59 20 generated generate VERB iajs-2428 59 21 by by ADP iajs-2428 59 22 𝒥 𝒥 PROPN iajs-2428 59 23 and and CCONJ iajs-2428 59 24 denoted denote VERB iajs-2428 59 25 by by ADP iajs-2428 59 26 λ λ PROPN iajs-2428 59 27 𝒥 𝒥 PROPN iajs-2428 59 28 , , PUNCT iajs-2428 59 29 that that ADV iajs-2428 59 30 is is ADV iajs-2428 59 31 , , PUNCT iajs-2428 60 1 λ λ PROPN iajs-2428 60 2 𝒥 𝒥 NOUN iajs-2428 61 1 = = SYM iajs-2428 62 1 ⋂ ⋂ PROPN iajs-2428 63 1 𝒦 𝒦 NOUN iajs-2428 63 2 : : PUNCT iajs-2428 63 3 𝒦 𝒦 PROPN iajs-2428 63 4 is be AUX iajs-2428 63 5 a a DET iajs-2428 63 6 λ λ NOUN iajs-2428 63 7 – – PUNCT iajs-2428 63 8 algebra algebra NOUN iajs-2428 63 9 of of ADP iajs-2428 63 10 𝔓 𝔓 PROPN iajs-2428 63 11 and and CCONJ iajs-2428 63 12 ⊆ ⊆ NUM iajs-2428 63 13 𝒦 𝒦 PROPN iajs-2428 63 14 , , PUNCT iajs-2428 63 15 ∀α ∀α VERB iajs-2428 63 16 ∈ ∈ PROPN iajs-2428 63 17 ι ι X iajs-2428 63 18 . . PUNCT iajs-2428 64 1 proposition proposition NOUN iajs-2428 64 2 7 7 NUM iajs-2428 64 3 let let VERB iajs-2428 64 4 𝒥 𝒥 PRON iajs-2428 64 5 ⊆ ⊆ NUM iajs-2428 64 6 p p PROPN iajs-2428 64 7 𝔓 𝔓 PROPN iajs-2428 64 8 . . PUNCT iajs-2428 65 1 then then ADV iajs-2428 65 2 λ λ X iajs-2428 65 3 𝒥 𝒥 PROPN iajs-2428 65 4 is be AUX iajs-2428 65 5 the the DET iajs-2428 65 6 smallest small ADJ iajs-2428 65 7 λ λ NOUN iajs-2428 65 8 – – PUNCT iajs-2428 65 9 algebra algebra NOUN iajs-2428 65 10 of of ADP iajs-2428 65 11 𝔓 𝔓 PROPN iajs-2428 65 12 which which PRON iajs-2428 65 13 includes include VERB iajs-2428 65 14 𝒥. 𝒥. NOUN iajs-2428 65 15 proof proof NOUN iajs-2428 65 16 since since SCONJ iajs-2428 65 17 λ λ PROPN iajs-2428 65 18 𝒥 𝒥 PROPN iajs-2428 65 19 = = PROPN iajs-2428 65 20 ⋂ ⋂ PROPN iajs-2428 65 21 𝒦 𝒦 NOUN iajs-2428 65 22 : : PUNCT iajs-2428 65 23 𝒦 𝒦 PROPN iajs-2428 65 24 is be AUX iajs-2428 65 25 a a DET iajs-2428 65 26 λ λ NOUN iajs-2428 65 27 – – PUNCT iajs-2428 65 28 algebra algebra NOUN iajs-2428 65 29 of of ADP iajs-2428 65 30 𝔓 𝔓 PROPN iajs-2428 65 31 and and CCONJ iajs-2428 65 32 𝒥 𝒥 PROPN iajs-2428 65 33 ⊆ ⊆ NUM iajs-2428 65 34 𝒦 𝒦 PROPN iajs-2428 65 35 , , PUNCT iajs-2428 65 36 ∀α ∀α VERB iajs-2428 65 37 ∈ ∈ PROPN iajs-2428 65 38 ι ι X iajs-2428 65 39 . . PUNCT iajs-2428 66 1 then then ADV iajs-2428 66 2 λ λ X iajs-2428 66 3 𝒥 𝒥 PROPN iajs-2428 66 4 is be AUX iajs-2428 66 5 λ λ NOUN iajs-2428 66 6 – – PUNCT iajs-2428 66 7 algebra algebra NOUN iajs-2428 66 8 of of ADP iajs-2428 66 9 𝔓 𝔓 PROPN iajs-2428 66 10 by by ADP iajs-2428 66 11 lemma lemma PROPN iajs-2428 66 12 5 5 NUM iajs-2428 66 13 . . PUNCT iajs-2428 66 14 to to PART iajs-2428 66 15 prove prove VERB iajs-2428 66 16 λ λ PROPN iajs-2428 66 17 𝒥 𝒥 PROPN iajs-2428 66 18 ⊇ ⊇ PROPN iajs-2428 66 19 𝒥 𝒥 PROPN iajs-2428 66 20 , , PUNCT iajs-2428 66 21 let let VERB iajs-2428 66 22 each each PRON iajs-2428 66 23 of of ADP iajs-2428 66 24 𝒦 𝒦 PROPN iajs-2428 66 25 is be AUX iajs-2428 66 26 a a DET iajs-2428 66 27 λ λ NOUN iajs-2428 66 28 – – PUNCT iajs-2428 66 29 algebra algebra NOUN iajs-2428 66 30 of of ADP iajs-2428 66 31 𝔓 𝔓 PROPN iajs-2428 66 32 and and CCONJ iajs-2428 66 33 𝒥 𝒥 PROPN iajs-2428 66 34 ⊆ ⊆ NUM iajs-2428 66 35 𝒦 𝒦 PROPN iajs-2428 66 36 , , PUNCT iajs-2428 66 37 ∀α ∀α VERB iajs-2428 66 38 ∈ ∈ PROPN iajs-2428 66 39 ι ι X iajs-2428 66 40 . . PUNCT iajs-2428 67 1 then then ADV iajs-2428 67 2 𝒥 𝒥 PROPN iajs-2428 67 3 ⊆ ⊆ NUM iajs-2428 67 4 ⋂ ⋂ PROPN iajs-2428 67 5 𝒦∈ 𝒦∈ NOUN iajs-2428 67 6 , , PUNCT iajs-2428 67 7 therefore therefore ADV iajs-2428 67 8 𝒥 𝒥 PROPN iajs-2428 67 9 ⊆ ⊆ NUM iajs-2428 67 10 λ λ PROPN iajs-2428 67 11 𝒥 𝒥 PROPN iajs-2428 67 12 . . PUNCT iajs-2428 68 1 now now ADV iajs-2428 68 2 , , PUNCT iajs-2428 68 3 let let VERB iajs-2428 68 4 𝒦∗ 𝒦∗ PRON iajs-2428 68 5 is be AUX iajs-2428 68 6 a a DET iajs-2428 68 7 λ λ NOUN iajs-2428 68 8 – – PUNCT iajs-2428 68 9 algebra algebra NOUN iajs-2428 68 10 of of ADP iajs-2428 68 11 𝔓 𝔓 PROPN iajs-2428 69 1 such such ADJ iajs-2428 69 2 that that SCONJ iajs-2428 69 3 𝒦∗ 𝒦∗ DET iajs-2428 69 4 ⊇ ⊇ PROPN iajs-2428 69 5 𝒥. 𝒥. PROPN iajs-2428 69 6 then then ADV iajs-2428 69 7 ⋂ ⋂ PROPN iajs-2428 69 8 𝒦 𝒦 NOUN iajs-2428 69 9 : : PUNCT iajs-2428 69 10 𝒦 𝒦 PROPN iajs-2428 69 11 is be AUX iajs-2428 69 12 a a DET iajs-2428 69 13 λ λ NOUN iajs-2428 69 14 – – PUNCT iajs-2428 69 15 algebra algebra NOUN iajs-2428 69 16 of of ADP iajs-2428 69 17 𝔓 𝔓 PROPN iajs-2428 69 18 and and CCONJ iajs-2428 69 19 𝒥 𝒥 PROPN iajs-2428 69 20 ⊆ ⊆ NUM iajs-2428 69 21 𝒦 𝒦 PROPN iajs-2428 69 22 , , PUNCT iajs-2428 69 23 ∀α ∀α VERB iajs-2428 69 24 ∈ ∈ PROPN iajs-2428 69 25 ι ι ADP iajs-2428 69 26 ⊆ ⊆ NUM iajs-2428 69 27 𝒦∗ 𝒦∗ NOUN iajs-2428 69 28 , , PUNCT iajs-2428 69 29 hence hence ADV iajs-2428 69 30 λ λ X iajs-2428 69 31 𝒥 𝒥 PROPN iajs-2428 69 32 ⊆ ⊆ NUM iajs-2428 69 33 𝒦∗. 𝒦∗. ADP iajs-2428 69 34 therefore therefore ADV iajs-2428 69 35 , , PUNCT iajs-2428 69 36 λ λ PROPN iajs-2428 69 37 𝒥 𝒥 PROPN iajs-2428 69 38 is be AUX iajs-2428 69 39 the the DET iajs-2428 69 40 smallest small ADJ iajs-2428 69 41 λ λ NOUN iajs-2428 69 42 – – PUNCT iajs-2428 69 43 algebra algebra NOUN iajs-2428 69 44 of of ADP iajs-2428 69 45 𝔓 𝔓 PROPN iajs-2428 69 46 which which PRON iajs-2428 69 47 includes include VERB iajs-2428 69 48 𝒥. 𝒥. NOUN iajs-2428 69 49 if if SCONJ iajs-2428 69 50 we we PRON iajs-2428 69 51 take take VERB iajs-2428 69 52 example example NOUN iajs-2428 69 53 3 3 NUM iajs-2428 69 54 and and CCONJ iajs-2428 69 55 if if SCONJ iajs-2428 69 56 we we PRON iajs-2428 69 57 assume assume VERB iajs-2428 69 58 𝒥 𝒥 PRON iajs-2428 69 59 = = NOUN iajs-2428 69 60 { { PUNCT iajs-2428 69 61 { { PUNCT iajs-2428 69 62 1},{2 1},{2 NUM iajs-2428 69 63 } } PUNCT iajs-2428 69 64 } } PUNCT iajs-2428 69 65 , , PUNCT iajs-2428 69 66 then then ADV iajs-2428 69 67 λ λ X iajs-2428 69 68 𝒥 𝒥 PROPN iajs-2428 69 69 = = NOUN iajs-2428 69 70 { { PUNCT iajs-2428 69 71 φ,{1},{2},{1,2 φ,{1},{2},{1,2 NOUN iajs-2428 69 72 } } PUNCT iajs-2428 69 73 , , PUNCT iajs-2428 69 74 𝔓 𝔓 NOUN iajs-2428 69 75 } } PUNCT iajs-2428 69 76 is be AUX iajs-2428 69 77 the the DET iajs-2428 69 78 smallest small ADJ iajs-2428 69 79 λ λ NOUN iajs-2428 69 80 – – PUNCT iajs-2428 69 81 algebra algebra NOUN iajs-2428 69 82 of of ADP iajs-2428 69 83 a a DET iajs-2428 69 84 set set NOUN iajs-2428 69 85 𝔓 𝔓 NOUN iajs-2428 69 86 which which PRON iajs-2428 69 87 includes include VERB iajs-2428 69 88 𝒥. 𝒥. NOUN iajs-2428 69 89 theorem theorem ADJ iajs-2428 69 90 8 8 NUM iajs-2428 69 91 let let VERB iajs-2428 69 92 𝒥 𝒥 PRON iajs-2428 69 93 ⊆ ⊆ NUM iajs-2428 69 94 p p PROPN iajs-2428 69 95 𝔓 𝔓 PROPN iajs-2428 69 96 . . PUNCT iajs-2428 70 1 then then ADV iajs-2428 70 2 ( ( PUNCT iajs-2428 70 3 𝔓 𝔓 PROPN iajs-2428 70 4 , , PUNCT iajs-2428 70 5 𝒥 𝒥 PROPN iajs-2428 70 6 ) ) PUNCT iajs-2428 70 7 is be AUX iajs-2428 70 8 measurable measurable ADJ iajs-2428 70 9 space space NOUN iajs-2428 70 10 relative relative ADJ iajs-2428 70 11 to to ADP iajs-2428 70 12 the the DET iajs-2428 70 13 λ λ NOUN iajs-2428 70 14 – – PUNCT iajs-2428 70 15 algebra algebra NOUN iajs-2428 70 16 𝒥. 𝒥. NOUN iajs-2428 70 17 if if SCONJ iajs-2428 71 1 and and CCONJ iajs-2428 71 2 only only ADV iajs-2428 71 3 if if SCONJ iajs-2428 71 4 𝒥 𝒥 PROPN iajs-2428 71 5 λ λ VERB iajs-2428 71 6 𝒥 𝒥 PROPN iajs-2428 71 7 . . PUNCT iajs-2428 72 1 proof proof NOUN iajs-2428 72 2 suppose suppose VERB iajs-2428 72 3 that that SCONJ iajs-2428 72 4 ( ( PUNCT iajs-2428 72 5 𝔓 𝔓 PROPN iajs-2428 72 6 , , PUNCT iajs-2428 72 7 𝒥 𝒥 PROPN iajs-2428 72 8 ) ) PUNCT iajs-2428 72 9 is be AUX iajs-2428 72 10 ( ( PUNCT iajs-2428 72 11 a a PRON iajs-2428 72 12 ) ) PUNCT iajs-2428 72 13 measurable measurable ADJ iajs-2428 72 14 space space NOUN iajs-2428 72 15 relative relative ADJ iajs-2428 72 16 to to ADP iajs-2428 72 17 the the DET iajs-2428 72 18 λ λ NOUN iajs-2428 72 19 – – PUNCT iajs-2428 72 20 algebra algebra NOUN iajs-2428 72 21 𝒥. 𝒥. NOUN iajs-2428 72 22 from from ADP iajs-2428 72 23 proposition proposition NOUN iajs-2428 72 24 7 7 NUM iajs-2428 72 25 , , PUNCT iajs-2428 72 26 we we PRON iajs-2428 72 27 have have VERB iajs-2428 72 28 λ λ X iajs-2428 72 29 𝒥 𝒥 PROPN iajs-2428 72 30 is be AUX iajs-2428 72 31 the the DET iajs-2428 72 32 smallest small ADJ iajs-2428 72 33 λ λ NOUN iajs-2428 72 34 – – PUNCT iajs-2428 72 35 algebra algebra NOUN iajs-2428 72 36 of of ADP iajs-2428 72 37 a a DET iajs-2428 72 38 set set NOUN iajs-2428 72 39 𝔓 𝔓 NOUN iajs-2428 72 40 which which PRON iajs-2428 72 41 includes include VERB iajs-2428 72 42 𝒥 𝒥 PROPN iajs-2428 72 43 implies imply VERB iajs-2428 72 44 that𝒥 that𝒥 PROPN iajs-2428 72 45 ⊆ ⊆ NUM iajs-2428 72 46 λ λ X iajs-2428 72 47 𝒥 𝒥 PROPN iajs-2428 72 48 . . PUNCT iajs-2428 73 1 by by ADP iajs-2428 73 2 hypothesis hypothesis NOUN iajs-2428 73 3 , , PUNCT iajs-2428 73 4 we we PRON iajs-2428 73 5 have have VERB iajs-2428 73 6 𝒥 𝒥 PROPN iajs-2428 73 7 is be AUX iajs-2428 73 8 a a DET iajs-2428 73 9 λ λ NOUN iajs-2428 73 10 – – PUNCT iajs-2428 73 11 algebra algebra NOUN iajs-2428 73 12 of of ADP iajs-2428 73 13 a a DET iajs-2428 73 14 set𝔓 set𝔓 NOUN iajs-2428 73 15 , , PUNCT iajs-2428 73 16 but but CCONJ iajs-2428 73 17 𝒥 𝒥 PROPN iajs-2428 73 18 ⊆ ⊆ NUM iajs-2428 73 19 𝒥 𝒥 PROPN iajs-2428 73 20 and and CCONJ iajs-2428 73 21 λ λ X iajs-2428 73 22 𝒥 𝒥 PROPN iajs-2428 73 23 is be AUX iajs-2428 73 24 the the DET iajs-2428 73 25 smallest small ADJ iajs-2428 73 26 λ λ NOUN iajs-2428 73 27 – – PUNCT iajs-2428 73 28 algebra algebra NOUN iajs-2428 73 29 of of ADP iajs-2428 73 30 a a DET iajs-2428 73 31 set set NOUN iajs-2428 73 32 𝔓 𝔓 NOUN iajs-2428 73 33 which which PRON iajs-2428 73 34 includes include VERB iajs-2428 73 35 𝒥 𝒥 PROPN iajs-2428 73 36 , , PUNCT iajs-2428 73 37 then then ADV iajs-2428 73 38 λ λ X iajs-2428 73 39 𝒥 𝒥 PROPN iajs-2428 73 40 ⊆ ⊆ NUM iajs-2428 73 41 𝒥 𝒥 PROPN iajs-2428 73 42 , , PUNCT iajs-2428 73 43 hence hence ADV iajs-2428 73 44 𝒥 𝒥 PROPN iajs-2428 74 1 λ λ PROPN iajs-2428 74 2 𝒥 𝒥 PROPN iajs-2428 74 3 . . PUNCT iajs-2428 75 1 conversely conversely ADV iajs-2428 75 2 ) ) PUNCT iajs-2428 75 3 let let VERB iajs-2428 75 4 𝒥 𝒥 PRON iajs-2428 75 5 ⊆ ⊆ NUM iajs-2428 75 6 p p NOUN iajs-2428 75 7 𝔓 𝔓 PROPN iajs-2428 75 8 and and CCONJ iajs-2428 75 9 let let VERB iajs-2428 75 10 𝒥 𝒥 PRON iajs-2428 75 11 λ λ VERB iajs-2428 75 12 𝒥 𝒥 PROPN iajs-2428 75 13 . . PUNCT iajs-2428 76 1 since since SCONJ iajs-2428 76 2 λ λ PROPN iajs-2428 76 3 𝒥 𝒥 PROPN iajs-2428 76 4 is be AUX iajs-2428 76 5 a a DET iajs-2428 76 6 λ λ NOUN iajs-2428 76 7 – – PUNCT iajs-2428 76 8 algebra algebra NOUN iajs-2428 76 9 of of ADP iajs-2428 76 10 a a DET iajs-2428 76 11 set set ADJ iajs-2428 76 12 𝔓 𝔓 NOUN iajs-2428 76 13 , , PUNCT iajs-2428 76 14 then then ADV iajs-2428 76 15 𝒥 𝒥 PROPN iajs-2428 76 16 is be AUX iajs-2428 76 17 λ λ NOUN iajs-2428 76 18 – – PUNCT iajs-2428 76 19 algebra algebra NOUN iajs-2428 76 20 of of ADP iajs-2428 76 21 a a DET iajs-2428 76 22 set set NOUN iajs-2428 76 23 𝔓. 𝔓. NOUN iajs-2428 76 24 if if SCONJ iajs-2428 76 25 we we PRON iajs-2428 76 26 take take VERB iajs-2428 76 27 example example NOUN iajs-2428 76 28 3 3 NUM iajs-2428 76 29 and and CCONJ iajs-2428 76 30 if if SCONJ iajs-2428 76 31 we we PRON iajs-2428 76 32 assume assume VERB iajs-2428 76 33 𝒥 𝒥 PROPN iajs-2428 76 34 = = PROPN iajs-2428 76 35 { { PUNCT iajs-2428 76 36 φ,{1},𝔓 φ,{1},𝔓 PROPN iajs-2428 76 37 } } PUNCT iajs-2428 76 38 , , PUNCT iajs-2428 76 39 then then ADV iajs-2428 76 40 we we PRON iajs-2428 76 41 conclude conclude VERB iajs-2428 76 42 that that PRON iajs-2428 77 1 λ λ PROPN iajs-2428 77 2 𝒥 𝒥 NOUN iajs-2428 77 3 = = SYM iajs-2428 77 4 𝒥. 𝒥. PROPN iajs-2428 77 5 now now ADV iajs-2428 77 6 , , PUNCT iajs-2428 77 7 we we PRON iajs-2428 77 8 introduce introduce VERB iajs-2428 77 9 the the DET iajs-2428 77 10 notion notion NOUN iajs-2428 77 11 of of ADP iajs-2428 77 12 restriction restriction NOUN iajs-2428 77 13 and and CCONJ iajs-2428 77 14 study study VERB iajs-2428 77 15 the the DET iajs-2428 77 16 basic basic ADJ iajs-2428 77 17 properties property NOUN iajs-2428 77 18 of of ADP iajs-2428 77 19 this this DET iajs-2428 77 20 notion notion NOUN iajs-2428 77 21 . . PUNCT iajs-2428 78 1 definition definition NOUN iajs-2428 78 2 9 9 NUM iajs-2428 78 3 let let VERB iajs-2428 78 4 𝒦 𝒦 PROPN iajs-2428 78 5 ⊆ ⊆ NUM iajs-2428 78 6 p p NOUN iajs-2428 78 7 𝔓 𝔓 PROPN iajs-2428 79 1 and and CCONJ iajs-2428 79 2 φ φ PRON iajs-2428 79 3 𝔇 𝔇 PROPN iajs-2428 79 4 ⊆ ⊆ NUM iajs-2428 79 5 𝔓 𝔓 PROPN iajs-2428 79 6 . . PUNCT iajs-2428 80 1 then then ADV iajs-2428 80 2 , , PUNCT iajs-2428 80 3 the the DET iajs-2428 80 4 restriction restriction NOUN iajs-2428 80 5 of of ADP iajs-2428 80 6 𝒦 𝒦 PROPN iajs-2428 80 7 over over ADP iajs-2428 80 8 the the DET iajs-2428 80 9 set set NOUN iajs-2428 80 10 𝔇 𝔇 PROPN iajs-2428 80 11 is be AUX iajs-2428 80 12 denoted denote VERB iajs-2428 80 13 by by ADP iajs-2428 80 14 𝒦|𝔇 𝒦|𝔇 PRON iajs-2428 80 15 and and CCONJ iajs-2428 80 16 defined define VERB iajs-2428 80 17 as as SCONJ iajs-2428 80 18 follows follow VERB iajs-2428 80 19 : : PUNCT iajs-2428 80 20 𝒦|𝔇 𝒦|𝔇 X iajs-2428 81 1 = = PUNCT iajs-2428 81 2 { { PUNCT iajs-2428 81 3 b b NOUN iajs-2428 81 4 : : PUNCT iajs-2428 81 5 b b NOUN iajs-2428 81 6 = = NOUN iajs-2428 81 7 e⋂𝔇 e⋂𝔇 NOUN iajs-2428 81 8 , , PUNCT iajs-2428 81 9 for for ADP iajs-2428 81 10 some some PRON iajs-2428 81 11 eϵ eϵ NOUN iajs-2428 81 12 𝒦 𝒦 NOUN iajs-2428 81 13 } } PUNCT iajs-2428 81 14 . . PUNCT iajs-2428 82 1 proposition10 proposition10 PROPN iajs-2428 82 2 let let VERB iajs-2428 82 3 ( ( PUNCT iajs-2428 82 4 𝔓 𝔓 NOUN iajs-2428 82 5 , , PUNCT iajs-2428 82 6 𝒦 𝒦 PROPN iajs-2428 82 7 ) ) PUNCT iajs-2428 82 8 is be AUX iajs-2428 82 9 measurable measurable ADJ iajs-2428 82 10 space space NOUN iajs-2428 82 11 relative relative ADJ iajs-2428 82 12 to to ADP iajs-2428 82 13 the the DET iajs-2428 82 14 λ λ NOUN iajs-2428 82 15 – – PUNCT iajs-2428 82 16 algebra algebra NOUN iajs-2428 82 17 𝒦 𝒦 PROPN iajs-2428 82 18 and and CCONJ iajs-2428 82 19 φ φ NUM iajs-2428 82 20 𝔇 𝔇 PROPN iajs-2428 82 21 ⊆ ⊆ NUM iajs-2428 82 22 𝔓. 