id sid tid token lemma pos ejpam-4528 1 1 european european PROPN ejpam-4528 1 2 journal journal PROPN ejpam-4528 1 3 of of ADP ejpam-4528 1 4 pure pure ADJ ejpam-4528 1 5 and and CCONJ ejpam-4528 1 6 applied apply VERB ejpam-4528 1 7 mathematics mathematic NOUN ejpam-4528 1 8 vol vol NOUN ejpam-4528 1 9 . . PROPN ejpam-4528 2 1 15 15 NUM ejpam-4528 2 2 , , PUNCT ejpam-4528 2 3 no no INTJ ejpam-4528 2 4 . . NOUN ejpam-4528 2 5 4 4 NUM ejpam-4528 2 6 , , PUNCT ejpam-4528 2 7 2022 2022 NUM ejpam-4528 2 8 , , PUNCT ejpam-4528 2 9 1512 1512 NUM ejpam-4528 2 10 - - SYM ejpam-4528 2 11 1520 1520 NUM ejpam-4528 2 12 issn issn PROPN ejpam-4528 2 13 1307 1307 NUM ejpam-4528 2 14 - - SYM ejpam-4528 2 15 5543 5543 NUM ejpam-4528 2 16 – – PUNCT ejpam-4528 2 17 ejpam.com ejpam.com X ejpam-4528 2 18 published publish VERB ejpam-4528 2 19 by by ADP ejpam-4528 2 20 new new PROPN ejpam-4528 2 21 york york PROPN ejpam-4528 2 22 business business PROPN ejpam-4528 2 23 global global PROPN ejpam-4528 2 24 pettis pettis PROPN ejpam-4528 2 25 integrability integrability NOUN ejpam-4528 2 26 in in ADP ejpam-4528 2 27 l1 l1 PROPN ejpam-4528 2 28 e′[e e′[e PROPN ejpam-4528 2 29 ] ] X ejpam-4528 2 30 related relate VERB ejpam-4528 2 31 to to ADP ejpam-4528 2 32 the the DET ejpam-4528 2 33 truncation truncation NOUN ejpam-4528 2 34 noureddine noureddine ADP ejpam-4528 2 35 sabiri1,∗ sabiri1,∗ NOUN ejpam-4528 2 36 , , PUNCT ejpam-4528 2 37 mohamed mohamed ADJ ejpam-4528 2 38 guessous1 guessous1 PROPN ejpam-4528 2 39 1 1 NUM ejpam-4528 2 40 department department NOUN ejpam-4528 2 41 of of ADP ejpam-4528 2 42 mathematics mathematic NOUN ejpam-4528 2 43 and and CCONJ ejpam-4528 2 44 computer computer NOUN ejpam-4528 2 45 science science NOUN ejpam-4528 2 46 , , PUNCT ejpam-4528 2 47 faculty faculty NOUN ejpam-4528 2 48 of of ADP ejpam-4528 2 49 sciences sciences PROPN ejpam-4528 2 50 ben ben PROPN ejpam-4528 2 51 m’sik m’sik PROPN ejpam-4528 2 52 , , PUNCT ejpam-4528 2 53 hassan hassan PROPN ejpam-4528 2 54 ii ii PROPN ejpam-4528 2 55 university university PROPN ejpam-4528 2 56 of of ADP ejpam-4528 2 57 casablanca casablanca PROPN ejpam-4528 2 58 , , PUNCT ejpam-4528 2 59 casablanca casablanca PROPN ejpam-4528 2 60 , , PUNCT ejpam-4528 2 61 morocco morocco PROPN ejpam-4528 2 62 . . PUNCT ejpam-4528 3 1 abstract abstract ADJ ejpam-4528 3 2 . . PUNCT ejpam-4528 4 1 we we PRON ejpam-4528 4 2 study study VERB ejpam-4528 4 3 the the DET ejpam-4528 4 4 pettis pettis PROPN ejpam-4528 4 5 integrability integrability NOUN ejpam-4528 4 6 in in ADP ejpam-4528 4 7 terms term NOUN ejpam-4528 4 8 of of ADP ejpam-4528