𝔓. PROPN iajs-2428 82 23 then then ADV iajs-2428 83 1 𝒦|𝔇 𝒦|𝔇 X iajs-2428 83 2 = = PUNCT iajs-2428 83 3 { { PUNCT iajs-2428 83 4 e e NOUN iajs-2428 83 5 ⊆ ⊆ NUM iajs-2428 83 6 𝔇 𝔇 PROPN iajs-2428 83 7 : : PUNCT iajs-2428 83 8 eϵ eϵ X iajs-2428 83 9 𝒦 𝒦 PROPN iajs-2428 83 10 } } PUNCT iajs-2428 83 11 . . PUNCT iajs-2428 84 1 proof proof NOUN iajs-2428 84 2 let let VERB iajs-2428 84 3 bϵ bϵ NOUN iajs-2428 84 4 𝒦|𝔇 𝒦|𝔇 X iajs-2428 84 5 . . PUNCT iajs-2428 85 1 then then ADV iajs-2428 85 2 b b X iajs-2428 85 3 = = NOUN iajs-2428 85 4 e⋂𝔇 e⋂𝔇 NOUN iajs-2428 85 5 , , PUNCT iajs-2428 85 6 for for ADP iajs-2428 85 7 some some PRON iajs-2428 85 8 eϵ𝒦. eϵ𝒦. PUNCT iajs-2428 85 9 since since SCONJ iajs-2428 85 10 e⋂𝔇 e⋂𝔇 VERB iajs-2428 85 11 ⊆ ⊆ NUM iajs-2428 85 12 e e NOUN iajs-2428 85 13 and and CCONJ iajs-2428 85 14 𝒦 𝒦 PROPN iajs-2428 85 15 is be AUX iajs-2428 85 16 λ λ NOUN iajs-2428 85 17 – – PUNCT iajs-2428 85 18 algebra algebra NOUN iajs-2428 85 19 of of ADP iajs-2428 85 20 a a DET iajs-2428 85 21 set set ADJ iajs-2428 85 22 𝔓 𝔓 NOUN iajs-2428 85 23 , , PUNCT iajs-2428 85 24 then then ADV iajs-2428 85 25 e⋂𝔇ϵ𝒦 e⋂𝔇ϵ𝒦 ADV iajs-2428 85 26 , , PUNCT iajs-2428 85 27 hence hence ADV iajs-2428 85 28 bϵ bϵ ADP iajs-2428 85 29 𝒦. 𝒦. PROPN iajs-2428 85 30 since since SCONJ iajs-2428 85 31 , , PUNCT iajs-2428 85 32 e⋂𝔇 e⋂𝔇 VERB iajs-2428 85 33 ⊆ ⊆ NUM iajs-2428 85 34 𝔇 𝔇 PROPN iajs-2428 85 35 , , PUNCT iajs-2428 85 36 then then ADV iajs-2428 85 37 b⊆ b⊆ PROPN iajs-2428 85 38 𝔇. 𝔇. PROPN iajs-2428 85 39 therefore therefore ADV iajs-2428 85 40 bϵ{𝐸 bϵ{𝐸 VERB iajs-2428 85 41 ⊆ ⊆ NUM iajs-2428 85 42 𝔇:eϵ 𝔇:eϵ ADJ iajs-2428 85 43 𝒦 𝒦 PROPN iajs-2428 85 44 } } PUNCT iajs-2428 85 45 and and CCONJ iajs-2428 85 46 𝒦|𝔇 𝒦|𝔇 ADJ iajs-2428 85 47 ⊆{a ⊆{a NOUN iajs-2428 85 48 ⊆ ⊆ NUM iajs-2428 85 49 𝔇:aϵ 𝔇:aϵ PUNCT iajs-2428 85 50 𝒦 𝒦 NOUN iajs-2428 85 51 } } PUNCT iajs-2428 85 52 . . PUNCT iajs-2428 86 1 let let VERB iajs-2428 86 2 cϵ{𝐸 cϵ{𝐸 VERB iajs-2428 86 3 ⊆ ⊆ NUM iajs-2428 86 4 𝔇 𝔇 NOUN iajs-2428 86 5 : : PUNCT iajs-2428 86 6 eϵ eϵ PROPN iajs-2428 87 1 𝒦 𝒦 PROPN iajs-2428 87 2 } } PUNCT iajs-2428 87 3 . . PUNCT iajs-2428 88 1 then then ADV iajs-2428 88 2 , , PUNCT iajs-2428 88 3 c c PROPN iajs-2428 88 4 ⊆ ⊆ NUM iajs-2428 88 5 𝔇 𝔇 PROPN iajs-2428 88 6 , , PUNCT iajs-2428 88 7 and and CCONJ iajs-2428 88 8 c c X iajs-2428 88 9 ϵ ϵ X iajs-2428 88 10 𝒦 𝒦 PROPN iajs-2428 88 11 , , PUNCT iajs-2428 88 12 hence hence ADV iajs-2428 88 13 , , PUNCT iajs-2428 88 14     SPACE iajs-2428 88 15 75 75 NUM iajs-2428 88 16 ibn ibn PROPN iajs-2428 88 17 al al PROPN iajs-2428 88 18 - - PUNCT iajs-2428 88 19 haitham haitham PROPN iajs-2428 88 20 jour jour X iajs-2428 88 21 . . PROPN iajs-2428 88 22 for for ADP iajs-2428 88 23 pure pure ADJ iajs-2428 88 24 & & CCONJ iajs-2428 88 25 appl appl PROPN iajs-2428 88 26 . . PUNCT iajs-2428 89 1 sci sci PROPN iajs-2428 89 2 . . PROPN iajs-2428 90 1 33 33 NUM iajs-2428 90 2 ( ( PUNCT iajs-2428 90 3 2 2 NUM iajs-2428 90 4 ) ) PUNCT iajs-2428 90 5 2020 2020 NUM iajs-2428 91 1 c c X iajs-2428 91 2 = = SYM iajs-2428 91 3 c⋂ c⋂ PROPN iajs-2428 91 4 𝔇 𝔇 PROPN iajs-2428 91 5 , , PUNCT iajs-2428 91 6 but but CCONJ iajs-2428 91 7 cϵ cϵ PROPN iajs-2428 91 8 𝒦 𝒦 PROPN iajs-2428 91 9 , , PUNCT iajs-2428 91 10 then then ADV iajs-2428 91 11 cϵ cϵ VERB iajs-2428 91 12 𝒦|𝔇 𝒦|𝔇 X iajs-2428 91 13 which which PRON iajs-2428 91 14 implies imply VERB iajs-2428 91 15 that{𝐸 that{𝐸 PROPN iajs-2428 91 16 ⊆ ⊆ X iajs-2428 91 17 𝔇:eϵ 𝔇:eϵ ADJ iajs-2428 91 18 𝒦}⊆ 𝒦}⊆ ADP iajs-2428 91 19 𝒦|𝔇 𝒦|𝔇 INTJ iajs-2428 91 20 , , PUNCT iajs-2428 91 21 therefore therefore ADV iajs-2428 91 22 𝒦|𝔇 𝒦|𝔇 X iajs-2428 92 1 = = NOUN iajs-2428 92 2 { { PUNCT iajs-2428 92 3 a a DET iajs-2428 92 4 ⊆ ⊆ NUM iajs-2428 92 5 𝔇:aϵ 𝔇:aϵ PUNCT iajs-2428 92 6 𝒦 𝒦 NOUN iajs-2428 92 7 } } PUNCT iajs-2428 92 8 . . PUNCT iajs-2428 93 1 corollary corollary ADJ iajs-2428 93 2 11 11 NUM iajs-2428 93 3 let let VERB iajs-2428 93 4 ( ( PUNCT iajs-2428 93 5 𝔓 𝔓 NOUN iajs-2428 93 6 , , PUNCT iajs-2428 93 7 𝒦 𝒦 PROPN iajs-2428 93 8 ) ) PUNCT iajs-2428 93 9 is be AUX iajs-2428 93 10 measurable measurable ADJ iajs-2428 93 11 space space NOUN iajs-2428 93 12 relative relative ADJ iajs-2428 93 13 to to ADP iajs-2428 93 14 the the DET iajs-2428 93 15 λ λ NOUN iajs-2428 93 16 – – PUNCT iajs-2428 93 17 algebra algebra NOUN iajs-2428 93 18 𝒦 𝒦 PROPN iajs-2428 93 19 and and CCONJ iajs-2428 93 20 φ φ NUM iajs-2428 93 21 𝔇 𝔇 PROPN iajs-2428 93 22 ⊆ ⊆ NUM iajs-2428 93 23 𝔓. 𝔓. PROPN iajs-2428 93 24 then then ADV iajs-2428 93 25 𝒦|𝔇 𝒦|𝔇 NOUN iajs-2428 94 1 ⊆ ⊆ NUM iajs-2428 94 2 𝒦. 𝒦. PROPN iajs-2428 94 3 proof proof NOUN iajs-2428 94 4 the the DET iajs-2428 94 5 result result NOUN iajs-2428 94 6 follows follow VERB iajs-2428 94 7 from from ADP iajs-2428 94 8 proposition10 proposition10 ADJ iajs-2428 94 9 proposition proposition NOUN iajs-2428 94 10 12 12 NUM iajs-2428 94 11 let let VERB iajs-2428 94 12 ( ( PUNCT iajs-2428 94 13 𝔓 𝔓 NOUN iajs-2428 94 14 , , PUNCT iajs-2428 94 15 𝒦 𝒦 PROPN iajs-2428 94 16 ) ) PUNCT iajs-2428 94 17 is be AUX iajs-2428 94 18 measurable measurable ADJ iajs-2428 94 19 space space NOUN iajs-2428 94 20 relative relative ADJ iajs-2428 94 21 to to ADP iajs-2428 94 22 the the DET iajs-2428 94 23 λ λ NOUN iajs-2428 94 24 – – PUNCT iajs-2428 94 25 algebra algebra NOUN iajs-2428 94 26 𝒦,and 𝒦,and NOUN iajs-2428 94 27 𝔇 𝔇 PROPN iajs-2428 94 28 ⊆ ⊆ NUM iajs-2428 94 29 𝔓. 𝔓. PROPN iajs-2428 94 30 then then ADV iajs-2428 94 31 ( ( PUNCT iajs-2428 94 32 𝔇 𝔇 PROPN iajs-2428 94 33 , , PUNCT iajs-2428 94 34 𝒦|𝔇 𝒦|𝔇 X iajs-2428 94 35 ) ) PUNCT iajs-2428 94 36 is be AUX iajs-2428 94 37 measurable measurable ADJ iajs-2428 94 38 space space NOUN iajs-2428 94 39 relative relative ADJ iajs-2428 94 40 to to ADP iajs-2428 94 41 the the DET iajs-2428 94 42 λ λ NOUN iajs-2428 94 43 – – PUNCT iajs-2428 94 44 algebra algebra NOUN iajs-2428 94 45 𝒦𝔇 𝒦𝔇 ADJ iajs-2428 94 46 proof proof NOUN iajs-2428 94 47 since since SCONJ iajs-2428 94 48 ( ( PUNCT iajs-2428 94 49 𝔓 𝔓 NOUN iajs-2428 94 50 , , PUNCT iajs-2428 94 51 𝒦 𝒦 PROPN iajs-2428 94 52 ) ) PUNCT iajs-2428 94 53 is be AUX iajs-2428 94 54 measurable measurable ADJ iajs-2428 94 55 space space NOUN iajs-2428 94 56 relative relative ADJ iajs-2428 94 57 to to ADP iajs-2428 94 58 theλ theλ NOUN iajs-2428 94 59 – – PUNCT iajs-2428 94 60 algebra algebra NOUN iajs-2428 94 61 𝒦 𝒦 PROPN iajs-2428 94 62 , , PUNCT iajs-2428 94 63 then then ADV iajs-2428 94 64 𝔓 𝔓 PROPN iajs-2428 94 65 ϵ ϵ ADP iajs-2428 94 66 𝒦. 𝒦. PROPN iajs-2428 94 67 since since SCONJ iajs-2428 94 68 ⊆ ⊆ NUM iajs-2428 94 69 𝔓 𝔓 PROPN iajs-2428 94 70 , , PUNCT iajs-2428 94 71 then then ADV iajs-2428 94 72 𝔇 𝔇 PROPN iajs-2428 94 73 𝔓 𝔓 PROPN iajs-2428 94 74 ⋂𝔇 ⋂𝔇 NUM iajs-2428 94 75 and and CCONJ iajs-2428 94 76 𝔇 𝔇 PROPN iajs-2428 94 77 ϵ ϵ X iajs-2428 94 78 𝒦|𝔇. 𝒦|𝔇. PUNCT iajs-2428 94 79 let let VERB iajs-2428 94 80 bϵ bϵ NOUN iajs-2428 94 81 𝒦|𝔇and 𝒦|𝔇and VERB iajs-2428 95 1 f f PROPN iajs-2428 96 1 ⊂ ⊂ PROPN iajs-2428 96 2 b b PROPN iajs-2428 96 3 ⊂ ⊂ PROPN iajs-2428 96 4 𝔇. 𝔇. PROPN iajs-2428 96 5 then then ADV iajs-2428 96 6 by by ADP iajs-2428 96 7 corollary corollary ADJ iajs-2428 96 8 11 11 NUM iajs-2428 96 9 , , PUNCT iajs-2428 96 10 we we PRON iajs-2428 96 11 get get VERB iajs-2428 96 12 bϵ bϵ ADJ iajs-2428 96 13 𝒦. 𝒦. PROPN iajs-2428 97 1 but but CCONJ iajs-2428 98 1 f f PROPN iajs-2428 98 2 ⊂ ⊂ PROPN iajs-2428 98 3 b b X iajs-2428 98 4 ⊂ ⊂ PROPN iajs-2428 98 5 𝔇 𝔇 PROPN iajs-2428 98 6 ⊂ ⊂ PROPN iajs-2428 98 7 𝔓 𝔓 PROPN iajs-2428 98 8 and and CCONJ iajs-2428 98 9 ( ( PUNCT iajs-2428 98 10 𝔓 𝔓 PROPN iajs-2428 98 11 , , PUNCT iajs-2428 98 12 𝒦 𝒦 PROPN iajs-2428 98 13 ) ) PUNCT iajs-2428 98 14 is be AUX iajs-2428 98 15 measurable measurable ADJ iajs-2428 98 16 space space NOUN iajs-2428 98 17 relative relative ADJ iajs-2428 98 18 to to ADP iajs-2428 98 19 the the DET iajs-2428 98 20 λ λ NOUN iajs-2428 98 21 – – PUNCT iajs-2428 98 22 algebra algebra NOUN iajs-2428 98 23 𝒦 𝒦 PROPN iajs-2428 98 24 , , PUNCT iajs-2428 98 25 then then ADV iajs-2428 98 26 fϵ fϵ ADP iajs-2428 98 27 𝒦. 𝒦. PROPN iajs-2428 98 28 now now ADV iajs-2428 98 29 , , PUNCT iajs-2428 98 30 f f PROPN iajs-2428 98 31 ⊂ ⊂ PROPN iajs-2428 98 32 𝔇 𝔇 PROPN iajs-2428 98 33 , , PUNCT iajs-2428 98 34 and and CCONJ iajs-2428 98 35 fϵ fϵ X iajs-2428 99 1 𝒦 𝒦 PROPN iajs-2428 99 2 , , PUNCT iajs-2428 99 3 then then ADV iajs-2428 99 4 by by ADP iajs-2428 99 5 proposition proposition NOUN iajs-2428 99 6 10 10 NUM iajs-2428 99 7 , , PUNCT iajs-2428 99 8 we we PRON iajs-2428 99 9 have have AUX iajs-2428 99 10 fϵ fϵ AUX iajs-2428 99 11 𝒦|𝔇. 𝒦|𝔇. ADV iajs-2428 99 12 let let VERB iajs-2428 99 13 b b NOUN iajs-2428 99 14 , , PUNCT iajs-2428 99 15 b b PROPN iajs-2428 99 16 , , PUNCT iajs-2428 99 17 … … PUNCT iajs-2428 100 1 ϵ ϵ X iajs-2428 100 2 𝒦|𝔇. 𝒦|𝔇. X iajs-2428 100 3 then then ADV iajs-2428 100 4 there there PRON iajs-2428 100 5 exist exist VERB iajs-2428 100 6 e e NOUN iajs-2428 100 7 , , PUNCT iajs-2428 100 8 e e X iajs-2428 100 9 , , PUNCT iajs-2428 100 10 … … PUNCT iajs-2428 100 11 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 101 1 such such ADJ iajs-2428 101 2 that that DET iajs-2428 101 3 b b NOUN iajs-2428 101 4 = = SYM iajs-2428 101 5 e e X iajs-2428 101 6 ⋂ ⋂ PROPN iajs-2428 101 7 𝔇 𝔇 PROPN iajs-2428 101 8 where where SCONJ iajs-2428 101 9 i=1,2 i=1,2 ADJ iajs-2428 101 10 , , PUNCT iajs-2428 101 11 … … PUNCT iajs-2428 101 12 , , PUNCT iajs-2428 101 13 hence hence ADV iajs-2428 101 14 ⋃ ⋃ NOUN iajs-2428 101 15 b b NOUN iajs-2428 101 16 = = NOUN iajs-2428 101 17 ⋃ ⋃ PROPN iajs-2428 101 18 e e X iajs-2428 101 19 ⋂ ⋂ PROPN iajs-2428 101 20 𝔇 𝔇 NOUN iajs-2428 101 21 = = PUNCT iajs-2428 101 22 ⋃ ⋃ PROPN iajs-2428 101 23 e e NOUN iajs-2428 101 24 ⋂ ⋂ PROPN iajs-2428 101 25 𝔇. 𝔇. PROPN iajs-2428 101 26 but but CCONJ iajs-2428 101 27 ( ( PUNCT iajs-2428 101 28 𝔓 𝔓 NOUN iajs-2428 101 29 , , PUNCT iajs-2428 101 30 𝒦 𝒦 PROPN iajs-2428 101 31 ) ) PUNCT iajs-2428 101 32 is be AUX iajs-2428 101 33 measurable measurable ADJ iajs-2428 101 34 space space NOUN iajs-2428 101 35 relative relative ADJ iajs-2428 101 36 to to ADP iajs-2428 101 37 the the DET iajs-2428 101 38 λ λ NOUN iajs-2428 101 39 – – PUNCT iajs-2428 101 40 algebra algebra NOUN iajs-2428 101 41 𝒦 𝒦 PROPN iajs-2428 101 42 and and CCONJ iajs-2428 101 43 e e NOUN iajs-2428 101 44 , , PUNCT iajs-2428 101 45 e e X iajs-2428 101 46 , , PUNCT iajs-2428 101 47 … … PUNCT iajs-2428 102 1 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 102 2 , , PUNCT iajs-2428 102 3 then then ADV iajs-2428 102 4 , , PUNCT iajs-2428 102 5 ⋃ ⋃ PROPN iajs-2428 102 6 e e NOUN iajs-2428 102 7 ϵ𝒦. ϵ𝒦. PROPN iajs-2428 102 8 hence hence ADV iajs-2428 102 9 , , PUNCT iajs-2428 102 10 ⋃ ⋃ PROPN iajs-2428 102 11 b b X iajs-2428 102 12 ϵ ϵ X iajs-2428 102 13 𝒦|𝔇. 𝒦|𝔇. X iajs-2428 102 14 therefore therefore ADV iajs-2428 102 15 , , PUNCT iajs-2428 102 16 ( ( PUNCT iajs-2428 102 17 𝔇 𝔇 PROPN iajs-2428 102 18 , , PUNCT iajs-2428 102 19 𝒦|𝔇 𝒦|𝔇 X iajs-2428 102 20 ) ) PUNCT iajs-2428 102 21 is be AUX iajs-2428 102 22 measurable measurable ADJ iajs-2428 102 23 space space NOUN iajs-2428 102 24 relative relative ADJ iajs-2428 102 25 to to ADP iajs-2428 102 26 the the DET iajs-2428 102 27 λ λ NOUN iajs-2428 102 28 – – PUNCT iajs-2428 102 29 algebra algebra NOUN iajs-2428 102 30 𝒦|𝔇. 𝒦|𝔇. PUNCT iajs-2428 102 31 example example NOUN iajs-2428 103 1 13 13 NUM iajs-2428 103 2 let let VERB iajs-2428 103 3 𝔓 𝔓 PROPN iajs-2428 103 4 = = NOUN iajs-2428 103 5 { { PUNCT iajs-2428 103 6 1,2,3,4,5 1,2,3,4,5 NUM iajs-2428 103 7 } } PUNCT iajs-2428 103 8 and and CCONJ iajs-2428 103 9 𝒦 𝒦 PROPN iajs-2428 103 10 { { PUNCT iajs-2428 103 11 φ,{1},{3},{5},{1,3},{1,5},{3,5},{1,3,5},𝔓 φ,{1},{3},{5},{1,3},{1,5},{3,5},{1,3,5},𝔓 PROPN iajs-2428 103 12 } } PUNCT iajs-2428 103 13 . . PUNCT iajs-2428 104 1 then then ADV iajs-2428 104 2 ( ( PUNCT iajs-2428 104 3 𝔓 𝔓 NOUN iajs-2428 104 4 , , PUNCT iajs-2428 104 5 𝒦 𝒦 PROPN iajs-2428 104 6 ) ) PUNCT iajs-2428 104 7 is be AUX iajs-2428 104 8 measurable measurable ADJ iajs-2428 104 9 space space NOUN iajs-2428 104 10 relative relative ADJ iajs-2428 104 11 to to ADP iajs-2428 104 12 the the DET iajs-2428 104 13 λ λ NOUN iajs-2428 104 14 – – PUNCT iajs-2428 104 15 algebra algebra NOUN iajs-2428 104 16 𝒦. 𝒦. PROPN iajs-2428 104 17 if if SCONJ iajs-2428 104 18 𝔇 𝔇 PROPN iajs-2428 104 19 { { PUNCT iajs-2428 104 20 1,2,4 1,2,4 NUM iajs-2428 104 21 } } PUNCT iajs-2428 104 22 , , PUNCT iajs-2428 104 23 then then ADV iajs-2428 104 24 𝒦|𝔇={φ,{1},𝔇 𝒦|𝔇={φ,{1},𝔇 PROPN iajs-2428 104 25 } } PUNCT iajs-2428 104 26 , , PUNCT iajs-2428 104 27 hence hence ADV iajs-2428 104 28 ( ( PUNCT iajs-2428 104 29 𝔇 𝔇 PROPN iajs-2428 104 30 , , PUNCT iajs-2428 104 31 𝒦|𝔇 𝒦|𝔇 X iajs-2428 104 32 ) ) PUNCT iajs-2428 104 33 is be AUX iajs-2428 104 34 measurable measurable ADJ iajs-2428 104 35 space space NOUN iajs-2428 104 36 relative relative ADJ iajs-2428 104 37 to to ADP iajs-2428 104 38 the the DET iajs-2428 104 39 λ λ NOUN iajs-2428 104 40 – – PUNCT iajs-2428 104 41 algebra algebra NOUN iajs-2428 104 42 𝒦|𝔇 𝒦|𝔇 X iajs-2428 105 1 and and CCONJ iajs-2428 105 2 𝒦|𝔇 𝒦|𝔇 PRON iajs-2428 105 3 ⊆ ⊆ NUM iajs-2428 105 4 𝒦. 𝒦. PROPN iajs-2428 105 5 proposition proposition NOUN iajs-2428 105 6 14 14 NUM iajs-2428 105 7 let let VERB iajs-2428 105 8 𝒥 𝒥 PRON iajs-2428 105 9 ⊆ ⊆ NUM iajs-2428 105 10 p p PROPN iajs-2428 105 11 𝔓 𝔓 PROPN iajs-2428 105 12 and and CCONJ iajs-2428 105 13 φ φ NUM iajs-2428 105 14 𝔇 𝔇 PROPN iajs-2428 105 15 ⊆ ⊆ NUM iajs-2428 105 16 𝔓. 𝔓. PROPN iajs-2428 105 17 if if SCONJ iajs-2428 105 18 𝒦 𝒦 PROPN iajs-2428 105 19 is be AUX iajs-2428 105 20 a a DET iajs-2428 105 21 λ λ NOUN iajs-2428 105 22 – – PUNCT iajs-2428 105 23 algebra algebra NOUN iajs-2428 105 24 of of ADP iajs-2428 105 25 𝔓 𝔓 PROPN iajs-2428 105 26 which which PRON iajs-2428 105 27 includes include VERB iajs-2428 105 28 𝒥 𝒥 PROPN iajs-2428 105 29 , , PUNCT iajs-2428 105 30 then then ADV iajs-2428 105 31 λ λ PROPN iajs-2428 105 32 𝒥 𝒥 PROPN iajs-2428 105 33 |𝔇 |𝔇 PUNCT iajs-2428 105 34 is be AUX iajs-2428 105 35 a a DET iajs-2428 105 36 λ λ NOUN iajs-2428 105 37 – – PUNCT iajs-2428 105 38 algebra algebra NOUN iajs-2428 105 39 of of ADP iajs-2428 105 40 a a DET iajs-2428 105 41 set set ADJ iajs-2428 105 42 𝔇. 𝔇. PROPN iajs-2428 105 43 proof proof NOUN iajs-2428 105 44 the the DET iajs-2428 105 45 result result NOUN iajs-2428 105 46 follows follow VERB iajs-2428 105 47 from from ADP iajs-2428 105 48 proposition proposition NOUN iajs-2428 105 49 7 7 NUM iajs-2428 105 50 and and CCONJ iajs-2428 105 51 proposition proposition NOUN iajs-2428 105 52 12 12 NUM iajs-2428 105 53 . . PUNCT iajs-2428 106 1 proposition proposition NOUN iajs-2428 106 2 15 15 NUM iajs-2428 106 3 let let VERB iajs-2428 106 4 𝒥 𝒥 PRON iajs-2428 106 5 ⊆ ⊆ NUM iajs-2428 106 6 p p PROPN iajs-2428 106 7 𝔓 𝔓 PROPN iajs-2428 106 8 and and CCONJ iajs-2428 106 9 φ φ PRON iajs-2428 106 10 𝔇 𝔇 PROPN iajs-2428 106 11 ⊆ ⊆ NUM iajs-2428 106 12 𝔓 𝔓 PROPN iajs-2428 106 13 and and CCONJ iajs-2428 106 14 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 106 15 is be AUX iajs-2428 106 16 the the DET iajs-2428 106 17 restriction restriction NOUN iajs-2428 106 18 of of ADP iajs-2428 106 19 𝒥 𝒥 PROPN iajs-2428 106 20 over over ADP iajs-2428 106 21 the the DET iajs-2428 106 22 set set NOUN iajs-2428 107 1 𝔇. 𝔇. PROPN iajs-2428 107 2 then then ADV iajs-2428 107 3 λ λ PROPN iajs-2428 107 4 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 107 5 is be AUX iajs-2428 107 6 the the DET iajs-2428 107 7 smallest small ADJ iajs-2428 107 8 λ λ NOUN iajs-2428 107 9 – – PUNCT iajs-2428 107 10 algebra algebra NOUN iajs-2428 107 11 of of ADP iajs-2428 107 12 a a DET iajs-2428 107 13 set𝔇 set𝔇 NOUN iajs-2428 107 14 , , PUNCT iajs-2428 107 15 which which PRON iajs-2428 107 16 includes include VERB iajs-2428 107 17 𝒥|𝔇 𝒥|𝔇 PROPN iajs-2428 107 18 , , PUNCT iajs-2428 107 19 where where SCONJ iajs-2428 107 20 λ λ X iajs-2428 107 21 𝒥|𝔇 𝒥|𝔇 VERB iajs-2428 107 22 = = SYM iajs-2428 107 23 ⋂ ⋂ PROPN iajs-2428 107 24 𝒦 𝒦 PROPN iajs-2428 107 25 |𝔇 |𝔇 NUM iajs-2428 107 26 : : PUNCT iajs-2428 107 27 𝒦 𝒦 PROPN iajs-2428 107 28 |𝔇 |𝔇 NUM iajs-2428 107 29 is be AUX iajs-2428 107 30 a a DET iajs-2428 107 31 λ λ NOUN iajs-2428 107 32 -algebra -algebra NOUN iajs-2428 107 33 of of ADP iajs-2428 107 34 𝔇 𝔇 PROPN iajs-2428 107 35 , , PUNCT iajs-2428 107 36 and and CCONJ iajs-2428 107 37 𝒦 𝒦 PROPN iajs-2428 107 38 |𝔇 |𝔇 NUM iajs-2428 107 39 ⊇ ⊇ NOUN iajs-2428 107 40 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 107 41 , , PUNCT iajs-2428 107 42 ∀i ∀i NOUN iajs-2428 107 43 ∈ ∈ NOUN iajs-2428 107 44 ι ι X iajs-2428 107 45 } } PUNCT iajs-2428 107 46 . . PUNCT iajs-2428 108 1 proof proof NOUN iajs-2428 108 2 from from ADP iajs-2428 108 3 lemma lemma PROPN iajs-2428 108 4 5 5 NUM iajs-2428 108 5 , , PUNCT iajs-2428 108 6 we we PRON iajs-2428 108 7 get get VERB iajs-2428 108 8 λ λ PROPN iajs-2428 108 9 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 108 10 is be AUX iajs-2428 108 11 a a DET iajs-2428 108 12 λ λ NOUN iajs-2428 108 13 – – PUNCT iajs-2428 108 14 algebra algebra NOUN iajs-2428 108 15 of of ADP iajs-2428 108 16 a a DET iajs-2428 108 17 set𝔇. set𝔇. NOUN iajs-2428 108 18 to to PART iajs-2428 108 19 prove prove VERB iajs-2428 108 20 that that SCONJ iajs-2428 108 21 λ λ PROPN iajs-2428 108 22 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 108 23 ⊇ ⊇ PROPN iajs-2428 108 24 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 108 25 , , PUNCT iajs-2428 108 26 suppose suppose VERB iajs-2428 108 27 that that SCONJ iajs-2428 108 28 each each PRON iajs-2428 108 29 of of ADP iajs-2428 108 30 𝒦 𝒦 PROPN iajs-2428 108 31 |𝔇 |𝔇 PUNCT iajs-2428 108 32 is be AUX iajs-2428 108 33 a a DET iajs-2428 108 34 λ λ NOUN iajs-2428 108 35 -algebra -algebra NOUN iajs-2428 108 36 of of ADP iajs-2428 108 37 a a DET iajs-2428 108 38 set set NOUN iajs-2428 108 39 𝔇 𝔇 PROPN iajs-2428 108 40 and𝒦 and𝒦 ADJ iajs-2428 108 41 |𝔇 |𝔇 PUNCT iajs-2428 108 42 ⊇ ⊇ NOUN iajs-2428 108 43 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 108 44 , , PUNCT iajs-2428 108 45 ∀i ∀i NOUN iajs-2428 108 46 ∈ ∈ PROPN iajs-2428 108 47 ι ι NOUN iajs-2428 108 48 , , PUNCT iajs-2428 108 49 then then ADV iajs-2428 108 50 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 108 51 ⊆ ⊆ NUM iajs-2428 108 52 ⋂ ⋂ PROPN iajs-2428 108 53 𝒦 𝒦 PROPN iajs-2428 108 54 |𝔇 |𝔇 PUNCT iajs-2428 108 55 ∈ ∈ PROPN iajs-2428 108 56 , , PUNCT iajs-2428 108 57 hence hence ADV iajs-2428 108 58 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 108 59 ⊆ ⊆ NUM iajs-2428 108 60 λ λ X iajs-2428 108 61 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 108 62 . . PUNCT iajs-2428 109 1 now now ADV iajs-2428 109 2 , , PUNCT iajs-2428 109 3 let let VERB iajs-2428 109 4 𝒦∗|𝔇 𝒦∗|𝔇 VERB iajs-2428 109 5 is be AUX iajs-2428 109 6 a a DET iajs-2428 109 7 λ λ NOUN iajs-2428 109 8 – – PUNCT iajs-2428 109 9 algebra algebra NOUN iajs-2428 109 10 of of ADP iajs-2428 109 11 a a DET iajs-2428 109 12 set set NOUN iajs-2428 109 13 𝔇 𝔇 PRON iajs-2428 109 14 such such DET iajs-2428 109 15 that that SCONJ iajs-2428 109 16 𝒦∗|𝔇 𝒦∗|𝔇 PUNCT iajs-2428 109 17 ⊇ ⊇ PROPN iajs-2428 109 18 𝒥|𝔇. 𝒥|𝔇. PROPN iajs-2428 109 19 then then ADV iajs-2428 109 20 𝒦∗|𝔇 𝒦∗|𝔇 X iajs-2428 109 21 ⊇ ⊇ PROPN iajs-2428 109 22 λ λ PROPN iajs-2428 109 23 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 109 24 . . PUNCT iajs-2428 110 1 therefore therefore ADV iajs-2428 110 2 , , PUNCT iajs-2428 110 3 λ λ PROPN iajs-2428 110 4 𝒥|𝔇 𝒥|𝔇 VERB iajs-2428 110 5 is be AUX iajs-2428 110 6 the the DET iajs-2428 110 7 smallest small ADJ iajs-2428 110 8 λ λ NOUN iajs-2428 110 9 – – PUNCT iajs-2428 110 10 algebra algebra NOUN iajs-2428 110 11 of of ADP iajs-2428 110 12 a a DET iajs-2428 110 13 set set NOUN iajs-2428 110 14 𝔇 𝔇 PROPN iajs-2428 110 15 includes𝒥|𝔇. includes𝒥|𝔇. NOUN iajs-2428 110 16     SPACE iajs-2428 110 17 76 76 NUM iajs-2428 110 18 ibn ibn PROPN iajs-2428 110 19 al al PROPN iajs-2428 110 20 - - PUNCT iajs-2428 110 21 haitham haitham PROPN iajs-2428 110 22 jour jour X iajs-2428 110 23 . . PROPN iajs-2428 110 24 for for ADP iajs-2428 110 25 pure pure ADJ iajs-2428 110 26 & & CCONJ iajs-2428 110 27 appl appl PROPN iajs-2428 110 28 . . PUNCT iajs-2428 111 1 sci sci PROPN iajs-2428 111 2 . . PROPN iajs-2428 112 1 33 33 NUM iajs-2428 112 2 ( ( PUNCT iajs-2428 112 3 2 2 NUM iajs-2428 112 4 ) ) PUNCT iajs-2428 112 5 2020 2020 NUM iajs-2428 112 6 proposition proposition NOUN iajs-2428 112 7 16 16 NUM iajs-2428 112 8 let let VERB iajs-2428 112 9 𝒥 𝒥 PRON iajs-2428 112 10 ⊆ ⊆ NUM iajs-2428 112 11 p p PROPN iajs-2428 112 12 𝔓 𝔓 PROPN iajs-2428 112 13 and and CCONJ iajs-2428 112 14 φ φ NUM iajs-2428 112 15 𝔇 𝔇 PROPN iajs-2428 112 16 ⊆ ⊆ NUM iajs-2428 112 17 𝔓 𝔓 NOUN iajs-2428 112 18 , , PUNCT iajs-2428 112 19 define define VERB iajs-2428 112 20 the the DET iajs-2428 112 21 collection collection NOUN iajs-2428 112 22 𝒦 𝒦 PROPN iajs-2428 112 23 as as ADP iajs-2428 112 24 : : PUNCT iajs-2428 112 25 𝒦 𝒦 PROPN iajs-2428 112 26 = = PRON iajs-2428 112 27 { { PUNCT iajs-2428 112 28 e e NOUN iajs-2428 112 29 ⊆ ⊆ NUM iajs-2428 112 30 𝔓 𝔓 PROPN iajs-2428 112 31 : : PUNCT iajs-2428 112 32 ( ( PUNCT iajs-2428 112 33 e⋂ e⋂ NOUN iajs-2428 112 34 𝔇 𝔇 PROPN iajs-2428 112 35 ϵ ϵ X iajs-2428 112 36 λ λ PROPN iajs-2428 112 37 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 112 38 } } PUNCT iajs-2428 112 39 . . PUNCT iajs-2428 113 1 then then ADV iajs-2428 113 2 ( ( PUNCT iajs-2428 113 3 𝔓 𝔓 NOUN iajs-2428 113 4 , , PUNCT iajs-2428 113 5 𝒦 𝒦 PROPN iajs-2428 113 6 ) ) PUNCT iajs-2428 113 7 is be AUX iajs-2428 113 8 measurable measurable ADJ iajs-2428 113 9 space space NOUN iajs-2428 113 10 relative relative ADJ iajs-2428 113 11 to to ADP iajs-2428 113 12 the the DET iajs-2428 113 13 λ λ NOUN iajs-2428 113 14 – – PUNCT iajs-2428 113 15 algebra algebra NOUN iajs-2428 113 16 𝒦. 𝒦. NOUN iajs-2428 113 17 proof proof NOUN iajs-2428 113 18 since since SCONJ iajs-2428 113 19 λ λ PROPN iajs-2428 113 20 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 113 21 } } PUNCT iajs-2428 113 22 is be AUX iajs-2428 113 23 a a DET iajs-2428 113 24 λ λ NOUN iajs-2428 113 25 – – PUNCT iajs-2428 113 26 algebra algebra NOUN iajs-2428 113 27 of of ADP iajs-2428 113 28 a a DET iajs-2428 113 29 set set NOUN iajs-2428 113 30 𝔇 𝔇 PROPN iajs-2428 113 31 , , PUNCT iajs-2428 113 32 then then ADV iajs-2428 113 33 φ φ NUM iajs-2428 113 34 , , PUNCT iajs-2428 113 35 𝔇 𝔇 PROPN iajs-2428 113 36 ϵ ϵ X iajs-2428 113 37 λ λ PROPN iajs-2428 113 38 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 113 39 . . PUNCT iajs-2428 114 1 since since SCONJ iajs-2428 114 2 𝔇 𝔇 PROPN iajs-2428 114 3 ⊆ ⊆ NUM iajs-2428 114 4 𝔓 𝔓 PROPN iajs-2428 114 5 , , PUNCT iajs-2428 114 6 then then ADV iajs-2428 114 7 𝔇 𝔇 PROPN iajs-2428 114 8 = = SYM iajs-2428 114 9 𝔓 𝔓 PROPN iajs-2428 114 10 ⋂ ⋂ PROPN iajs-2428 114 11 𝔇 𝔇 PROPN iajs-2428 114 12 and and CCONJ iajs-2428 114 13 𝔓 𝔓 PROPN iajs-2428 115 1 ϵ𝒦. ϵ𝒦. PROPN iajs-2428 115 2 let let VERB iajs-2428 115 3 eϵ𝒦 eϵ𝒦 NOUN iajs-2428 115 4 and and CCONJ iajs-2428 115 5 f f PROPN iajs-2428 115 6 ⊂ ⊂ PROPN iajs-2428 115 7 e e PROPN iajs-2428 115 8 ⊂ ⊂ PROPN iajs-2428 115 9 𝔓. 𝔓. PROPN iajs-2428 115 10 then then ADV iajs-2428 115 11 , , PUNCT iajs-2428 115 12 e⋂ e⋂ PUNCT iajs-2428 115 13 𝔇 𝔇 PROPN iajs-2428 115 14 ϵλ ϵλ PROPN iajs-2428 115 15 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 115 16 . . PUNCT iajs-2428 116 1 since since SCONJ iajs-2428 116 2 , , PUNCT iajs-2428 116 3 f f PROPN iajs-2428 116 4 ⊂ ⊂ PROPN iajs-2428 116 5 e e PROPN iajs-2428 116 6 , , PUNCT iajs-2428 116 7 then then ADV iajs-2428 116 8 f⋂ f⋂ ADP iajs-2428 116 9 𝔇 𝔇 PROPN iajs-2428 116 10 ⊂ ⊂ PROPN iajs-2428 116 11 e⋂ e⋂ PUNCT iajs-2428 116 12 𝔇 𝔇 PROPN iajs-2428 116 13 . . PUNCT iajs-2428 117 1 but but CCONJ iajs-2428 117 2 λ λ PROPN iajs-2428 117 3 𝒥|𝔇 𝒥|𝔇 VERB iajs-2428 117 4 is be AUX iajs-2428 117 5 a a DET iajs-2428 117 6 λ λ NOUN iajs-2428 117 7 – – PUNCT iajs-2428 117 8 algebra algebra NOUN iajs-2428 117 9 of of ADP iajs-2428 117 10 a a DET iajs-2428 117 11 set𝔇 set𝔇 NOUN iajs-2428 117 12 , , PUNCT iajs-2428 117 13 which which PRON iajs-2428 117 14 implies imply VERB iajs-2428 117 15 that that SCONJ iajs-2428 117 16 f⋂𝔇 f⋂𝔇 PROPN iajs-2428 117 17 ϵ ϵ X iajs-2428 117 18 λ λ X iajs-2428 117 19 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 117 20 and and CCONJ iajs-2428 117 21 fϵ𝒦. fϵ𝒦. AUX iajs-2428 117 22 let let VERB iajs-2428 117 23 e e NOUN iajs-2428 117 24 , , PUNCT iajs-2428 117 25 e e X iajs-2428 117 26 , , PUNCT iajs-2428 117 27 … … PUNCT iajs-2428 118 1 ϵ ϵ X iajs-2428 118 2 𝒦. 𝒦. PROPN iajs-2428 118 3 then then ADV iajs-2428 118 4 e e X iajs-2428 118 5 ⋂𝔇 ⋂𝔇 NUM iajs-2428 118 6 ϵ ϵ X iajs-2428 118 7 λ λ PROPN iajs-2428 118 8 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 118 9 , , PUNCT iajs-2428 118 10 for for ADP iajs-2428 118 11 all all DET iajs-2428 118 12 i=1,2 i=1,2 ADJ iajs-2428 118 13 , , PUNCT iajs-2428 118 14 … … PUNCT iajs-2428 118 15 , , PUNCT iajs-2428 118 16 hence hence ADV iajs-2428 118 17 ⋃ ⋃ PUNCT iajs-2428 118 18 e e NOUN iajs-2428 118 19 ⋂ ⋂ PROPN iajs-2428 118 20 𝔇 𝔇 PROPN iajs-2428 118 21 ϵλ ϵλ PROPN iajs-2428 118 22 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 118 23 and and CCONJ iajs-2428 118 24 ⋃ ⋃ PROPN iajs-2428 118 25 e e NOUN iajs-2428 118 26 ⋂ ⋂ PROPN iajs-2428 118 27 𝔇 𝔇 PROPN iajs-2428 118 28 ϵλ ϵλ PROPN iajs-2428 118 29 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 118 30 implies imply VERB iajs-2428 118 31 that that SCONJ iajs-2428 118 32 ⋃ ⋃ PUNCT iajs-2428 118 33 e e NOUN iajs-2428 118 34 ϵ ϵ X iajs-2428 118 35 𝒦. 𝒦. PROPN iajs-2428 118 36 therefore therefore ADV iajs-2428 118 37 𝒦 𝒦 PROPN iajs-2428 118 38 is be AUX iajs-2428 118 39 λ λ NOUN iajs-2428 118 40 – – PUNCT iajs-2428 118 41 algebra algebra NOUN iajs-2428 118 42 of of ADP iajs-2428 118 43 a a DET iajs-2428 118 44 set set ADJ iajs-2428 118 45 𝔓. 𝔓. NOUN iajs-2428 118 46 theorem theorem NOUN iajs-2428 118 47 17 17 NUM iajs-2428 118 48 let let VERB iajs-2428 118 49 𝒥 𝒥 PRON iajs-2428 118 50 ⊆ ⊆ NUM iajs-2428 118 51 p p PROPN iajs-2428 118 52 𝔓 𝔓 PROPN iajs-2428 118 53 and and CCONJ iajs-2428 118 54 φ φ NUM iajs-2428 118 55 𝔇 𝔇 PROPN iajs-2428 118 56 ⊆ ⊆ NUM iajs-2428 118 57 𝔓. 𝔓. PROPN iajs-2428 118 58 then then ADV iajs-2428 118 59 λ λ X iajs-2428 118 60 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 118 61 = = SYM iajs-2428 118 62 λ λ PROPN iajs-2428 118 63 𝒥 𝒥 PROPN iajs-2428 118 64 |𝔇. |𝔇. PROPN iajs-2428 118 65 proof proof NOUN iajs-2428 118 66 let let AUX iajs-2428 118 67 bϵ𝒥|𝔇 bϵ𝒥|𝔇 VERB iajs-2428 118 68 , , PUNCT iajs-2428 118 69 then then ADV iajs-2428 118 70 b b X iajs-2428 118 71 = = NOUN iajs-2428 118 72 e⋂ e⋂ NOUN iajs-2428 118 73 𝔇 𝔇 PROPN iajs-2428 118 74 , , PUNCT iajs-2428 118 75 for for ADP iajs-2428 118 76 someeϵ someeϵ NOUN iajs-2428 118 77 𝒥. 𝒥. PROPN iajs-2428 118 78 but but CCONJ iajs-2428 118 79 𝒥 𝒥 PROPN iajs-2428 119 1 ⊆ ⊆ NUM iajs-2428 119 2 λ λ SYM iajs-2428 119 3 𝒥 𝒥 PROPN iajs-2428 119 4 , , PUNCT iajs-2428 119 5 then then ADV iajs-2428 119 6 eϵ eϵ ADP iajs-2428 119 7 λ λ PROPN iajs-2428 119 8 𝒥 𝒥 PROPN iajs-2428 119 9 , , PUNCT iajs-2428 119 10 thus thus ADV iajs-2428 119 11 bϵ bϵ ADP iajs-2428 119 12 λ λ PROPN iajs-2428 119 13 𝒥 𝒥 PROPN iajs-2428 119 14 |𝔇 |𝔇 SYM iajs-2428 119 15 , , PUNCT iajs-2428 119 16 hence hence ADV iajs-2428 119 17 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 119 18 ⊆ ⊆ NUM iajs-2428 119 19 λ λ X iajs-2428 119 20 𝒥 𝒥 PROPN iajs-2428 119 21 |𝔇 |𝔇 PUNCT iajs-2428 119 22 , , PUNCT iajs-2428 119 23 but but CCONJ iajs-2428 119 24 λ λ PROPN iajs-2428 119 25 𝒥|𝔇 𝒥|𝔇 VERB iajs-2428 119 26 is be AUX iajs-2428 119 27 smallest small ADJ iajs-2428 119 28 λ λ NOUN iajs-2428 119 29 – – PUNCT iajs-2428 119 30 algebra algebra NOUN iajs-2428 119 31 of of ADP iajs-2428 119 32 a a DET iajs-2428 119 33 set set NOUN iajs-2428 119 34 𝔇 𝔇 PROPN iajs-2428 119 35 , , PUNCT iajs-2428 119 36 which which PRON iajs-2428 119 37 include include VERB iajs-2428 119 38 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 119 39 and and CCONJ iajs-2428 119 40 λ λ PROPN iajs-2428 119 41 𝒥 𝒥 PROPN iajs-2428 119 42 |𝔇is |𝔇is NUM iajs-2428 119 43 a a DET iajs-2428 119 44 λ λ NOUN iajs-2428 119 45 – – PUNCT iajs-2428 119 46 algebra algebra NOUN iajs-2428 119 47 of of ADP iajs-2428 119 48 a a DET iajs-2428 119 49 set set NOUN iajs-2428 119 50 𝔇 𝔇 PROPN iajs-2428 119 51 which which PRON iajs-2428 119 52 include include VERB iajs-2428 119 53 𝒥|𝔇 𝒥|𝔇 PROPN iajs-2428 119 54 , , PUNCT iajs-2428 119 55 then then ADV iajs-2428 119 56 λ λ X iajs-2428 119 57 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 119 58 ⊆ ⊆ NUM iajs-2428 119 59 λ λ SYM iajs-2428 119 60 𝒥 𝒥 PROPN iajs-2428 119 61 |𝔇. |𝔇. PROPN iajs-2428 119 62 now now ADV iajs-2428 119 63 , , PUNCT iajs-2428 119 64 define define VERB iajs-2428 119 65 collection collection NOUN iajs-2428 119 66 𝒦 𝒦 PROPN iajs-2428 119 67 as as ADP iajs-2428 119 68 : : PUNCT iajs-2428 119 69 𝒦= 𝒦= PROPN iajs-2428 119 70 { { PUNCT iajs-2428 119 71 e e NOUN iajs-2428 119 72 ⊆ ⊆ NUM iajs-2428 119 73 𝔓 𝔓 PROPN iajs-2428 119 74 : : PUNCT iajs-2428 119 75 e⋂ e⋂ VERB iajs-2428 119 76 𝔇 𝔇 PROPN iajs-2428 119 77 ϵλ ϵλ PROPN iajs-2428 119 78 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 119 79 } } PUNCT iajs-2428 119 80 , , PUNCT iajs-2428 119 81 then then ADV iajs-2428 119 82 from from ADP iajs-2428 119 83 proposition proposition NOUN iajs-2428 119 84 16 16 NUM iajs-2428 119 85 , , PUNCT iajs-2428 119 86 we we PRON iajs-2428 119 87 obtain obtain VERB iajs-2428 119 88 𝒦 𝒦 PROPN iajs-2428 119 89 is be AUX iajs-2428 119 90 a a DET iajs-2428 119 91 λ λ NOUN iajs-2428 119 92 – – PUNCT iajs-2428 119 93 algebra algebra NOUN iajs-2428 119 94 of of ADP iajs-2428 119 95 a a DET iajs-2428 119 96 set set ADJ iajs-2428 119 97 𝔓. 𝔓. NOUN iajs-2428 119 98 let let VERB iajs-2428 119 99 cϵ cϵ VERB iajs-2428 119 100 𝒥 𝒥 PROPN iajs-2428 119 101 , , PUNCT iajs-2428 119 102 then then ADV iajs-2428 119 103 c c PROPN iajs-2428 119 104 ∩ ∩ PROPN iajs-2428 119 105 𝔇 𝔇 NOUN iajs-2428 119 106 ϵ𝒥|𝔇 ϵ𝒥|𝔇 NOUN iajs-2428 119 107 , , PUNCT iajs-2428 119 108 but but CCONJ iajs-2428 119 109 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 119 110 ⊆ ⊆ NUM iajs-2428 119 111 λ λ PROPN iajs-2428 119 112 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 119 113 implies imply VERB iajs-2428 119 114 that that SCONJ iajs-2428 119 115 c c PROPN iajs-2428 119 116 ∩ ∩ NOUN iajs-2428 119 117 𝔇 𝔇 PROPN iajs-2428 119 118 ϵ ϵ X iajs-2428 119 119 λ λ PROPN iajs-2428 119 120 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 119 121 , , PUNCT iajs-2428 119 122 hence hence ADV iajs-2428 119 123 cϵ cϵ VERB iajs-2428 119 124 𝒦 𝒦 PROPN iajs-2428 119 125 and and CCONJ iajs-2428 119 126 𝒥 𝒥 PROPN iajs-2428 119 127 ⊆ ⊆ NUM iajs-2428 119 128 𝒦. 𝒦. PROPN iajs-2428 119 129 let let VERB iajs-2428 119 130 bϵ bϵ PROPN iajs-2428 119 131 λ λ PROPN iajs-2428 119 132 𝒥 𝒥 PROPN iajs-2428 119 133 |𝔇 |𝔇 PROPN iajs-2428 119 134 , , PUNCT iajs-2428 119 135 then then ADV iajs-2428 119 136 b= b= VERB iajs-2428 119 137 f f PROPN iajs-2428 119 138 ∩ ∩ PROPN iajs-2428 119 139 𝔇 𝔇 PROPN iajs-2428 119 140 , , PUNCT iajs-2428 119 141 for for ADP iajs-2428 119 142 some some DET iajs-2428 119 143 fϵ fϵ NOUN iajs-2428 119 144 λ λ X iajs-2428 119 145 𝒥 𝒥 PROPN iajs-2428 119 146 . . PUNCT iajs-2428 120 1 but but CCONJ iajs-2428 120 2 λ λ X iajs-2428 120 3 𝒥 𝒥 PROPN iajs-2428 120 4 ⊆ ⊆ NUM iajs-2428 120 5 𝒦 𝒦 PROPN iajs-2428 120 6 , , PUNCT iajs-2428 120 7 then then ADV iajs-2428 120 8 fϵ fϵ ADP iajs-2428 120 9 𝒦 𝒦 PROPN iajs-2428 120 10 , , PUNCT iajs-2428 120 11 hence hence ADV iajs-2428 120 12 bϵ bϵ ADP iajs-2428 120 13 λ λ PROPN iajs-2428 120 14 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 120 15 and and CCONJ iajs-2428 120 16 λ λ X iajs-2428 120 17 𝒥 𝒥 PROPN iajs-2428 120 18 |𝔇 |𝔇 PUNCT iajs-2428 120 19 ⊆ ⊆ NUM iajs-2428 120 20 λ λ NOUN iajs-2428 120 21 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 120 22 , , PUNCT iajs-2428 120 23 consequently consequently ADV iajs-2428 120 24 λ λ X iajs-2428 120 25 𝒥|𝔇 𝒥|𝔇 NUM iajs-2428 120 26 = = PUNCT iajs-2428 121 1 λ λ NOUN iajs-2428 121 2 𝒥 𝒥 PROPN iajs-2428 121 3 |𝔇. |𝔇. PROPN iajs-2428 121 4 we we PRON iajs-2428 121 5 end end VERB iajs-2428 121 6 this this DET iajs-2428 121 7 section section NOUN iajs-2428 121 8 by by ADP iajs-2428 121 9 introduce introduce VERB iajs-2428 121 10 the the DET iajs-2428 121 11 relationships relationship NOUN iajs-2428 121 12 between between ADP iajs-2428 121 13 α α PROPN iajs-2428 121 14 – – PUNCT iajs-2428 121 15 σ σ NOUN iajs-2428 121 16 – – PUNCT iajs-2428 121 17 field field NOUN iajs-2428 121 18 , , PUNCT iajs-2428 121 19 monotone monotone ADJ iajs-2428 121 20 class class NOUN iajs-2428 121 21 , , PUNCT iajs-2428 121 22 β β X iajs-2428 121 23 – – PUNCT iajs-2428 121 24 σ σ NOUN iajs-2428 121 25 – – PUNCT iajs-2428 121 26 field field NOUN iajs-2428 121 27 and and CCONJ iajs-2428 121 28 λ λ NOUN iajs-2428 121 29 – – PUNCT iajs-2428 121 30 algebra algebra NOUN iajs-2428 121 31 . . PUNCT iajs-2428 122 1 proposition proposition NOUN iajs-2428 122 2 18 18 NUM iajs-2428 122 3 every every DET iajs-2428 122 4 λ λ NOUN iajs-2428 122 5 – – PUNCT iajs-2428 122 6 algebra algebra NOUN iajs-2428 122 7 is be AUX iajs-2428 122 8 a a DET iajs-2428 122 9 α α PROPN iajs-2428 122 10 – – PUNCT iajs-2428 122 11 σ σ NOUN iajs-2428 122 12 – – PUNCT iajs-2428 122 13 field field NOUN iajs-2428 122 14 . . PUNCT iajs-2428 123 1 proof proof NOUN iajs-2428 123 2 let let VERB iajs-2428 123 3 𝒦 𝒦 PRON iajs-2428 123 4 be be AUX iajs-2428 123 5 a a DET iajs-2428 123 6 λ λ NOUN iajs-2428 123 7 – – PUNCT iajs-2428 123 8 algebra algebra NOUN iajs-2428 123 9 of of ADP iajs-2428 123 10 a a DET iajs-2428 123 11 set set NOUN iajs-2428 123 12 𝔓. 𝔓. NOUN iajs-2428 123 13 then then ADV iajs-2428 123 14 by by ADP iajs-2428 123 15 definition definition NOUN iajs-2428 123 16 of of ADP iajs-2428 123 17 λ λ PROPN iajs-2428 123 18 – – PUNCT iajs-2428 123 19 algebra algebra NOUN iajs-2428 123 20 , , PUNCT iajs-2428 123 21 we we PRON iajs-2428 123 22 have have VERB iajs-2428 123 23 φ φ NUM iajs-2428 123 24 , , PUNCT iajs-2428 123 25 𝔓ϵ𝒦. 𝔓ϵ𝒦. VERB iajs-2428 123 26 let let VERB iajs-2428 123 27 d d NOUN iajs-2428 123 28 , , PUNCT iajs-2428 123 29 d d PROPN iajs-2428 123 30 , , PUNCT iajs-2428 123 31 … … PUNCT iajs-2428 123 32 ϵ𝒦. ϵ𝒦. NOUN iajs-2428 123 33 since since SCONJ iajs-2428 123 34 𝒦 𝒦 PROPN iajs-2428 123 35 is be AUX iajs-2428 123 36 a a DET iajs-2428 123 37 λ λ NOUN iajs-2428 123 38 – – PUNCT iajs-2428 123 39 algebra algebra NOUN iajs-2428 123 40 , , PUNCT iajs-2428 123 41 then then ADV iajs-2428 123 42 by by ADP iajs-2428 123 43 definition definition NOUN iajs-2428 123 44 of of ADP iajs-2428 123 45 𝒦 𝒦 PROPN iajs-2428 123 46 , , PUNCT iajs-2428 123 47 we we PRON iajs-2428 123 48 have have VERB iajs-2428 123 49 ⋃ ⋃ NOUN iajs-2428 123 50 d d PROPN iajs-2428 123 51 ϵ ϵ X iajs-2428 123 52 𝒦. 𝒦. PROPN iajs-2428 123 53 therefore therefore ADV iajs-2428 123 54 𝒦 𝒦 PROPN iajs-2428 123 55 is be AUX iajs-2428 123 56 a a DET iajs-2428 123 57 α α PROPN iajs-2428 123 58 – – PUNCT iajs-2428 123 59 σ σ NOUN iajs-2428 123 60 – – PUNCT iajs-2428 123 61 field field NOUN iajs-2428 123 62 . . PUNCT iajs-2428 124 1 in in ADP iajs-2428 124 2 general general ADJ iajs-2428 124 3 , , PUNCT iajs-2428 124 4 the the DET iajs-2428 124 5 converse converse NOUN iajs-2428 124 6 of of ADP iajs-2428 124 7 above above ADJ iajs-2428 124 8 proposition proposition NOUN iajs-2428 124 9 is be AUX iajs-2428 124 10 not not PART iajs-2428 124 11 true true ADJ iajs-2428 124 12 . . PUNCT iajs-2428 125 1 for for ADP iajs-2428 125 2 example example NOUN iajs-2428 125 3 , , PUNCT iajs-2428 125 4 if if SCONJ iajs-2428 125 5 𝔓 𝔓 PROPN iajs-2428 125 6 = = NOUN iajs-2428 125 7 { { PUNCT iajs-2428 125 8 1,2,3 1,2,3 NOUN iajs-2428 125 9 } } PUNCT iajs-2428 125 10 and and CCONJ iajs-2428 125 11 𝒦 𝒦 PROPN iajs-2428 125 12 { { PUNCT iajs-2428 125 13 φ φ PROPN iajs-2428 125 14 , , PUNCT iajs-2428 125 15 { { PUNCT iajs-2428 125 16 1},{1,3},𝔓 1},{1,3},𝔓 NUM iajs-2428 125 17 } } PUNCT iajs-2428 125 18 , , PUNCT iajs-2428 125 19 then then ADV iajs-2428 125 20 𝒦 𝒦 PROPN iajs-2428 125 21 is be AUX iajs-2428 125 22 α α NUM iajs-2428 125 23 – – PUNCT iajs-2428 125 24 σ σ NOUN iajs-2428 125 25 – – PUNCT iajs-2428 125 26 field field NOUN iajs-2428 125 27 but but CCONJ iajs-2428 125 28 not not PART iajs-2428 125 29 λ λ PROPN iajs-2428 125 30 – – PUNCT iajs-2428 125 31 algebra algebra NOUN iajs-2428 125 32 , , PUNCT iajs-2428 125 33 because because SCONJ iajs-2428 125 34 { { PUNCT iajs-2428 125 35 1,3}∈ 1,3}∈ NOUN iajs-2428 125 36 𝒦 𝒦 VERB iajs-2428 125 37 and and CCONJ iajs-2428 125 38 { { PUNCT iajs-2428 125 39 3}⊂{1,3 3}⊂{1,3 NUM iajs-2428 125 40 } } PUNCT iajs-2428 125 41 , , PUNCT iajs-2428 125 42 but but CCONJ iajs-2428 125 43 { { PUNCT iajs-2428 125 44 3}∉ 3}∉ NUM iajs-2428 125 45 𝒦. 𝒦. PROPN iajs-2428 125 46 proposition proposition NOUN iajs-2428 125 47 19 19 NUM iajs-2428 125 48 every every DET iajs-2428 125 49 λ λ NOUN iajs-2428 125 50 – – PUNCT iajs-2428 125 51 algebra algebra NOUN iajs-2428 125 52 is be AUX iajs-2428 125 53 a a DET iajs-2428 125 54 β β X iajs-2428 125 55 – – PUNCT iajs-2428 125 56 σ σ NOUN iajs-2428 125 57 – – PUNCT iajs-2428 125 58 field field NOUN iajs-2428 125 59 . . PUNCT iajs-2428 126 1 proof proof NOUN iajs-2428 126 2 the the DET iajs-2428 126 3 proof proof NOUN iajs-2428 126 4 follows follow VERB iajs-2428 126 5 from from ADP iajs-2428 126 6 proposition proposition NOUN iajs-2428 126 7 4 4 NUM iajs-2428 126 8 and and CCONJ iajs-2428 126 9 definition definition NOUN iajs-2428 126 10 of of ADP iajs-2428 126 11 λ λ PROPN iajs-2428 126 12 – – PUNCT iajs-2428 126 13 algebra algebra NOUN iajs-2428 126 14 . . PUNCT iajs-2428 127 1 in in ADP iajs-2428 127 2 general general ADJ iajs-2428 127 3 , , PUNCT iajs-2428 127 4 the the DET iajs-2428 127 5 converse converse NOUN iajs-2428 127 6 of of ADP iajs-2428 127 7 above above ADJ iajs-2428 127 8 proposition proposition NOUN iajs-2428 127 9 is be AUX iajs-2428 127 10 not not PART iajs-2428 127 11 true true ADJ iajs-2428 127 12 as as SCONJ iajs-2428 127 13 shown show VERB iajs-2428 127 14 in in ADP iajs-2428 127 15 following follow VERB iajs-2428 127 16 example example NOUN iajs-2428 127 17 . . PUNCT iajs-2428 127 18     SPACE iajs-2428 128 1 77 77 NUM iajs-2428 128 2 ibn ibn PROPN iajs-2428 128 3 al al PROPN iajs-2428 128 4 - - PUNCT iajs-2428 128 5 haitham haitham PROPN iajs-2428 128 6 jour jour X iajs-2428 128 7 . . PROPN iajs-2428 128 8 for for ADP iajs-2428 128 9 pure pure ADJ iajs-2428 128 10 & & CCONJ iajs-2428 128 11 appl appl PROPN iajs-2428 128 12 . . PUNCT iajs-2428 129 1 sci sci PROPN iajs-2428 129 2 . . PROPN iajs-2428 130 1 33 33 NUM iajs-2428 130 2 ( ( PUNCT iajs-2428 130 3 2 2 NUM iajs-2428 130 4 ) ) PUNCT iajs-2428 130 5 2020 2020 NUM iajs-2428 130 6 example example NOUN iajs-2428 130 7 20 20 NUM iajs-2428 130 8 let let VERB iajs-2428 130 9 𝔓 𝔓 PRON iajs-2428 130 10 = = NOUN iajs-2428 130 11 { { PUNCT iajs-2428 130 12 1,2,3,4 1,2,3,4 NUM iajs-2428 130 13 } } PUNCT iajs-2428 130 14 and and CCONJ iajs-2428 130 15 𝒦 𝒦 PROPN iajs-2428 130 16 { { PUNCT iajs-2428 130 17 φ φ PROPN iajs-2428 130 18 , , PUNCT iajs-2428 130 19 { { PUNCT iajs-2428 130 20 1},{1,3,4},{3,4},𝔓 1},{1,3,4},{3,4},𝔓 NUM iajs-2428 130 21 } } PUNCT iajs-2428 130 22 . . PUNCT iajs-2428 131 1 then then ADV iajs-2428 131 2 , , PUNCT iajs-2428 131 3 𝒦 𝒦 PROPN iajs-2428 131 4 is be AUX iajs-2428 131 5 β β X iajs-2428 131 6 – – PUNCT iajs-2428 131 7 σ σ NOUN iajs-2428 131 8 – – PUNCT iajs-2428 131 9 field field NOUN iajs-2428 131 10 but but CCONJ iajs-2428 131 11 not not PART iajs-2428 131 12 λ λ PROPN iajs-2428 131 13 – – PUNCT iajs-2428 131 14 algebra algebra NOUN iajs-2428 131 15 , , PUNCT iajs-2428 131 16 because because SCONJ iajs-2428 131 17 { { PUNCT iajs-2428 131 18 1,3,4}∈ 1,3,4}∈ NUM iajs-2428 131 19 𝒦 𝒦 NOUN iajs-2428 131 20 and and CCONJ iajs-2428 131 21 { { PUNCT iajs-2428 131 22 3,4}⊂{1,3,4 3,4}⊂{1,3,4 NOUN iajs-2428 131 23 } } PUNCT iajs-2428 131 24 , , PUNCT iajs-2428 131 25 but but CCONJ iajs-2428 131 26 { { PUNCT iajs-2428 131 27 3,4}∉ 3,4}∉ NUM iajs-2428 131 28 𝒦. 𝒦. PROPN iajs-2428 131 29 proposition proposition NOUN iajs-2428 131 30 21 21 NUM iajs-2428 131 31 every every DET iajs-2428 131 32 λ λ NOUN iajs-2428 131 33 – – PUNCT iajs-2428 131 34 algebra algebra NOUN iajs-2428 131 35 is be AUX iajs-2428 131 36 a a DET iajs-2428 131 37 monotone monotone ADJ iajs-2428 131 38 class class NOUN iajs-2428 131 39 . . PUNCT iajs-2428 132 1 proof proof NOUN iajs-2428 132 2 let let VERB iajs-2428 132 3 𝒦 𝒦 PRON iajs-2428 132 4 be be AUX iajs-2428 132 5 a a DET iajs-2428 132 6 λ λ NOUN iajs-2428 132 7 – – PUNCT iajs-2428 132 8 algebra algebra NOUN iajs-2428 132 9 of of ADP iajs-2428 132 10 a a DET iajs-2428 132 11 set set ADJ iajs-2428 132 12 𝔓 𝔓 PROPN iajs-2428 132 13 and and CCONJ iajs-2428 132 14 d d NOUN iajs-2428 132 15 , , PUNCT iajs-2428 132 16 d d X iajs-2428 132 17 , , PUNCT iajs-2428 132 18 … … PUNCT iajs-2428 132 19 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 132 20 such such ADJ iajs-2428 132 21 that that SCONJ iajs-2428 132 22 d d PROPN iajs-2428 132 23 ↑ ↑ PROPN iajs-2428 132 24 d. d. PROPN iajs-2428 132 25 then then ADV iajs-2428 132 26 ⋃ ⋃ PROPN iajs-2428 132 27 d d PROPN iajs-2428 132 28 d d PROPN iajs-2428 132 29 since since SCONJ iajs-2428 132 30 𝒦 𝒦 PROPN iajs-2428 132 31 is be AUX iajs-2428 132 32 a a DET iajs-2428 132 33 λ λ NOUN iajs-2428 132 34 – – PUNCT iajs-2428 132 35 algebra algebra NOUN iajs-2428 132 36 , , PUNCT iajs-2428 132 37 then then ADV iajs-2428 132 38 by by ADP iajs-2428 132 39 definition definition NOUN iajs-2428 132 40 of of ADP iajs-2428 132 41 𝒦 𝒦 PROPN iajs-2428 132 42 , , PUNCT iajs-2428 132 43 we we PRON iajs-2428 132 44 have have VERB iajs-2428 132 45 ⋃ ⋃ NOUN iajs-2428 132 46 d d NOUN iajs-2428 132 47 ϵ ϵ X iajs-2428 132 48 𝒦 𝒦 PROPN iajs-2428 132 49 which which PRON iajs-2428 132 50 implies imply VERB iajs-2428 132 51 that that SCONJ iajs-2428 132 52 dϵ dϵ PROPN iajs-2428 132 53 𝒦. 𝒦. PROPN iajs-2428 132 54 let let VERB iajs-2428 132 55 d d NOUN iajs-2428 132 56 , , PUNCT iajs-2428 132 57 d d PROPN iajs-2428 132 58 , , PUNCT iajs-2428 132 59 … … PUNCT iajs-2428 132 60 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 133 1 such such ADJ iajs-2428 133 2 that that SCONJ iajs-2428 133 3 d d PROPN iajs-2428 133 4 ↓ ↓ PROPN iajs-2428 133 5 d. d. PROPN iajs-2428 133 6 then then ADV iajs-2428 133 7 , , PUNCT iajs-2428 133 8 ⋂ ⋂ PROPN iajs-2428 133 9 d d X iajs-2428 133 10 d d PROPN iajs-2428 133 11 , , PUNCT iajs-2428 133 12 but but CCONJ iajs-2428 133 13 𝒦 𝒦 PROPN iajs-2428 133 14 is be AUX iajs-2428 133 15 a a DET iajs-2428 133 16 λ λ NOUN iajs-2428 133 17 – – PUNCT iajs-2428 133 18 algebra algebra NOUN iajs-2428 133 19 , , PUNCT iajs-2428 133 20 implies imply VERB iajs-2428 133 21 that that SCONJ iajs-2428 134 1 ⋂ ⋂ PROPN iajs-2428 135 1 d d PROPN iajs-2428 136 1 ϵ𝒦 ϵ𝒦 PROPN iajs-2428 136 2 and and CCONJ iajs-2428 136 3 d d NOUN iajs-2428 136 4 ϵ𝒦. ϵ𝒦. PROPN iajs-2428 136 5 hence hence ADV iajs-2428 136 6 𝒦 𝒦 PROPN iajs-2428 136 7 is be AUX iajs-2428 136 8 a a DET iajs-2428 136 9 monotone monotone ADJ iajs-2428 136 10 class class NOUN iajs-2428 136 11 . . PUNCT iajs-2428 137 1 in in ADP iajs-2428 137 2 general general ADJ iajs-2428 137 3 , , PUNCT iajs-2428 137 4 the the DET iajs-2428 137 5 converse converse NOUN iajs-2428 137 6 of of ADP iajs-2428 137 7 above above ADJ iajs-2428 137 8 proposition proposition NOUN iajs-2428 137 9 is be AUX iajs-2428 137 10 not not PART iajs-2428 137 11 true true ADJ iajs-2428 137 12 . . PUNCT iajs-2428 138 1 for for ADP iajs-2428 138 2 example example NOUN iajs-2428 138 3 , , PUNCT iajs-2428 138 4 if if SCONJ iajs-2428 138 5 𝔓 𝔓 PROPN iajs-2428 138 6 = = NOUN iajs-2428 138 7 { { PUNCT iajs-2428 138 8 1,2,3 1,2,3 NOUN iajs-2428 138 9 } } PUNCT iajs-2428 138 10 and and CCONJ iajs-2428 138 11 𝕄 𝕄 PROPN iajs-2428 138 12 { { PUNCT iajs-2428 138 13 φ,{1},{1,2 φ,{1},{1,2 NUM iajs-2428 138 14 } } PUNCT iajs-2428 138 15 } } PUNCT iajs-2428 138 16 , , PUNCT iajs-2428 138 17 then then ADV iajs-2428 138 18 𝕄 𝕄 PROPN iajs-2428 138 19 is be AUX iajs-2428 138 20 a a DET iajs-2428 138 21 monotone monotone ADJ iajs-2428 138 22 class class NOUN iajs-2428 138 23 , , PUNCT iajs-2428 138 24 but but CCONJ iajs-2428 138 25 not not PART iajs-2428 138 26 λ λ PROPN iajs-2428 138 27 – – PUNCT iajs-2428 138 28 algebra algebra NOUN iajs-2428 138 29 , , PUNCT iajs-2428 138 30 because because SCONJ iajs-2428 138 31 { { PUNCT iajs-2428 138 32 1,2}∈ 1,2}∈ PROPN iajs-2428 138 33 𝕄 𝕄 PROPN iajs-2428 138 34 and and CCONJ iajs-2428 138 35 { { PUNCT iajs-2428 138 36 2}⊂{1,2 2}⊂{1,2 NUM iajs-2428 138 37 } } PUNCT iajs-2428 138 38 , , PUNCT iajs-2428 138 39 but but CCONJ iajs-2428 138 40 { { PUNCT iajs-2428 138 41 2}∉ 2}∉ NUM iajs-2428 138 42 𝕄. 𝕄. NOUN iajs-2428 138 43 definition definition NOUN iajs-2428 138 44 22 22 NUM iajs-2428 138 45 [ [ X iajs-2428 138 46 6 6 NUM iajs-2428 138 47 ] ] PUNCT iajs-2428 138 48 let let VERB iajs-2428 138 49 𝒥 𝒥 PROPN iajs-2428 138 50 ⊆ ⊆ NUM iajs-2428 138 51 p p PROPN iajs-2428 138 52 𝔓 𝔓 PROPN iajs-2428 138 53 . . PUNCT iajs-2428 139 1 then then ADV iajs-2428 139 2 the the DET iajs-2428 139 3 intersection intersection NOUN iajs-2428 139 4 of of ADP iajs-2428 139 5 all all DET iajs-2428 139 6 monotone monotone ADJ iajs-2428 139 7 classes class NOUN iajs-2428 139 8 of of ADP iajs-2428 139 9 𝔓 𝔓 PROPN iajs-2428 139 10 which which PRON iajs-2428 139 11 include include VERB iajs-2428 139 12 𝒥 𝒥 PROPN iajs-2428 139 13 is be AUX iajs-2428 139 14 called call VERB iajs-2428 139 15 the the DET iajs-2428 139 16 monotone monotone ADJ iajs-2428 139 17 class class NOUN iajs-2428 139 18 generated generate VERB iajs-2428 139 19 by by ADP iajs-2428 139 20 𝒥 𝒥 PROPN iajs-2428 139 21 and and CCONJ iajs-2428 139 22 denoted denote VERB iajs-2428 139 23 by by ADP iajs-2428 139 24 𝕄 𝕄 PROPN iajs-2428 139 25 𝒥 𝒥 PROPN iajs-2428 139 26 , , PUNCT iajs-2428 139 27 that that ADV iajs-2428 139 28 is is ADV iajs-2428 139 29 , , PUNCT iajs-2428 139 30 𝕄 𝕄 PROPN iajs-2428 139 31 𝒥 𝒥 PROPN iajs-2428 139 32 = = SYM iajs-2428 139 33 ⋂ ⋂ PROPN iajs-2428 139 34 𝕄 𝕄 PROPN iajs-2428 139 35 : : PUNCT iajs-2428 139 36 𝕄 𝕄 PROPN iajs-2428 139 37 is be AUX iajs-2428 139 38 a a DET iajs-2428 139 39 monotone monotone ADJ iajs-2428 139 40 class class NOUN iajs-2428 139 41 of of ADP iajs-2428 139 42 𝔓 𝔓 PROPN iajs-2428 139 43 and and CCONJ iajs-2428 139 44 𝒥 𝒥 PROPN iajs-2428 139 45 ⊆ ⊆ NUM iajs-2428 139 46 𝕄 𝕄 PROPN iajs-2428 139 47 , , PUNCT iajs-2428 139 48 ∀i ∀i NOUN iajs-2428 139 49 ∈ ∈ PROPN iajs-2428 139 50 ι ι X iajs-2428 139 51 . . PUNCT iajs-2428 140 1 lemma lemma PROPN iajs-2428 140 2 23 23 NUM iajs-2428 141 1 [ [ X iajs-2428 141 2 6 6 NUM iajs-2428 141 3 ] ] PUNCT iajs-2428 141 4 let let VERB iajs-2428 141 5 𝕄 𝕄 PROPN iajs-2428 141 6 ∈ ∈ PROPN iajs-2428 141 7 be be AUX iajs-2428 141 8 a a DET iajs-2428 141 9 collection collection NOUN iajs-2428 141 10 of of ADP iajs-2428 141 11 monotone monotone ADJ iajs-2428 141 12 classes class NOUN iajs-2428 141 13 on on ADP iajs-2428 141 14 𝔓. 𝔓. PROPN iajs-2428 141 15 then then ADV iajs-2428 141 16 ⋂ ⋂ PROPN iajs-2428 141 17 𝕄∈ 𝕄∈ NOUN iajs-2428 141 18 is be AUX iajs-2428 141 19 a a DET iajs-2428 141 20 monotone monotone ADJ iajs-2428 141 21 class class NOUN iajs-2428 141 22 on on ADP iajs-2428 141 23 𝔓. 𝔓. ADJ iajs-2428 141 24 proposition proposition NOUN iajs-2428 141 25 24 24 NUM iajs-2428 141 26 [ [ X iajs-2428 141 27 6 6 NUM iajs-2428 141 28 ] ] PUNCT iajs-2428 141 29 let let VERB iajs-2428 141 30 𝒥 𝒥 PROPN iajs-2428 141 31 ⊆ ⊆ NUM iajs-2428 141 32 p p PROPN iajs-2428 141 33 𝔓 𝔓 PROPN iajs-2428 141 34 . . PUNCT iajs-2428 142 1 then then ADV iajs-2428 142 2 𝕄 𝕄 PROPN iajs-2428 142 3 𝒥 𝒥 PROPN iajs-2428 142 4 is be AUX iajs-2428 142 5 the the DET iajs-2428 142 6 smallest small ADJ iajs-2428 142 7 monotone monotone ADJ iajs-2428 142 8 class class NOUN iajs-2428 142 9 of of ADP iajs-2428 142 10 𝔓 𝔓 PROPN iajs-2428 142 11 which which PRON iajs-2428 142 12 includes include VERB iajs-2428 142 13 𝒥. 𝒥. NOUN iajs-2428 142 14 theorem theorem VERB iajs-2428 142 15 25 25 NUM iajs-2428 142 16 let let VERB iajs-2428 142 17 𝒥 𝒥 PROPN iajs-2428 142 18 ⊆ ⊆ NUM iajs-2428 142 19 p p PROPN iajs-2428 142 20 𝔓 𝔓 PROPN iajs-2428 142 21 . . PUNCT iajs-2428 143 1 then then ADV iajs-2428 143 2 𝕄 𝕄 PROPN iajs-2428 143 3 𝒥 𝒥 PROPN iajs-2428 143 4 ⊆ ⊆ NUM iajs-2428 143 5 λ λ SYM iajs-2428 143 6 𝒥 𝒥 PROPN iajs-2428 143 7 . . PUNCT iajs-2428 144 1 proof proof NOUN iajs-2428 144 2 let let VERB iajs-2428 144 3 𝒥 𝒥 PROPN iajs-2428 144 4 ⊆ ⊆ NUM iajs-2428 144 5 p p PROPN iajs-2428 144 6 𝔓 𝔓 PROPN iajs-2428 144 7 . . PUNCT iajs-2428 145 1 then then ADV iajs-2428 145 2 by by ADP iajs-2428 145 3 proposition proposition NOUN iajs-2428 145 4 7 7 NUM iajs-2428 145 5 , , PUNCT iajs-2428 145 6 we we PRON iajs-2428 145 7 have have VERB iajs-2428 145 8 λ λ X iajs-2428 145 9 𝒥 𝒥 PROPN iajs-2428 145 10 is be AUX iajs-2428 145 11 a a DET iajs-2428 145 12 λ λ NOUN iajs-2428 145 13 – – PUNCT iajs-2428 145 14 algebra algebra NOUN iajs-2428 145 15 of of ADP iajs-2428 145 16 𝔓 𝔓 PROPN iajs-2428 145 17 which which PRON iajs-2428 145 18 includes include VERB iajs-2428 145 19 𝒥. 𝒥. NOUN iajs-2428 145 20 from from ADP iajs-2428 145 21 proposition proposition NOUN iajs-2428 145 22 21 21 NUM iajs-2428 145 23 , , PUNCT iajs-2428 145 24 we we PRON iajs-2428 145 25 have have VERB iajs-2428 145 26 , , PUNCT iajs-2428 145 27 every every DET iajs-2428 145 28 λ λ NOUN iajs-2428 145 29 – – PUNCT iajs-2428 145 30 algebra algebra NOUN iajs-2428 145 31 is be AUX iajs-2428 145 32 a a DET iajs-2428 145 33 monotone monotone ADJ iajs-2428 145 34 class class NOUN iajs-2428 145 35 , , PUNCT iajs-2428 145 36 implies imply VERB iajs-2428 145 37 that that SCONJ iajs-2428 145 38 λ λ PROPN iajs-2428 145 39 𝒥 𝒥 PROPN iajs-2428 145 40 is be AUX iajs-2428 145 41 a a DET iajs-2428 145 42 monotone monotone ADJ iajs-2428 145 43 class class NOUN iajs-2428 145 44 which which PRON iajs-2428 145 45 includes include VERB iajs-2428 145 46 𝒥. 𝒥. PROPN iajs-2428 145 47 but but CCONJ iajs-2428 145 48 𝕄 𝕄 PROPN iajs-2428 145 49 𝒥 𝒥 PROPN iajs-2428 145 50 is be AUX iajs-2428 145 51 the the DET iajs-2428 145 52 smallest small ADJ iajs-2428 145 53 monotone monotone ADJ iajs-2428 145 54 class class NOUN iajs-2428 145 55 which which PRON iajs-2428 145 56 includes include VERB iajs-2428 145 57 𝒥 𝒥 PRON iajs-2428 145 58 by by ADP iajs-2428 145 59 proposition proposition NOUN iajs-2428 145 60 24 24 NUM iajs-2428 145 61 , , PUNCT iajs-2428 145 62 then then ADV iajs-2428 145 63 𝕄 𝕄 PROPN iajs-2428 145 64 𝒥 𝒥 PROPN iajs-2428 145 65 ⊆ ⊆ NUM iajs-2428 145 66 λ λ SYM iajs-2428 145 67 𝒥 𝒥 PROPN iajs-2428 145 68 . . PUNCT iajs-2428 146 1 3 3 X iajs-2428 146 2 . . X iajs-2428 146 3 measure measure NOUN iajs-2428 146 4 defined define VERB iajs-2428 146 5 on on ADP iajs-2428 146 6 𝛌 𝛌 PROPN iajs-2428 146 7 – – PUNCT iajs-2428 146 8 𝐚𝐥𝐠𝐞𝐛𝐫𝐚 𝐚𝐥𝐠𝐞𝐛𝐫𝐚 NOUN iajs-2428 146 9 our our PRON iajs-2428 146 10 aim aim NOUN iajs-2428 146 11 in in ADP iajs-2428 146 12 this this DET iajs-2428 146 13 section section NOUN iajs-2428 146 14 is be AUX iajs-2428 146 15 to to PART iajs-2428 146 16 prove prove VERB iajs-2428 146 17 that that SCONJ iajs-2428 146 18 any any DET iajs-2428 146 19 measure measure NOUN iajs-2428 146 20 defined define VERB iajs-2428 146 21 on on ADP iajs-2428 146 22 λ λ PROPN iajs-2428 146 23 – – PUNCT iajs-2428 146 24 algebra algebra NOUN iajs-2428 146 25 is be AUX iajs-2428 146 26 complete complete ADJ iajs-2428 146 27 . . PUNCT iajs-2428 147 1 we we PRON iajs-2428 147 2 begin begin VERB iajs-2428 147 3 with with ADP iajs-2428 147 4 the the DET iajs-2428 147 5 notions notion NOUN iajs-2428 147 6 of of ADP iajs-2428 147 7 measure measure NOUN iajs-2428 147 8 on on ADP iajs-2428 147 9 λ λ PROPN iajs-2428 147 10 – – PUNCT iajs-2428 147 11 algebra algebra NOUN iajs-2428 147 12 . . PUNCT iajs-2428 148 1 definition definition NOUN iajs-2428 148 2 26 26 NUM iajs-2428 148 3 let let VERB iajs-2428 148 4 ( ( PUNCT iajs-2428 148 5 𝔓 𝔓 NOUN iajs-2428 148 6 , , PUNCT iajs-2428 148 7 𝒦 𝒦 PROPN iajs-2428 148 8 ) ) PUNCT iajs-2428 148 9 is be AUX iajs-2428 148 10 measurable measurable ADJ iajs-2428 148 11 space space NOUN iajs-2428 148 12 relative relative ADJ iajs-2428 148 13 to to ADP iajs-2428 148 14 the the DET iajs-2428 148 15 λ λ NOUN iajs-2428 148 16 – – PUNCT iajs-2428 148 17 algebra algebra NOUN iajs-2428 148 18 𝒦. 𝒦. PROPN iajs-2428 148 19 then then ADV iajs-2428 148 20 , , PUNCT iajs-2428 148 21 a a DET iajs-2428 148 22 set set NOUN iajs-2428 148 23 function function NOUN iajs-2428 148 24 𝔐 𝔐 PROPN iajs-2428 148 25 , , PUNCT iajs-2428 148 26 𝔐 𝔐 PROPN iajs-2428 148 27 : : PUNCT iajs-2428 149 1 𝒦 𝒦 PROPN iajs-2428 149 2 → → SYM iajs-2428 149 3 0 0 NUM iajs-2428 149 4 , , PUNCT iajs-2428 149 5 ∞ ∞ PROPN iajs-2428 149 6 is be AUX iajs-2428 149 7 called call VERB iajs-2428 149 8 measure measure NOUN iajs-2428 149 9 relative relative ADJ iajs-2428 149 10 to to ADP iajs-2428 149 11 the the DET iajs-2428 149 12 λ λ NOUN iajs-2428 149 13 – – PUNCT iajs-2428 149 14 algebra algebra NOUN iajs-2428 149 15 𝒦 𝒦 X iajs-2428 149 16 if if SCONJ iajs-2428 149 17 whenever whenever SCONJ iajs-2428 149 18 𝐷 𝐷 PROPN iajs-2428 149 19 , , PUNCT iajs-2428 149 20 𝐷 𝐷 PROPN iajs-2428 149 21 , , PUNCT iajs-2428 149 22 … … PUNCT iajs-2428 149 23 form form VERB iajs-2428 149 24 a a DET iajs-2428 149 25 finite finite NOUN iajs-2428 149 26 or or CCONJ iajs-2428 149 27 countably countably ADV iajs-2428 149 28 infinite infinite ADJ iajs-2428 149 29 collection collection NOUN iajs-2428 149 30 of of ADP iajs-2428 149 31 disjoint disjoint NOUN iajs-2428 149 32 sets set NOUN iajs-2428 149 33 in in ADP iajs-2428 149 34 𝒦 𝒦 PROPN iajs-2428 149 35 , , PUNCT iajs-2428 149 36 we we PRON iajs-2428 149 37 have have VERB iajs-2428 149 38 𝔐 𝔐 PRON iajs-2428 149 39 ⋃ ⋃ NOUN iajs-2428 149 40 𝐷 𝐷 NOUN iajs-2428 149 41 ∑ ∑ PROPN iajs-2428 149 42 𝔐 𝔐 PROPN iajs-2428 149 43 𝐷 𝐷 NOUN iajs-2428 149 44 and and CCONJ iajs-2428 149 45 𝔐 𝔐 PROPN iajs-2428 149 46 φ φ NOUN iajs-2428 149 47 0 0 NUM iajs-2428 149 48 . . PUNCT iajs-2428 149 49     SPACE iajs-2428 150 1 78 78 NUM iajs-2428 150 2 ibn ibn PROPN iajs-2428 150 3 al al PROPN iajs-2428 150 4 - - PUNCT iajs-2428 150 5 haitham haitham PROPN iajs-2428 150 6 jour jour X iajs-2428 150 7 . . PROPN iajs-2428 150 8 for for ADP iajs-2428 150 9 pure pure ADJ iajs-2428 150 10 & & CCONJ iajs-2428 150 11 appl appl PROPN iajs-2428 150 12 . . PUNCT iajs-2428 151 1 sci sci PROPN iajs-2428 151 2 . . PROPN iajs-2428 152 1 33 33 NUM iajs-2428 152 2 ( ( PUNCT iajs-2428 152 3 2 2 NUM iajs-2428 152 4 ) ) PUNCT iajs-2428 152 5 2020 2020 NUM iajs-2428 152 6 example example NOUN iajs-2428 152 7 27 27 NUM iajs-2428 152 8 let let VERB iajs-2428 152 9 𝔓 𝔓 PRON iajs-2428 152 10 = = NOUN iajs-2428 152 11 { { PUNCT iajs-2428 152 12 1,2,3 1,2,3 NOUN iajs-2428 152 13 } } PUNCT iajs-2428 152 14 and and CCONJ iajs-2428 152 15 𝒦 𝒦 PROPN iajs-2428 152 16 { { PUNCT iajs-2428 152 17 φ,{1},{3},{1,3},𝔓 φ,{1},{3},{1,3},𝔓 NUM iajs-2428 152 18 } } PUNCT iajs-2428 152 19 . . PUNCT iajs-2428 153 1 then then ADV iajs-2428 153 2 ( ( PUNCT iajs-2428 153 3 𝔓 𝔓 NOUN iajs-2428 153 4 , , PUNCT iajs-2428 153 5 𝒦 𝒦 PROPN iajs-2428 153 6 ) ) PUNCT iajs-2428 153 7 is be AUX iajs-2428 153 8 measurable measurable ADJ iajs-2428 153 9 space space NOUN iajs-2428 153 10 relative relative ADJ iajs-2428 153 11 to to ADP iajs-2428 153 12 the the DET iajs-2428 153 13 λ λ NOUN iajs-2428 153 14 – – PUNCT iajs-2428 153 15 algebra algebra NOUN iajs-2428 153 16 𝒦. 𝒦. PROPN iajs-2428 153 17 if if SCONJ iajs-2428 153 18 we we PRON iajs-2428 153 19 define define VERB iajs-2428 153 20 a a DET iajs-2428 153 21 set set NOUN iajs-2428 153 22 function function NOUN iajs-2428 153 23 𝔐 𝔐 NOUN iajs-2428 153 24 : : PUNCT iajs-2428 153 25 𝒦 𝒦 PROPN iajs-2428 153 26 → → SYM iajs-2428 153 27 0 0 NUM iajs-2428 153 28 , , PUNCT iajs-2428 153 29 ∞ ∞ NUM iajs-2428 153 30 by by ADP iajs-2428 153 31 𝔐(𝐷 𝔐(𝐷 PROPN iajs-2428 153 32 ) ) PUNCT iajs-2428 154 1 = = SYM iajs-2428 154 2 𝑜 𝑜 NOUN iajs-2428 154 3 ; ; PUNCT iajs-2428 154 4 𝑖𝑓 𝑖𝑓 NUM iajs-2428 154 5 𝐷 𝐷 PROPN iajs-2428 154 6 φ φ PROPN iajs-2428 154 7 ; ; PUNCT iajs-2428 154 8 𝑖𝑓 𝑖𝑓 NUM iajs-2428 154 9 𝐷 𝐷 PROPN iajs-2428 154 10 1 1 NUM iajs-2428 154 11 𝑜𝑟 𝑜𝑟 ADP iajs-2428 154 12 3 3 NUM iajs-2428 154 13 1 1 NUM iajs-2428 154 14 ; ; PUNCT iajs-2428 154 15 𝑜𝑡ℎ𝑒𝑟 𝑜𝑡ℎ𝑒𝑟 PROPN iajs-2428 154 16 𝑤𝑖𝑠𝑒 𝑤𝑖𝑠𝑒 NOUN iajs-2428 154 17 then then ADV iajs-2428 154 18 𝔐 𝔐 PROPN iajs-2428 154 19 is be AUX iajs-2428 154 20 a a DET iajs-2428 154 21 measure measure NOUN iajs-2428 154 22 relative relative ADJ iajs-2428 154 23 to to ADP iajs-2428 154 24 the the DET iajs-2428 154 25 λ λ NOUN iajs-2428 154 26 – – PUNCT iajs-2428 154 27 algebra algebra NOUN iajs-2428 154 28 𝒦. 𝒦. PROPN iajs-2428 154 29 definition definition NOUN iajs-2428 154 30 28 28 NUM iajs-2428 154 31 a a DET iajs-2428 154 32 measure measure NOUN iajs-2428 154 33 space space NOUN iajs-2428 154 34 relative relative ADJ iajs-2428 154 35 to to ADP iajs-2428 154 36 the the DET iajs-2428 154 37 λ λ NOUN iajs-2428 154 38 – – PUNCT iajs-2428 154 39 algebra algebra NOUN iajs-2428 154 40 𝒦 𝒦 PROPN iajs-2428 154 41 is be AUX iajs-2428 154 42 a a DET iajs-2428 154 43 triple triple ADJ iajs-2428 154 44 ( ( PUNCT iajs-2428 154 45 𝔓 𝔓 NOUN iajs-2428 154 46 , , PUNCT iajs-2428 154 47 𝒦 𝒦 PROPN iajs-2428 154 48 , , PUNCT iajs-2428 154 49 𝔐 𝔐 PROPN iajs-2428 154 50 ) ) PUNCT iajs-2428 154 51 where where SCONJ iajs-2428 154 52 ( ( PUNCT iajs-2428 154 53 𝔓 𝔓 NOUN iajs-2428 154 54 , , PUNCT iajs-2428 154 55 𝒦 𝒦 PROPN iajs-2428 154 56 ) ) PUNCT iajs-2428 154 57 is be AUX iajs-2428 154 58 measurable measurable ADJ iajs-2428 154 59 space space NOUN iajs-2428 154 60 relative relative ADJ iajs-2428 154 61 to to ADP iajs-2428 154 62 the the DET iajs-2428 154 63 λ λ NOUN iajs-2428 154 64 – – PUNCT iajs-2428 154 65 algebra algebra NOUN iajs-2428 154 66 𝒦 𝒦 PROPN iajs-2428 154 67 and and CCONJ iajs-2428 154 68 𝔐 𝔐 PROPN iajs-2428 154 69 is be AUX iajs-2428 154 70 a a DET iajs-2428 154 71 measure measure NOUN iajs-2428 154 72 relative relative ADJ iajs-2428 154 73 to to ADP iajs-2428 154 74 the the DET iajs-2428 154 75 λ λ NOUN iajs-2428 154 76 – – PUNCT iajs-2428 154 77 algebra algebra NOUN iajs-2428 154 78 𝒦. 𝒦. PROPN iajs-2428 154 79 in in ADP iajs-2428 154 80 the the DET iajs-2428 154 81 following following NOUN iajs-2428 154 82 theorem theorem NOUN iajs-2428 154 83 , , PUNCT iajs-2428 154 84 we we PRON iajs-2428 154 85 use use VERB iajs-2428 154 86 mathematical mathematical ADJ iajs-2428 154 87 induction induction NOUN iajs-2428 154 88 to to PART iajs-2428 154 89 prove prove VERB iajs-2428 154 90 that that SCONJ iajs-2428 154 91 the the DET iajs-2428 154 92 linear linear ADJ iajs-2428 154 93 combination combination NOUN iajs-2428 154 94 of of ADP iajs-2428 154 95 measure measure NOUN iajs-2428 154 96 relative relative ADJ iajs-2428 154 97 to to ADP iajs-2428 154 98 the the DET iajs-2428 154 99 λ λ NOUN iajs-2428 154 100 – – PUNCT iajs-2428 154 101 algebra algebra NOUN iajs-2428 154 102 𝒦 𝒦 PROPN iajs-2428 154 103 is be AUX iajs-2428 154 104 also also ADV iajs-2428 154 105 measure measure NOUN iajs-2428 154 106 relative relative ADJ iajs-2428 154 107 to to ADP iajs-2428 154 108 the the DET iajs-2428 154 109 λ λ NOUN iajs-2428 154 110 – – PUNCT iajs-2428 154 111 algebra algebra NOUN iajs-2428 154 112 𝒦. 𝒦. PROPN iajs-2428 154 113 theorem theorem VERB iajs-2428 154 114 29 29 NUM iajs-2428 154 115 let let VERB iajs-2428 154 116 ( ( PUNCT iajs-2428 154 117 𝔓 𝔓 NOUN iajs-2428 154 118 , , PUNCT iajs-2428 154 119 𝒦 𝒦 PROPN iajs-2428 154 120 , , PUNCT iajs-2428 154 121 𝔐 𝔐 PROPN iajs-2428 154 122 ) ) PUNCT iajs-2428 154 123 be be AUX iajs-2428 154 124 a a DET iajs-2428 154 125 measure measure NOUN iajs-2428 154 126 space space NOUN iajs-2428 154 127 relative relative ADJ iajs-2428 154 128 to to ADP iajs-2428 154 129 the the DET iajs-2428 154 130 λ λ NOUN iajs-2428 154 131 – – PUNCT iajs-2428 154 132 algebra algebra NOUN iajs-2428 154 133 𝒦 𝒦 PROPN iajs-2428 154 134 and and CCONJ iajs-2428 154 135 𝑐 𝑐 ADP iajs-2428 154 136 ∈ ∈ PROPN iajs-2428 154 137 0 0 NUM iajs-2428 154 138 , , PUNCT iajs-2428 154 139 ∞ ∞ PROPN iajs-2428 154 140 for for ADP iajs-2428 154 141 all all DET iajs-2428 154 142 𝑗 𝑗 DET iajs-2428 154 143 1,2 1,2 NUM iajs-2428 154 144 , , PUNCT iajs-2428 154 145 … … PUNCT iajs-2428 154 146 , , PUNCT iajs-2428 154 147 𝑘. 𝑘. VERB iajs-2428 154 148 if if SCONJ iajs-2428 154 149 a a DET iajs-2428 154 150 set set NOUN iajs-2428 154 151 function function NOUN iajs-2428 154 152 ∑ ∑ PROPN iajs-2428 154 153 𝑐 𝑐 PROPN iajs-2428 154 154 𝔐 𝔐 PROPN iajs-2428 154 155 : : PUNCT iajs-2428 154 156 ℘ ℘ PROPN iajs-2428 154 157 → → SYM iajs-2428 154 158 0 0 NUM iajs-2428 154 159 , , PUNCT iajs-2428 154 160 ∞ ∞ PROPN iajs-2428 154 161 is be AUX iajs-2428 154 162 defined define VERB iajs-2428 154 163 by by ADP iajs-2428 154 164 : : PUNCT iajs-2428 154 165 ∑ ∑ PUNCT iajs-2428 154 166 𝑐 𝑐 PROPN iajs-2428 154 167 𝔐 𝔐 PRON iajs-2428 154 168 𝐷 𝐷 PROPN iajs-2428 154 169 ∑ ∑ PROPN iajs-2428 154 170 𝑐 𝑐 PROPN iajs-2428 154 171 . . PUNCT iajs-2428 155 1 𝔐 𝔐 PROPN iajs-2428 155 2 𝐷 𝐷 NOUN iajs-2428 155 3 ∀𝐷𝜖℘ ∀𝐷𝜖℘ NOUN iajs-2428 155 4 , , PUNCT iajs-2428 155 5 then then ADV iajs-2428 155 6 ( ( PUNCT iajs-2428 155 7 𝔓 𝔓 NOUN iajs-2428 155 8 , , PUNCT iajs-2428 155 9 𝒦 𝒦 PROPN iajs-2428 155 10 , , PUNCT iajs-2428 155 11 ∑ ∑ PROPN iajs-2428 155 12 𝑐 𝑐 PROPN iajs-2428 155 13 𝔐 𝔐 PROPN iajs-2428 155 14 ) ) PUNCT iajs-2428 155 15 is be AUX iajs-2428 155 16 measure measure NOUN iajs-2428 155 17 space space NOUN iajs-2428 155 18 relative relative ADJ iajs-2428 155 19 to to ADP iajs-2428 155 20 the the DET iajs-2428 155 21 λ λ NOUN iajs-2428 155 22 – – PUNCT iajs-2428 155 23 algebra algebra NOUN iajs-2428 155 24 𝒦. 𝒦. NOUN iajs-2428 155 25 proof proof NOUN iajs-2428 155 26 if if SCONJ iajs-2428 155 27 𝑘 𝑘 PROPN iajs-2428 155 28 2 2 NUM iajs-2428 155 29 , , PUNCT iajs-2428 155 30 then then ADV iajs-2428 155 31 𝑐 𝑐 PROPN iajs-2428 155 32 𝔐 𝔐 INTJ iajs-2428 155 33 𝑐 𝑐 NOUN iajs-2428 155 34 𝔐 𝔐 PROPN iajs-2428 155 35 φ φ PROPN iajs-2428 155 36 𝑐 𝑐 PROPN iajs-2428 155 37 . . PUNCT iajs-2428 156 1 𝔐 𝔐 PRON iajs-2428 156 2 φ φ PROPN iajs-2428 156 3 𝑐 𝑐 PROPN iajs-2428 156 4 . . PUNCT iajs-2428 157 1 𝔐 𝔐 PRON iajs-2428 157 2 φ φ PROPN iajs-2428 157 3 𝑐 𝑐 PROPN iajs-2428 157 4 . . PUNCT iajs-2428 157 5 0 0 NUM iajs-2428 158 1 𝑐 𝑐 PROPN iajs-2428 158 2 . . PUNCT iajs-2428 158 3 0 0 NUM iajs-2428 158 4 0 0 NUM iajs-2428 158 5 let let VERB iajs-2428 158 6 𝐷 𝐷 PROPN iajs-2428 158 7 , , PUNCT iajs-2428 158 8 𝐷 𝐷 PROPN iajs-2428 158 9 , , PUNCT iajs-2428 158 10 … … PUNCT iajs-2428 158 11 are be AUX iajs-2428 158 12 disjoint disjoint NOUN iajs-2428 158 13 sets set NOUN iajs-2428 158 14 in in ADP iajs-2428 158 15 𝒦. 𝒦. PROPN iajs-2428 158 16 since since SCONJ iajs-2428 158 17 𝔐 𝔐 PRON iajs-2428 158 18 is be AUX iajs-2428 158 19 measure measure NOUN iajs-2428 158 20 relative relative ADJ iajs-2428 158 21 to to ADP iajs-2428 158 22 the the DET iajs-2428 158 23 λ λ NOUN iajs-2428 158 24 – – PUNCT iajs-2428 158 25 algebra algebra NOUN iajs-2428 158 26 𝒦 𝒦 PROPN iajs-2428 158 27 , , PUNCT iajs-2428 158 28 𝑗 𝑗 PROPN iajs-2428 158 29 1,2 1,2 NUM iajs-2428 158 30 then then ADV iajs-2428 158 31 , , PUNCT iajs-2428 158 32 𝔐 𝔐 PROPN iajs-2428 158 33 ⋃ ⋃ NOUN iajs-2428 158 34 𝐷 𝐷 NOUN iajs-2428 158 35 ∑ ∑ PROPN iajs-2428 158 36 𝔐 𝔐 PROPN iajs-2428 158 37 𝐷 𝐷 NOUN iajs-2428 158 38 . . PUNCT iajs-2428 159 1 so so ADV iajs-2428 159 2 , , PUNCT iajs-2428 159 3 we we PRON iajs-2428 159 4 have have VERB iajs-2428 159 5 𝑐 𝑐 PRON iajs-2428 159 6 𝔐 𝔐 VERB iajs-2428 159 7 𝑐 𝑐 NOUN iajs-2428 159 8 𝔐 𝔐 PROPN iajs-2428 159 9 ⋃ ⋃ PROPN iajs-2428 159 10 𝐷 𝐷 NOUN iajs-2428 159 11 𝑐 𝑐 NOUN iajs-2428 159 12 . . PUNCT iajs-2428 160 1 𝔐 𝔐 PRON iajs-2428 160 2 ⋃ ⋃ PROPN iajs-2428 160 3 𝐷 𝐷 NOUN iajs-2428 160 4 𝑐 𝑐 NOUN iajs-2428 160 5 . . PUNCT iajs-2428 161 1 𝔐 𝔐 PRON iajs-2428 161 2 ⋃ ⋃ PROPN iajs-2428 161 3 𝐷 𝐷 NOUN iajs-2428 161 4 𝑐 𝑐 PROPN iajs-2428 161 5 . . PUNCT iajs-2428 162 1 ∑ ∑ PUNCT iajs-2428 162 2 𝔐 𝔐 PROPN iajs-2428 162 3 𝐷 𝐷 NOUN iajs-2428 162 4 𝑐 𝑐 PROPN iajs-2428 162 5 . . PUNCT iajs-2428 163 1 ∑ ∑ PUNCT iajs-2428 163 2 𝔐 𝔐 PRON iajs-2428 163 3 𝐷 𝐷 PROPN iajs-2428 163 4 ∑ ∑ NOUN iajs-2428 163 5 𝑐 𝑐 PROPN iajs-2428 163 6 . . PUNCT iajs-2428 164 1 𝔐 𝔐 PRON iajs-2428 164 2 𝐷 𝐷 PROPN iajs-2428 164 3 ∑ ∑ PROPN iajs-2428 164 4 𝑐 𝑐 PROPN iajs-2428 164 5 . . PUNCT iajs-2428 165 1 𝔐 𝔐 PRON iajs-2428 165 2 𝐷 𝐷 PROPN iajs-2428 165 3 ∑ ∑ PROPN iajs-2428 165 4 𝑐 𝑐 PROPN iajs-2428 165 5 . . PUNCT iajs-2428 166 1 𝔐 𝔐 PRON iajs-2428 166 2 𝐷 𝐷 PROPN iajs-2428 166 3 𝑐 𝑐 PROPN iajs-2428 166 4 . . PUNCT iajs-2428 167 1 𝔐 𝔐 PRON iajs-2428 167 2 𝐷 𝐷 PROPN iajs-2428 167 3 ∑ ∑ PROPN iajs-2428 167 4 𝑐 𝑐 PROPN iajs-2428 167 5 𝔐 𝔐 NOUN iajs-2428 167 6 𝑐 𝑐 NOUN iajs-2428 167 7 𝔐 𝔐 NOUN iajs-2428 167 8 𝐷 𝐷 NOUN iajs-2428 167 9 hence hence ADV iajs-2428 167 10 , , PUNCT iajs-2428 167 11 ( ( PUNCT iajs-2428 167 12 𝔓 𝔓 NOUN iajs-2428 167 13 , , PUNCT iajs-2428 167 14 𝒦 𝒦 PROPN iajs-2428 167 15 , , PUNCT iajs-2428 167 16 𝑐 𝑐 PROPN iajs-2428 167 17 𝔐 𝔐 PRON iajs-2428 167 18 𝑐 𝑐 PROPN iajs-2428 167 19 𝔐 𝔐 PROPN iajs-2428 167 20 ) ) PUNCT iajs-2428 167 21 is be AUX iajs-2428 167 22 measure measure NOUN iajs-2428 167 23 space space NOUN iajs-2428 167 24 relative relative ADJ iajs-2428 167 25 to to ADP iajs-2428 167 26 the the DET iajs-2428 167 27 λ λ NOUN iajs-2428 167 28 – – PUNCT iajs-2428 167 29 algebra algebra NOUN iajs-2428 167 30 𝒦. 𝒦. PROPN iajs-2428 167 31 now now ADV iajs-2428 167 32 , , PUNCT iajs-2428 167 33 we we PRON iajs-2428 167 34 assume assume VERB iajs-2428 167 35 that that SCONJ iajs-2428 167 36 ( ( PUNCT iajs-2428 167 37 𝔓 𝔓 NOUN iajs-2428 167 38 , , PUNCT iajs-2428 167 39 𝒦 𝒦 PROPN iajs-2428 167 40 , , PUNCT iajs-2428 167 41 ∑ ∑ PROPN iajs-2428 167 42 𝑐 𝑐 PROPN iajs-2428 167 43 𝔐 𝔐 PROPN iajs-2428 167 44 ) ) PUNCT iajs-2428 167 45 is be AUX iajs-2428 167 46 measure measure NOUN iajs-2428 167 47 space space NOUN iajs-2428 167 48 relative relative ADJ iajs-2428 167 49 to to ADP iajs-2428 167 50 the the DET iajs-2428 167 51 λ λ NOUN iajs-2428 167 52 – – PUNCT iajs-2428 167 53 algebra algebra NOUN iajs-2428 167 54 𝒦 𝒦 PROPN iajs-2428 167 55 , , PUNCT iajs-2428 167 56 when when SCONJ iajs-2428 167 57 𝑘 𝑘 PRON iajs-2428 167 58 m m VERB iajs-2428 167 59 and and CCONJ iajs-2428 167 60 we we PRON iajs-2428 167 61 prove prove VERB iajs-2428 167 62 this this DET iajs-2428 167 63 fact fact NOUN iajs-2428 167 64 when when SCONJ iajs-2428 167 65 𝑘 𝑘 PRON iajs-2428 167 66 m m VERB iajs-2428 167 67 1 1 NUM iajs-2428 167 68 . . PUNCT iajs-2428 168 1 let let VERB iajs-2428 168 2 ( ( PUNCT iajs-2428 168 3 𝔓 𝔓 NOUN iajs-2428 168 4 , , PUNCT iajs-2428 168 5 𝒦 𝒦 PROPN iajs-2428 168 6 , , PUNCT iajs-2428 168 7 𝔐 𝔐 PROPN iajs-2428 168 8 ) ) PUNCT iajs-2428 168 9 be be AUX iajs-2428 168 10 a a DET iajs-2428 168 11 measure measure NOUN iajs-2428 168 12 space space NOUN iajs-2428 168 13 relative relative ADJ iajs-2428 168 14 to to ADP iajs-2428 168 15 the the DET iajs-2428 168 16 λ λ NOUN iajs-2428 168 17 – – PUNCT iajs-2428 168 18 algebra algebra NOUN iajs-2428 168 19 𝒦 𝒦 PROPN iajs-2428 168 20 and and CCONJ iajs-2428 168 21 𝑐 𝑐 ADP iajs-2428 168 22 ∈ ∈ PROPN iajs-2428 168 23 0 0 NUM iajs-2428 168 24 , , PUNCT iajs-2428 168 25 ∞ ∞ PROPN iajs-2428 168 26 for for ADP iajs-2428 168 27 all all DET iajs-2428 168 28 𝑗 𝑗 DET iajs-2428 168 29 1,2 1,2 NUM iajs-2428 168 30 , , PUNCT iajs-2428 168 31 … … PUNCT iajs-2428 168 32 , , PUNCT iajs-2428 168 33 𝑚 𝑚 PROPN iajs-2428 168 34 , , PUNCT iajs-2428 168 35 𝑚 𝑚 PROPN iajs-2428 168 36 1 1 NUM iajs-2428 168 37 . . PUNCT iajs-2428 169 1 then then ADV iajs-2428 169 2 ∑ ∑ ADP iajs-2428 169 3 𝑐 𝑐 PROPN iajs-2428 169 4 𝔐 𝔐 PROPN iajs-2428 169 5 φ φ NOUN iajs-2428 169 6 ∑ ∑ PROPN iajs-2428 169 7 𝑐 𝑐 PROPN iajs-2428 169 8 𝔐 𝔐 PROPN iajs-2428 169 9 𝑐 𝑐 NOUN iajs-2428 169 10 𝔐 𝔐 PROPN iajs-2428 169 11 φ φ NOUN iajs-2428 169 12 ∑ ∑ PROPN iajs-2428 169 13 𝑐 𝑐 PROPN iajs-2428 169 14 . . PUNCT iajs-2428 170 1 𝔐 𝔐 PRON iajs-2428 170 2 φ φ PROPN iajs-2428 170 3 𝑐 𝑐 PROPN iajs-2428 170 4 . . PUNCT iajs-2428 171 1 𝔐 𝔐 PRON iajs-2428 171 2 φ φ NOUN iajs-2428 171 3 0 0 PUNCT iajs-2428 171 4 since since SCONJ iajs-2428 171 5 , , PUNCT iajs-2428 171 6 𝔐 𝔐 PROPN iajs-2428 171 7 is be AUX iajs-2428 171 8 measure measure NOUN iajs-2428 171 9 relative relative ADJ iajs-2428 171 10 to to ADP iajs-2428 171 11 the the DET iajs-2428 171 12 λ λ NOUN iajs-2428 171 13 – – PUNCT iajs-2428 171 14 algebra algebra NOUN iajs-2428 171 15 𝒦. 𝒦. PROPN iajs-2428 171 16 let let VERB iajs-2428 171 17 𝐷 𝐷 PROPN iajs-2428 171 18 , , PUNCT iajs-2428 171 19 𝐷 𝐷 PROPN iajs-2428 171 20 , , PUNCT iajs-2428 171 21 … … PUNCT iajs-2428 171 22 are be AUX iajs-2428 171 23 disjoint disjoint NOUN iajs-2428 171 24 sets set NOUN iajs-2428 171 25 in in ADP iajs-2428 171 26 𝒦. 𝒦. PROPN iajs-2428 171 27 since since SCONJ iajs-2428 171 28 ( ( PUNCT iajs-2428 171 29 𝔓 𝔓 PROPN iajs-2428 171 30 , , PUNCT iajs-2428 171 31 𝒦 𝒦 PROPN iajs-2428 171 32 , , PUNCT iajs-2428 171 33 ∑ ∑ PROPN iajs-2428 171 34 𝑐 𝑐 PROPN iajs-2428 171 35 𝔐 𝔐 PROPN iajs-2428 171 36 ) ) PUNCT iajs-2428 171 37 is be AUX iajs-2428 171 38 measure measure NOUN iajs-2428 171 39 space space NOUN iajs-2428 171 40 relative relative ADJ iajs-2428 171 41 to to ADP iajs-2428 171 42 the the DET iajs-2428 171 43 λ λ NOUN iajs-2428 171 44 – – PUNCT iajs-2428 171 45 algebra algebra NOUN iajs-2428 171 46 𝒦 𝒦 PROPN iajs-2428 171 47 , , PUNCT iajs-2428 171 48 then then ADV iajs-2428 171 49 ∑ ∑ ADP iajs-2428 171 50 𝑐 𝑐 VERB iajs-2428 171 51 𝔐 𝔐 PROPN iajs-2428 171 52 ⋃ ⋃ NOUN iajs-2428 171 53 𝐷 𝐷 NOUN iajs-2428 171 54 ∑ ∑ PROPN iajs-2428 171 55 ∑ ∑ PROPN iajs-2428 171 56 𝑐 𝑐 PROPN iajs-2428 171 57 𝔐 𝔐 PROPN iajs-2428 171 58 𝐷 𝐷 PROPN iajs-2428 171 59 . . PUNCT iajs-2428 172 1 so so ADV iajs-2428 172 2 , , PUNCT iajs-2428 172 3 we we PRON iajs-2428 172 4 have have VERB iajs-2428 172 5     SPACE iajs-2428 172 6 79 79 NUM iajs-2428 172 7 ibn ibn PROPN iajs-2428 172 8 al al PROPN iajs-2428 172 9 - - PUNCT iajs-2428 172 10 haitham haitham PROPN iajs-2428 172 11 jour jour X iajs-2428 172 12 . . PROPN iajs-2428 173 1 for for ADP iajs-2428 173 2 pure pure ADJ iajs-2428 173 3 & & CCONJ iajs-2428 173 4 appl appl PROPN iajs-2428 173 5 . . PUNCT iajs-2428 174 1 sci sci PROPN iajs-2428 174 2 . . PROPN iajs-2428 175 1 33 33 NUM iajs-2428 175 2 ( ( PUNCT iajs-2428 175 3 2 2 NUM iajs-2428 175 4 ) ) PUNCT iajs-2428 175 5 2020 2020 NUM iajs-2428 175 6 ∑ ∑ PUNCT iajs-2428 175 7 𝑐 𝑐 PROPN iajs-2428 175 8 𝔐 𝔐 PROPN iajs-2428 175 9 ⋃ ⋃ NOUN iajs-2428 175 10 𝐷 𝐷 PROPN iajs-2428 175 11 ∑ ∑ PROPN iajs-2428 175 12 𝑐 𝑐 PROPN iajs-2428 175 13 𝔐 𝔐 NOUN iajs-2428 175 14 𝑐 𝑐 NOUN iajs-2428 175 15 𝔐 𝔐 PROPN iajs-2428 175 16 ⋃ ⋃ NOUN iajs-2428 175 17 𝐷 𝐷 NOUN iajs-2428 175 18 ∑ ∑ PROPN iajs-2428 175 19 𝑐 𝑐 PROPN iajs-2428 175 20 . . PUNCT iajs-2428 176 1 𝔐 𝔐 PRON iajs-2428 176 2 ⋃ ⋃ PROPN iajs-2428 176 3 𝐷 𝐷 NOUN iajs-2428 176 4 𝑐 𝑐 NOUN iajs-2428 176 5 . . PUNCT iajs-2428 177 1 𝔐 𝔐 PRON iajs-2428 177 2 ⋃ ⋃ NOUN iajs-2428 177 3 𝐷 𝐷 PROPN iajs-2428 177 4 ∑ ∑ PROPN iajs-2428 177 5 𝑐 𝑐 PROPN iajs-2428 177 6 𝔐 𝔐 PROPN iajs-2428 177 7 ⋃ ⋃ PROPN iajs-2428 177 8 𝐷 𝐷 NOUN iajs-2428 177 9 𝑐 𝑐 NOUN iajs-2428 177 10 . . PUNCT iajs-2428 178 1 𝔐 𝔐 PRON iajs-2428 178 2 ⋃ ⋃ NOUN iajs-2428 178 3 𝐷 𝐷 NOUN iajs-2428 178 4 ∑ ∑ PROPN iajs-2428 178 5 ∑ ∑ PROPN iajs-2428 178 6 𝑐 𝑐 PROPN iajs-2428 178 7 𝔐 𝔐 PROPN iajs-2428 178 8 𝐷 𝐷 PROPN iajs-2428 178 9 𝑐 𝑐 PROPN iajs-2428 178 10 . . PUNCT iajs-2428 179 1 ∑ ∑ PUNCT iajs-2428 179 2 𝔐 𝔐 PRON iajs-2428 179 3 𝐷 𝐷 NOUN iajs-2428 179 4 ∑ ∑ PROPN iajs-2428 179 5 ∑ ∑ PROPN iajs-2428 179 6 𝑐 𝑐 PROPN iajs-2428 179 7 . . PUNCT iajs-2428 180 1 𝔐 𝔐 PRON iajs-2428 180 2 𝐷 𝐷 PROPN iajs-2428 180 3 ∑ ∑ PROPN iajs-2428 180 4 𝑐 𝑐 PROPN iajs-2428 180 5 . . PUNCT iajs-2428 181 1 𝔐 𝔐 PRON iajs-2428 181 2 𝐷 𝐷 PROPN iajs-2428 181 3 ∑ ∑ PROPN iajs-2428 181 4 ∑ ∑ PROPN iajs-2428 181 5 𝑐 𝑐 PROPN iajs-2428 181 6 . . PUNCT iajs-2428 182 1 𝔐 𝔐 PRON iajs-2428 182 2 𝐷 𝐷 PROPN iajs-2428 182 3 𝑐 𝑐 PROPN iajs-2428 182 4 . . PUNCT iajs-2428 183 1 𝔐 𝔐 PRON iajs-2428 183 2 𝐷 𝐷 PROPN iajs-2428 183 3 ∑ ∑ PROPN iajs-2428 183 4 ∑ ∑ PROPN iajs-2428 183 5 𝑐 𝑐 PROPN iajs-2428 183 6 𝔐 𝔐 NOUN iajs-2428 183 7 𝑐 𝑐 NOUN iajs-2428 183 8 𝔐 𝔐 PRON iajs-2428 183 9 𝐷 𝐷 PROPN iajs-2428 183 10 ∑ ∑ PROPN iajs-2428 183 11 ∑ ∑ PROPN iajs-2428 183 12 𝑐 𝑐 PROPN iajs-2428 183 13 𝔐 𝔐 PROPN iajs-2428 183 14 𝐷 𝐷 PROPN iajs-2428 183 15 . . PUNCT iajs-2428 184 1 hence hence ADV iajs-2428 184 2 , , PUNCT iajs-2428 184 3 ∑ ∑ PROPN iajs-2428 184 4 𝑐 𝑐 PROPN iajs-2428 184 5 𝔐 𝔐 PROPN iajs-2428 184 6 is be AUX iajs-2428 184 7 measure measure NOUN iajs-2428 184 8 relative relative ADJ iajs-2428 184 9 to to ADP iajs-2428 184 10 𝒦 𝒦 PROPN iajs-2428 184 11 , , PUNCT iajs-2428 184 12 therefore therefore ADV iajs-2428 184 13 ( ( PUNCT iajs-2428 184 14 𝔓 𝔓 NOUN iajs-2428 184 15 , , PUNCT iajs-2428 184 16 𝒦 𝒦 PROPN iajs-2428 184 17 , , PUNCT iajs-2428 184 18 ∑ ∑ PROPN iajs-2428 184 19 𝑐 𝑐 PROPN iajs-2428 184 20 𝔐 𝔐 PROPN iajs-2428 184 21 ) ) PUNCT iajs-2428 184 22 is be AUX iajs-2428 184 23 measure measure NOUN iajs-2428 184 24 space space NOUN iajs-2428 184 25 relative relative ADJ iajs-2428 184 26 to to ADP iajs-2428 184 27 the the DET iajs-2428 184 28 λ λ NOUN iajs-2428 184 29 – – PUNCT iajs-2428 184 30 algebra algebra NOUN iajs-2428 184 31 𝒦. 𝒦. PROPN iajs-2428 184 32 definition definition NOUN iajs-2428 184 33 30 30 NUM iajs-2428 185 1 [ [ X iajs-2428 185 2 1 1 X iajs-2428 185 3 ] ] PUNCT iajs-2428 185 4 a a DET iajs-2428 185 5 measure measure NOUN iajs-2428 185 6 on on ADP iajs-2428 185 7 a a DET iajs-2428 185 8 σ σ NOUN iajs-2428 185 9 – – PUNCT iajs-2428 185 10 field field NOUN iajs-2428 185 11 𝒦 𝒦 PROPN iajs-2428 185 12 is be AUX iajs-2428 185 13 a a DET iajs-2428 185 14 nonnegative nonnegative ADJ iajs-2428 185 15 , , PUNCT iajs-2428 185 16 extended extended ADJ iajs-2428 185 17 real real ADV iajs-2428 185 18 - - PUNCT iajs-2428 185 19 valued value VERB iajs-2428 185 20 set set NOUN iajs-2428 185 21 function function NOUN iajs-2428 185 22 𝔐 𝔐 PROPN iajs-2428 185 23 on on ADP iajs-2428 185 24 𝒦 𝒦 PROPN iajs-2428 185 25 such such ADJ iajs-2428 185 26 that that SCONJ iajs-2428 185 27 whenever whenever SCONJ iajs-2428 185 28 𝐴 𝐴 PROPN iajs-2428 185 29 , , PUNCT iajs-2428 185 30 𝐴 𝐴 PROPN iajs-2428 185 31 , , PUNCT iajs-2428 185 32 … … PUNCT iajs-2428 185 33 form form VERB iajs-2428 185 34 a a DET iajs-2428 185 35 finite finite NOUN iajs-2428 185 36 or or CCONJ iajs-2428 185 37 countably countably ADV iajs-2428 185 38 infinite infinite ADJ iajs-2428 185 39 collection collection NOUN iajs-2428 185 40 of of ADP iajs-2428 185 41 disjoint disjoint NOUN iajs-2428 185 42 sets set NOUN iajs-2428 185 43 in in ADP iajs-2428 185 44 𝒦 𝒦 PROPN iajs-2428 185 45 , , PUNCT iajs-2428 185 46 we we PRON iajs-2428 185 47 have have VERB iajs-2428 185 48 , , PUNCT iajs-2428 185 49 𝔐 𝔐 PROPN iajs-2428 185 50 ⋃ ⋃ NOUN iajs-2428 185 51 𝐴 𝐴 PROPN iajs-2428 185 52 ∑ ∑ PUNCT iajs-2428 185 53 𝔐 𝔐 PROPN iajs-2428 185 54 𝐴 𝐴 PROPN iajs-2428 185 55 . . PUNCT iajs-2428 186 1 definition definition NOUN iajs-2428 186 2 31 31 NUM iajs-2428 187 1 [ [ X iajs-2428 187 2 1 1 NUM iajs-2428 187 3 , , PUNCT iajs-2428 187 4 3 3 X iajs-2428 187 5 ] ] PUNCT iajs-2428 187 6 a a DET iajs-2428 187 7 measure measure NOUN iajs-2428 187 8 𝔐 𝔐 NOUN iajs-2428 187 9 on on ADP iajs-2428 187 10 a a DET iajs-2428 187 11 σ σ NOUN iajs-2428 187 12 – – PUNCT iajs-2428 187 13 field field NOUN iajs-2428 187 14 𝒦 𝒦 PROPN iajs-2428 187 15 is be AUX iajs-2428 187 16 said say VERB iajs-2428 187 17 to to PART iajs-2428 187 18 be be AUX iajs-2428 187 19 complete complete ADJ iajs-2428 187 20 iff iff PROPN iajs-2428 187 21 whenever whenever SCONJ iajs-2428 187 22 a a DET iajs-2428 187 23 ϵ𝒦and ϵ𝒦and PROPN iajs-2428 187 24 𝔐 𝔐 PROPN iajs-2428 187 25 𝐴 𝐴 PROPN iajs-2428 187 26 0 0 NUM iajs-2428 187 27 , , PUNCT iajs-2428 187 28 we we PRON iajs-2428 187 29 have have VERB iajs-2428 187 30 b b X iajs-2428 187 31 ϵ𝒦for ϵ𝒦for PROPN iajs-2428 187 32 all all DET iajs-2428 187 33 𝐵 𝐵 NOUN iajs-2428 187 34 ⊂ ⊂ X iajs-2428 187 35 𝐴. 𝐴. PROPN iajs-2428 188 1 the the DET iajs-2428 188 2 following follow VERB iajs-2428 188 3 example example NOUN iajs-2428 188 4 shows show VERB iajs-2428 188 5 that that SCONJ iajs-2428 188 6 , , PUNCT iajs-2428 188 7 if if SCONJ iajs-2428 188 8 𝔐 𝔐 PRON iajs-2428 188 9 is be AUX iajs-2428 188 10 a a DET iajs-2428 188 11 measure measure NOUN iajs-2428 188 12 on on ADP iajs-2428 188 13 σ σ PROPN iajs-2428 188 14 – – PUNCT iajs-2428 188 15 field field NOUN iajs-2428 188 16 𝒦 𝒦 PROPN iajs-2428 188 17 , , PUNCT iajs-2428 188 18 then then ADV iajs-2428 188 19 not not PART iajs-2428 188 20 necessarily necessarily ADV iajs-2428 188 21 that that SCONJ iajs-2428 188 22 𝔐 𝔐 PRON iajs-2428 188 23 is be AUX iajs-2428 188 24 complete complete ADJ iajs-2428 188 25 . . PUNCT iajs-2428 189 1 example example NOUN iajs-2428 189 2 32 32 NUM iajs-2428 189 3 let let VERB iajs-2428 189 4 𝔓 𝔓 PRON iajs-2428 189 5 = = NOUN iajs-2428 189 6 { { PUNCT iajs-2428 189 7 1,2,3 1,2,3 NOUN iajs-2428 189 8 } } PUNCT iajs-2428 189 9 and and CCONJ iajs-2428 189 10 𝒦 𝒦 PROPN iajs-2428 189 11 { { PUNCT iajs-2428 189 12 φ,{1},{2,3},𝔓 φ,{1},{2,3},𝔓 PROPN iajs-2428 189 13 } } PUNCT iajs-2428 189 14 . . PUNCT iajs-2428 190 1 then then ADV iajs-2428 190 2 𝒦 𝒦 PROPN iajs-2428 190 3 is be AUX iajs-2428 190 4 σ σ NOUN iajs-2428 190 5 – – PUNCT iajs-2428 190 6 field field NOUN iajs-2428 190 7 of of ADP iajs-2428 190 8 a a DET iajs-2428 190 9 set set ADJ iajs-2428 190 10 𝔓 𝔓 NOUN iajs-2428 190 11 . . PUNCT iajs-2428 191 1 if if SCONJ iajs-2428 191 2 we we PRON iajs-2428 191 3 define define VERB iajs-2428 191 4 a a DET iajs-2428 191 5 set set NOUN iajs-2428 191 6 function function NOUN iajs-2428 191 7 𝔐 𝔐 NOUN iajs-2428 191 8 : : PUNCT iajs-2428 191 9 𝒦 𝒦 PROPN iajs-2428 191 10 → → SYM iajs-2428 191 11 0 0 NUM iajs-2428 191 12 , , PUNCT iajs-2428 191 13 ∞ ∞ NUM iajs-2428 191 14 by by ADP iajs-2428 191 15 𝔐(𝐷 𝔐(𝐷 PROPN iajs-2428 191 16 ) ) PUNCT iajs-2428 192 1 = = SYM iajs-2428 192 2 𝑜 𝑜 NOUN iajs-2428 192 3 ; ; PUNCT iajs-2428 192 4 𝑖𝑓 𝑖𝑓 NUM iajs-2428 192 5 𝐷 𝐷 PROPN iajs-2428 192 6 φ φ PROPN iajs-2428 192 7 𝑜𝑟 𝑜𝑟 ADP iajs-2428 192 8 𝐷 𝐷 PROPN iajs-2428 192 9 2,3 2,3 NUM iajs-2428 192 10 1 1 NUM iajs-2428 192 11 ; ; PUNCT iajs-2428 192 12 𝑜𝑡ℎ𝑒𝑟 𝑜𝑡ℎ𝑒𝑟 PROPN iajs-2428 192 13 𝑤𝑖𝑠𝑒 𝑤𝑖𝑠𝑒 NOUN iajs-2428 192 14 then then ADV iajs-2428 192 15 𝔐 𝔐 PROPN iajs-2428 192 16 is be AUX iajs-2428 192 17 a a DET iajs-2428 192 18 measure measure NOUN iajs-2428 192 19 on on ADP iajs-2428 192 20 σ σ PROPN iajs-2428 192 21 – – PUNCT iajs-2428 192 22 field field NOUN iajs-2428 192 23 𝒦 𝒦 PROPN iajs-2428 192 24 , , PUNCT iajs-2428 192 25 it it PRON iajs-2428 192 26 is be AUX iajs-2428 192 27 clear clear ADJ iajs-2428 192 28 that that SCONJ iajs-2428 192 29 𝔐 𝔐 PRON iajs-2428 192 30 is be AUX iajs-2428 192 31 not not PART iajs-2428 192 32 complete complete ADJ iajs-2428 192 33 , , PUNCT iajs-2428 192 34 because because SCONJ iajs-2428 192 35 { { PUNCT iajs-2428 192 36 2,3}∈ 2,3}∈ NOUN iajs-2428 192 37 𝒦 𝒦 PROPN iajs-2428 192 38 and and CCONJ iajs-2428 192 39 𝔐 𝔐 PRON iajs-2428 192 40 2,3 2,3 NUM iajs-2428 192 41 0 0 NUM iajs-2428 192 42 , , PUNCT iajs-2428 192 43 now now ADV iajs-2428 192 44 { { PUNCT iajs-2428 192 45 2},{3}⊂{2,3 2},{3}⊂{2,3 NUM iajs-2428 192 46 } } PUNCT iajs-2428 192 47 , , PUNCT iajs-2428 192 48 but but CCONJ iajs-2428 192 49 { { PUNCT iajs-2428 192 50 2},{3}∉ 2},{3}∉ NUM iajs-2428 192 51 𝒦. 𝒦. PROPN iajs-2428 192 52 theorem theorem VERB iajs-2428 192 53 33 33 NUM iajs-2428 192 54 every every DET iajs-2428 192 55 measure measure NOUN iajs-2428 192 56 relative relative ADJ iajs-2428 192 57 to to ADP iajs-2428 192 58 the the DET iajs-2428 192 59 λ λ NOUN iajs-2428 192 60 – – PUNCT iajs-2428 192 61 algebra algebra NOUN iajs-2428 192 62 is be AUX iajs-2428 192 63 complete complete ADJ iajs-2428 192 64 . . PUNCT iajs-2428 193 1 proof proof NOUN iajs-2428 193 2 let let VERB iajs-2428 193 3 𝔐 𝔐 PRON iajs-2428 193 4 be be AUX iajs-2428 193 5 a a DET iajs-2428 193 6 measure measure NOUN iajs-2428 193 7 relative relative ADJ iajs-2428 193 8 to to ADP iajs-2428 193 9 the the DET iajs-2428 193 10 λ λ NOUN iajs-2428 193 11 – – PUNCT iajs-2428 193 12 algebra algebra NOUN iajs-2428 193 13 𝒦. 𝒦. PROPN iajs-2428 193 14 assume assume VERB iajs-2428 193 15 that that SCONJ iajs-2428 193 16 a a DET iajs-2428 193 17 ϵ𝒦 ϵ𝒦 NOUN iajs-2428 193 18 such such ADJ iajs-2428 193 19 that that SCONJ iajs-2428 193 20 𝔐 𝔐 PROPN iajs-2428 193 21 𝐴 𝐴 PROPN iajs-2428 193 22 0 0 NUM iajs-2428 193 23 , , PUNCT iajs-2428 193 24 since since SCONJ iajs-2428 193 25 𝒦 𝒦 PROPN iajs-2428 193 26 is be AUX iajs-2428 193 27 a a DET iajs-2428 193 28 λ λ NOUN iajs-2428 193 29 – – PUNCT iajs-2428 193 30 algebra algebra NOUN iajs-2428 193 31 , , PUNCT iajs-2428 193 32 then then ADV iajs-2428 193 33 b b X iajs-2428 193 34 ϵ𝒦for ϵ𝒦for PROPN iajs-2428 193 35 all all DET iajs-2428 193 36 𝐵 𝐵 NOUN iajs-2428 193 37 ⊂ ⊂ PROPN iajs-2428 193 38 𝐴. 𝐴. PROPN iajs-2428 193 39 therefore therefore ADV iajs-2428 193 40 𝔐 𝔐 PROPN iajs-2428 193 41 is be AUX iajs-2428 193 42 complete complete ADJ iajs-2428 193 43 measure measure NOUN iajs-2428 193 44 . . PUNCT iajs-2428 194 1 example example NOUN iajs-2428 195 1 34 34 NUM iajs-2428 195 2 let let VERB iajs-2428 195 3 𝔓 𝔓 PROPN iajs-2428 195 4 = = NOUN iajs-2428 195 5 { { PUNCT iajs-2428 195 6 a a PRON iajs-2428 195 7 , , PUNCT iajs-2428 195 8 b b NOUN iajs-2428 195 9 , , PUNCT iajs-2428 195 10 c c X iajs-2428 195 11 , , PUNCT iajs-2428 195 12 d d NOUN iajs-2428 195 13 } } PUNCT iajs-2428 195 14 and and CCONJ iajs-2428 195 15 𝒦 𝒦 PROPN iajs-2428 195 16 { { PUNCT iajs-2428 195 17 φ,{a},{c},{d},{a φ,{a},{c},{d},{a ADV iajs-2428 195 18 , , PUNCT iajs-2428 195 19 c},{c c},{c NOUN iajs-2428 195 20 , , PUNCT iajs-2428 195 21 d},{a d},{a ADV iajs-2428 195 22 , , PUNCT iajs-2428 195 23 d},{a d},{a ADV iajs-2428 195 24 , , PUNCT iajs-2428 195 25 c c NOUN iajs-2428 195 26 , , PUNCT iajs-2428 195 27 d},𝔓 d},𝔓 ADP iajs-2428 195 28 } } PUNCT iajs-2428 195 29 . . PUNCT iajs-2428 196 1 then then ADV iajs-2428 196 2 𝒦 𝒦 PROPN iajs-2428 196 3 is be AUX iajs-2428 196 4 λ λ NOUN iajs-2428 196 5 – – PUNCT iajs-2428 196 6 algebra algebra NOUN iajs-2428 196 7 of of ADP iajs-2428 196 8 a a DET iajs-2428 196 9 set set ADJ iajs-2428 196 10 𝔓 𝔓 NOUN iajs-2428 196 11 . . PUNCT iajs-2428 197 1 if if SCONJ iajs-2428 197 2 we we PRON iajs-2428 197 3 define define VERB iajs-2428 197 4 a a DET iajs-2428 197 5 set set NOUN iajs-2428 197 6 function function NOUN iajs-2428 197 7 𝔐 𝔐 NOUN iajs-2428 197 8 : : PUNCT iajs-2428 197 9 𝒦 𝒦 PROPN iajs-2428 197 10 → → SYM iajs-2428 197 11 0 0 NUM iajs-2428 197 12 , , PUNCT iajs-2428 197 13 ∞ ∞ NUM iajs-2428 197 14 by by ADP iajs-2428 197 15 𝔐(𝐷 𝔐(𝐷 PROPN iajs-2428 197 16 ) ) PUNCT iajs-2428 198 1 = = SYM iajs-2428 198 2 𝑜 𝑜 NOUN iajs-2428 198 3 ; ; PUNCT iajs-2428 198 4 𝑖𝑓 𝑖𝑓 ADP iajs-2428 198 5 𝐷 𝐷 PROPN iajs-2428 198 6 𝔓 𝔓 PROPN iajs-2428 198 7 1 1 NUM iajs-2428 198 8 ; ; PUNCT iajs-2428 198 9 𝑖𝑓 𝑖𝑓 NUM iajs-2428 198 10 𝐷 𝐷 PROPN iajs-2428 198 11 𝔓 𝔓 PROPN iajs-2428 198 12     SPACE iajs-2428 198 13 80 80 NUM iajs-2428 198 14 ibn ibn PROPN iajs-2428 198 15 al al PROPN iajs-2428 198 16 - - PUNCT iajs-2428 198 17 haitham haitham PROPN iajs-2428 198 18 jour jour X iajs-2428 198 19 . . PROPN iajs-2428 199 1 for for ADP iajs-2428 199 2 pure pure ADJ iajs-2428 199 3 & & CCONJ iajs-2428 199 4 appl appl PROPN iajs-2428 199 5 . . PUNCT iajs-2428 200 1 sci sci PROPN iajs-2428 200 2 . . PROPN iajs-2428 201 1 33 33 NUM iajs-2428 201 2 ( ( PUNCT iajs-2428 201 3 2 2 NUM iajs-2428 201 4 ) ) PUNCT iajs-2428 201 5 2020 2020 NUM iajs-2428 202 1 then then ADV iajs-2428 202 2 𝔐 𝔐 PRON iajs-2428 202 3 is be AUX iajs-2428 202 4 a a DET iajs-2428 202 5 measure measure NOUN iajs-2428 202 6 on on ADP iajs-2428 202 7 λ λ PROPN iajs-2428 202 8 – – PUNCT iajs-2428 202 9 algebra algebra NOUN iajs-2428 202 10 𝒦. 𝒦. PROPN iajs-2428 202 11 now now ADV iajs-2428 202 12 , , PUNCT iajs-2428 202 13 for for ADP iajs-2428 202 14 any any DET iajs-2428 202 15 aϵ𝒦 aϵ𝒦 NOUN iajs-2428 202 16 such such ADJ iajs-2428 202 17 that that SCONJ iajs-2428 202 18 𝔐 𝔐 PROPN iajs-2428 202 19 𝐴 𝐴 PROPN iajs-2428 202 20 0 0 NUM iajs-2428 202 21 , , PUNCT iajs-2428 202 22 then then ADV iajs-2428 202 23 bϵ𝒦 bϵ𝒦 NOUN iajs-2428 202 24 for for ADP iajs-2428 202 25 all all DET iajs-2428 202 26 𝐵 𝐵 PROPN iajs-2428 202 27 ⊂ ⊂ PROPN iajs-2428 202 28 𝐴. 𝐴. PROPN iajs-2428 202 29 therefore therefore ADV iajs-2428 202 30 𝔐 𝔐 PROPN iajs-2428 202 31 is be AUX iajs-2428 202 32 complete complete ADJ iajs-2428 202 33 measure measure NOUN iajs-2428 202 34 . . PUNCT iajs-2428 203 1 4 4 X iajs-2428 203 2 . . X iajs-2428 203 3 conclusions conclusion NOUN iajs-2428 203 4 the the DET iajs-2428 203 5 main main ADJ iajs-2428 203 6 results result NOUN iajs-2428 203 7 of of ADP iajs-2428 203 8 this this DET iajs-2428 203 9 paper paper NOUN iajs-2428 203 10 are be AUX iajs-2428 203 11 the the DET iajs-2428 203 12 following follow VERB iajs-2428 203 13 : : PUNCT iajs-2428 203 14 ( ( PUNCT iajs-2428 203 15 1 1 X iajs-2428 203 16 ) ) PUNCT iajs-2428 203 17 let let VERB iajs-2428 203 18 𝒦 𝒦 PROPN iajs-2428 203 19 ∈ ∈ PRON iajs-2428 203 20 be be AUX iajs-2428 203 21 a a DET iajs-2428 203 22 collection collection NOUN iajs-2428 203 23 of of ADP iajs-2428 203 24 λ λ PROPN iajs-2428 203 25 – – PUNCT iajs-2428 203 26 algebra algebra NOUN iajs-2428 203 27 on on ADP iajs-2428 203 28 𝔓. 𝔓. PROPN iajs-2428 203 29 then then ADV iajs-2428 203 30 ⋂ ⋂ PROPN iajs-2428 203 31 𝒦∈ 𝒦∈ PROPN iajs-2428 203 32 is be AUX iajs-2428 203 33 a a DET iajs-2428 203 34 λ λ NOUN iajs-2428 203 35 – – PUNCT iajs-2428 203 36 algebra algebra NOUN iajs-2428 203 37 on on ADP iajs-2428 203 38 𝔓. 𝔓. PROPN iajs-2428 203 39 ( ( PUNCT iajs-2428 203 40 2 2 X iajs-2428 203 41 ) ) PUNCT iajs-2428 203 42 let let VERB iajs-2428 203 43 𝒥 𝒥 PRON iajs-2428 203 44 ⊆ ⊆ NUM iajs-2428 203 45 p p PROPN iajs-2428 203 46 𝔓 𝔓 PROPN iajs-2428 203 47 . . PUNCT iajs-2428 204 1 then then ADV iajs-2428 204 2 λ λ X iajs-2428 204 3 𝒥 𝒥 PROPN iajs-2428 204 4 is be AUX iajs-2428 204 5 the the DET iajs-2428 204 6 smallest small ADJ iajs-2428 204 7 λ λ NOUN iajs-2428 204 8 – – PUNCT iajs-2428 204 9 algebra algebra NOUN iajs-2428 204 10 of of ADP iajs-2428 204 11 𝔓 𝔓 PROPN iajs-2428 204 12 which which PRON iajs-2428 204 13 includes include VERB iajs-2428 204 14 𝒥. 𝒥. PROPN iajs-2428 204 15 ( ( PUNCT iajs-2428 204 16 3 3 X iajs-2428 204 17 ) ) PUNCT iajs-2428 204 18 let let VERB iajs-2428 204 19 𝒥 𝒥 PRON iajs-2428 204 20 ⊆ ⊆ NUM iajs-2428 204 21 p p PROPN iajs-2428 204 22 𝔓 𝔓 PROPN iajs-2428 204 23 . . PUNCT iajs-2428 205 1 then then ADV iajs-2428 205 2 𝒥 𝒥 PRON iajs-2428 205 3 is be AUX iajs-2428 205 4 a a DET iajs-2428 205 5 λ λ NOUN iajs-2428 205 6 – – PUNCT iajs-2428 205 7 algebra algebra NOUN iajs-2428 205 8 of of ADP iajs-2428 205 9 a a DET iajs-2428 205 10 set set NOUN iajs-2428 205 11 𝔓 𝔓 NOUN iajs-2428 205 12 if if SCONJ iajs-2428 206 1 and and CCONJ iajs-2428 206 2 only only ADV iajs-2428 206 3 if if SCONJ iajs-2428 206 4 𝒥 𝒥 PROPN iajs-2428 206 5 λ λ VERB iajs-2428 206 6 𝒥 𝒥 PROPN iajs-2428 206 7 . . PUNCT iajs-2428 207 1 ( ( PUNCT iajs-2428 207 2 4 4 X iajs-2428 207 3 ) ) PUNCT iajs-2428 207 4 let let VERB iajs-2428 207 5 𝒥 𝒥 PRON iajs-2428 207 6 ⊆ ⊆ NUM iajs-2428 207 7 p p PROPN iajs-2428 207 8 𝔓 𝔓 PROPN iajs-2428 207 9 and and CCONJ iajs-2428 207 10 φ φ NUM iajs-2428 207 11 𝔇 𝔇 PROPN iajs-2428 207 12 ⊆ ⊆ NUM iajs-2428 207 13 𝔓. 𝔓. PROPN iajs-2428 207 14 if if SCONJ iajs-2428 207 15 𝒦 𝒦 PROPN iajs-2428 207 16 is be AUX iajs-2428 207 17 a a DET iajs-2428 207 18 λ λ NOUN iajs-2428 207 19 – – PUNCT iajs-2428 207 20 algebra algebra NOUN iajs-2428 207 21 of of ADP iajs-2428 207 22 𝔓 𝔓 PROPN iajs-2428 207 23 which which PRON iajs-2428 207 24 includes include VERB iajs-2428 207 25 𝒥 𝒥 PROPN iajs-2428 207 26 , , PUNCT iajs-2428 207 27 then then ADV iajs-2428 207 28 λ λ PROPN iajs-2428 207 29 𝒥 𝒥 PROPN iajs-2428 207 30 |𝔇 |𝔇 PUNCT iajs-2428 207 31 is be AUX iajs-2428 207 32 a a DET iajs-2428 207 33 λ λ NOUN iajs-2428 207 34 – – PUNCT iajs-2428 207 35 algebra algebra NOUN iajs-2428 207 36 of of ADP iajs-2428 207 37 a a DET iajs-2428 207 38 set set NOUN iajs-2428 207 39 𝔇. 𝔇. PROPN iajs-2428 207 40 ( ( PUNCT iajs-2428 207 41 5 5 X iajs-2428 207 42 ) ) PUNCT iajs-2428 207 43 let let VERB iajs-2428 207 44 𝒥 𝒥 PRON iajs-2428 207 45 ⊆ ⊆ NUM iajs-2428 207 46 p p PROPN iajs-2428 207 47 𝔓 𝔓 PROPN iajs-2428 207 48 and and CCONJ iajs-2428 207 49 φ φ NUM iajs-2428 207 50 𝔇 𝔇 PROPN iajs-2428 207 51 ⊆ ⊆ NUM iajs-2428 207 52 𝔓. 𝔓. PROPN iajs-2428 207 53 then then ADV iajs-2428 207 54 λ λ X iajs-2428 207 55 𝒥|𝔇 𝒥|𝔇 PUNCT iajs-2428 207 56 = = SYM iajs-2428 207 57 λ λ PROPN iajs-2428 207 58 𝒥 𝒥 PROPN iajs-2428 207 59 |𝔇. |𝔇. PROPN iajs-2428 207 60 ( ( PUNCT iajs-2428 207 61 6 6 NUM iajs-2428 207 62 ) ) PUNCT iajs-2428 207 63 every every DET iajs-2428 207 64 λ λ NOUN iajs-2428 207 65 – – PUNCT iajs-2428 207 66 algebra algebra NOUN iajs-2428 207 67 is be AUX iajs-2428 207 68 a a DET iajs-2428 207 69 α α PROPN iajs-2428 207 70 – – PUNCT iajs-2428 207 71 σ σ NOUN iajs-2428 207 72 – – PUNCT iajs-2428 207 73 field field NOUN iajs-2428 207 74 . . PUNCT iajs-2428 208 1 ( ( PUNCT iajs-2428 208 2 7 7 X iajs-2428 208 3 ) ) PUNCT iajs-2428 208 4 every every DET iajs-2428 208 5 λ λ NOUN iajs-2428 208 6 – – PUNCT iajs-2428 208 7 algebra algebra NOUN iajs-2428 208 8 is be AUX iajs-2428 208 9 a a DET iajs-2428 208 10 β β X iajs-2428 208 11 – – PUNCT iajs-2428 208 12 σ σ NOUN iajs-2428 208 13 – – PUNCT iajs-2428 208 14 field field NOUN iajs-2428 208 15 . . PUNCT iajs-2428 209 1 ( ( PUNCT iajs-2428 209 2 8) 8) NUM iajs-2428 209 3 every every DET iajs-2428 209 4 λ λ NOUN iajs-2428 209 5 – – PUNCT iajs-2428 209 6 algebra algebra NOUN iajs-2428 209 7 is be AUX iajs-2428 209 8 a a DET iajs-2428 209 9 monotone monotone ADJ iajs-2428 209 10 class class NOUN iajs-2428 209 11 . . PUNCT iajs-2428 210 1 ( ( PUNCT iajs-2428 210 2 9 9 X iajs-2428 210 3 ) ) PUNCT iajs-2428 210 4 let let VERB iajs-2428 210 5 𝒥 𝒥 PRON iajs-2428 210 6 be be AUX iajs-2428 210 7 a a DET iajs-2428 210 8 collection collection NOUN iajs-2428 210 9 of of ADP iajs-2428 210 10 subsets subset NOUN iajs-2428 210 11 of of ADP iajs-2428 210 12 a a DET iajs-2428 210 13 nonempty nonempty ADV iajs-2428 210 14 set set VERB iajs-2428 210 15 𝔓 𝔓 PROPN iajs-2428 210 16 . . PUNCT iajs-2428 211 1 then then ADV iajs-2428 211 2 𝕄 𝕄 PROPN iajs-2428 211 3 𝒥 𝒥 PROPN iajs-2428 211 4 ⊆ ⊆ NUM iajs-2428 211 5 λ λ SYM iajs-2428 211 6 𝒥 𝒥 PROPN iajs-2428 211 7 . . PUNCT iajs-2428 212 1 ( ( PUNCT iajs-2428 212 2 10 10 NUM iajs-2428 212 3 ) ) PUNCT iajs-2428 212 4 let let VERB iajs-2428 212 5 ( ( PUNCT iajs-2428 212 6 𝔓 𝔓 NOUN iajs-2428 212 7 , , PUNCT iajs-2428 212 8 𝒦 𝒦 PROPN iajs-2428 212 9 , , PUNCT iajs-2428 212 10 𝔐 𝔐 PROPN iajs-2428 212 11 ) ) PUNCT iajs-2428 212 12 be be AUX iajs-2428 212 13 a a DET iajs-2428 212 14 measure measure NOUN iajs-2428 212 15 space space NOUN iajs-2428 212 16 relative relative ADJ iajs-2428 212 17 to to ADP iajs-2428 212 18 the the DET iajs-2428 212 19 λ λ NOUN iajs-2428 212 20 – – PUNCT iajs-2428 212 21 algebra algebra NOUN iajs-2428 212 22 𝒦 𝒦 PROPN iajs-2428 212 23 and and CCONJ iajs-2428 212 24 𝑐 𝑐 ADP iajs-2428 212 25 ∈ ∈ PROPN iajs-2428 212 26 0 0 NUM iajs-2428 212 27 , , PUNCT iajs-2428 212 28 ∞ ∞ PROPN iajs-2428 212 29 for for ADP iajs-2428 212 30 all all DET iajs-2428 212 31 𝑗 𝑗 DET iajs-2428 212 32 1,2 1,2 NUM iajs-2428 212 33 , , PUNCT iajs-2428 212 34 … … PUNCT iajs-2428 212 35 , , PUNCT iajs-2428 212 36 𝑘. 𝑘. VERB iajs-2428 212 37 if if SCONJ iajs-2428 212 38 a a DET iajs-2428 212 39 set set NOUN iajs-2428 212 40 function function NOUN iajs-2428 212 41 ∑ ∑ PROPN iajs-2428 212 42 𝑐 𝑐 PROPN iajs-2428 212 43 𝔐 𝔐 PROPN iajs-2428 212 44 : : PUNCT iajs-2428 212 45 ℘ ℘ PROPN iajs-2428 212 46 → → SYM iajs-2428 212 47 0 0 NUM iajs-2428 212 48 , , PUNCT iajs-2428 212 49 ∞ ∞ PROPN iajs-2428 212 50 is be AUX iajs-2428 212 51 defined define VERB iajs-2428 212 52 by by ADP iajs-2428 212 53 : : PUNCT iajs-2428 212 54 ∑ ∑ PUNCT iajs-2428 212 55 𝑐 𝑐 PROPN iajs-2428 212 56 𝔐 𝔐 PRON iajs-2428 212 57 𝐷 𝐷 PROPN iajs-2428 212 58 ∑ ∑ PROPN iajs-2428 212 59 𝑐 𝑐 PROPN iajs-2428 212 60 . . PUNCT iajs-2428 213 1 𝔐 𝔐 PROPN iajs-2428 213 2 𝐷 𝐷 NOUN iajs-2428 213 3 ∀𝐷𝜖℘ ∀𝐷𝜖℘ NOUN iajs-2428 213 4 , , PUNCT iajs-2428 213 5 then then ADV iajs-2428 213 6 ( ( PUNCT iajs-2428 213 7 𝔓 𝔓 NOUN iajs-2428 213 8 , , PUNCT iajs-2428 213 9 𝒦 𝒦 PROPN iajs-2428 213 10 , , PUNCT iajs-2428 213 11 ∑ ∑ PROPN iajs-2428 213 12 𝑐 𝑐 PROPN iajs-2428 213 13 𝔐 𝔐 PROPN iajs-2428 213 14 ) ) PUNCT iajs-2428 213 15 is be AUX iajs-2428 213 16 measure measure NOUN iajs-2428 213 17 space space NOUN iajs-2428 213 18 relative relative ADJ iajs-2428 213 19 to to ADP iajs-2428 213 20 the the DET iajs-2428 213 21 λ λ NOUN iajs-2428 213 22 – – PUNCT iajs-2428 213 23 algebra algebra NOUN iajs-2428 213 24 𝒦. 𝒦. PROPN iajs-2428 213 25 ( ( PUNCT iajs-2428 213 26 11 11 NUM iajs-2428 213 27 ) ) PUNCT iajs-2428 213 28 every every DET iajs-2428 213 29 measure measure NOUN iajs-2428 213 30 relative relative ADJ iajs-2428 213 31 to to ADP iajs-2428 213 32 the the DET iajs-2428 213 33 λ λ NOUN iajs-2428 213 34 – – PUNCT iajs-2428 213 35 algebra algebra NOUN iajs-2428 213 36 is be AUX iajs-2428 213 37 complete complete ADJ iajs-2428 213 38 . . PUNCT iajs-2428 214 1 references reference NOUN iajs-2428 214 2 1 1 NUM iajs-2428 214 3 . . PUNCT iajs-2428 214 4 robert robert PROPN iajs-2428 214 5 , , PUNCT iajs-2428 214 6 b.a b.a PROPN iajs-2428 214 7 . . PROPN iajs-2428 214 8 real real ADJ iajs-2428 214 9 analysis analysis NOUN iajs-2428 214 10 and and CCONJ iajs-2428 214 11 probability probability NOUN iajs-2428 214 12 , , PUNCT iajs-2428 214 13 1st 1st ADJ iajs-2428 214 14 ed ed NOUN iajs-2428 214 15 ; ; PUNCT iajs-2428 214 16 academic academic ADJ iajs-2428 214 17 press press NOUN iajs-2428 214 18 , , PUNCT iajs-2428 214 19 inc inc PROPN iajs-2428 214 20 : : PUNCT iajs-2428 214 21 new new PROPN iajs-2428 214 22 york york PROPN iajs-2428 214 23 . . PUNCT iajs-2428 215 1 1972 1972 NUM iajs-2428 215 2 , , PUNCT iajs-2428 215 3 isbn isbn ADJ iajs-2428 215 4 : : PUNCT iajs-2428 215 5 75 75 NUM iajs-2428 215 6 - - SYM iajs-2428 215 7 182642 182642 NUM iajs-2428 215 8 . . PUNCT iajs-2428 216 1 2 2 X iajs-2428 216 2 . . X iajs-2428 216 3 dietmar dietmar PROPN iajs-2428 216 4 , , PUNCT iajs-2428 216 5 a.s a.s PROPN iajs-2428 216 6 . . PROPN iajs-2428 216 7 measure measure NOUN iajs-2428 216 8 and and CCONJ iajs-2428 216 9 integration integration NOUN iajs-2428 216 10 , , PUNCT iajs-2428 216 11 1st 1st ADJ iajs-2428 216 12 ed ed NOUN iajs-2428 216 13 ; ; PUNCT iajs-2428 216 14 eth eth PROPN iajs-2428 216 15 zürich zürich PROPN iajs-2428 216 16 , , PUNCT iajs-2428 216 17 switzerland.2016 switzerland.2016 PROPN iajs-2428 216 18 , , PUNCT iajs-2428 216 19 isbn isbn ADJ iajs-2428 216 20 : : PUNCT iajs-2428 216 21 978 978 NUM iajs-2428 216 22 - - SYM iajs-2428 216 23 3 3 NUM iajs-2428 216 24 - - PUNCT iajs-2428 216 25 03 03 NUM iajs-2428 216 26 - - PUNCT iajs-2428 216 27 719 719 NUM iajs-2428 216 28 - - PUNCT iajs-2428 216 29 659 659 NUM iajs-2428 216 30 - - SYM iajs-2428 216 31 5 5 NUM iajs-2428 216 32 . . NOUN iajs-2428 216 33 3 3 NUM iajs-2428 216 34 . . X iajs-2428 216 35 krishna krishna PROPN iajs-2428 216 36 , , PUNCT iajs-2428 216 37 b.a b.a PROPN iajs-2428 216 38 . . PROPN iajs-2428 216 39 ; ; PUNCT iajs-2428 216 40 soumendra soumendra PROPN iajs-2428 216 41 , , PUNCT iajs-2428 216 42 n.l n.l PROPN iajs-2428 216 43 . . PROPN iajs-2428 216 44 measure measure NOUN iajs-2428 216 45 theory theory NOUN iajs-2428 216 46 and and CCONJ iajs-2428 216 47 probability probability NOUN iajs-2428 216 48 theory theory NOUN iajs-2428 216 49 , , PUNCT iajs-2428 216 50 1st 1st ADJ iajs-2428 216 51 ed ed NOUN iajs-2428 216 52 ; ; PUNCT iajs-2428 216 53 springer springer NOUN iajs-2428 216 54 science science NOUN iajs-2428 216 55 + + CCONJ iajs-2428 216 56 business business NOUN iajs-2428 216 57 media medium NOUN iajs-2428 216 58 , , PUNCT iajs-2428 216 59 llc llc PROPN iajs-2428 216 60 : : PUNCT iajs-2428 216 61 new new PROPN iajs-2428 216 62 york.2006 york.2006 PROPN iajs-2428 216 63 , , PUNCT iajs-2428 216 64 isbn-10 isbn-10 PROPN iajs-2428 216 65 : : PUNCT iajs-2428 216 66 0 0 NUM iajs-2428 216 67 - - SYM iajs-2428 216 68 387 387 NUM iajs-2428 216 69 - - PUNCT iajs-2428 216 70 32903 32903 NUM iajs-2428 216 71 - - PUNCT iajs-2428 216 72 x. x. NOUN iajs-2428 216 73 4 4 PROPN iajs-2428 216 74 . . PUNCT iajs-2428 216 75 peipei peipei PROPN iajs-2428 216 76 , , PUNCT iajs-2428 216 77 w. w. PROPN iajs-2428 216 78 ; ; PUNCT iajs-2428 216 79 minhao minhao PROPN iajs-2428 216 80 , , PUNCT iajs-2428 216 81 yu yu PROPN iajs-2428 216 82 . . PROPN iajs-2428 216 83 ; ; PUNCT iajs-2428 216 84 jun jun PROPN iajs-2428 216 85 , , PUNCT iajs-2428 216 86 l. l. PROPN iajs-2428 216 87 monotone monotone NOUN iajs-2428 216 88 measures measure NOUN iajs-2428 216 89 defined define VERB iajs-2428 216 90 by by ADP iajs-2428 216 91 pan pan ADJ iajs-2428 216 92 - - ADJ iajs-2428 216 93 integral integral ADJ iajs-2428 216 94 , , PUNCT iajs-2428 216 95 advancesin advancesin NOUN iajs-2428 216 96 pure pure ADJ iajs-2428 216 97 mathematics.2018 mathematics.2018 PROPN iajs-2428 216 98 , , PUNCT iajs-2428 216 99 8 8 NUM iajs-2428 216 100 , , PUNCT iajs-2428 216 101 535–547 535–547 NUM iajs-2428 216 102 , , PUNCT iajs-2428 216 103 doi doi NOUN iajs-2428 216 104 : : PUNCT iajs-2428 216 105 10.4236 10.4236 NUM iajs-2428 216 106 / / SYM iajs-2428 216 107 apm.2018.86031 apm.2018.86031 NUM iajs-2428 216 108 . . NOUN iajs-2428 216 109 5 5 NUM iajs-2428 216 110 . . X iajs-2428 216 111 noboru noboru NOUN iajs-2428 216 112 , , PUNCT iajs-2428 216 113 e. e. PROPN iajs-2428 216 114 ; ; PUNCT iajs-2428 216 115 kazuhisa kazuhisa PROPN iajs-2428 216 116 , , PUNCT iajs-2428 216 117 n. n. PROPN iajs-2428 216 118 σ σ PROPN iajs-2428 216 119 – – PUNCT iajs-2428 216 120 ring ring NOUN iajs-2428 216 121 and and CCONJ iajs-2428 216 122 σ σ NOUN iajs-2428 216 123 – – PUNCT iajs-2428 216 124 algebra algebra NOUN iajs-2428 216 125 of of ADP iajs-2428 216 126 sets set NOUN iajs-2428 216 127 , , PUNCT iajs-2428 216 128 formalized formalize VERB iajs-2428 216 129 mathematics mathematic NOUN iajs-2428 216 130 , , PUNCT iajs-2428 216 131 2015 2015 NUM iajs-2428 216 132 , , PUNCT iajs-2428 216 133 23 23 NUM iajs-2428 216 134 , , PUNCT iajs-2428 216 135 1 1 NUM iajs-2428 216 136 , , PUNCT iajs-2428 216 137 51 51 NUM iajs-2428 216 138 - - SYM iajs-2428 216 139 57 57 NUM iajs-2428 216 140 , , PUNCT iajs-2428 216 141 doi doi NOUN iajs-2428 216 142 : : PUNCT iajs-2428 216 143 10.2478 10.2478 NUM iajs-2428 216 144 / / SYM iajs-2428 216 145 forma-2015 forma-2015 NOUN iajs-2428 216 146 - - PUNCT iajs-2428 216 147 0004 0004 NUM iajs-2428 216 148 . . PUNCT iajs-2428 217 1 6 6 NUM iajs-2428 217 2 . . X iajs-2428 218 1 zhenyuan zhenyuan PROPN iajs-2428 218 2 , , PUNCT iajs-2428 218 3 w. w. PROPN iajs-2428 218 4 ; ; PUNCT iajs-2428 218 5 george george PROPN iajs-2428 218 6 , , PUNCT iajs-2428 218 7 j.k j.k PROPN iajs-2428 218 8 . . 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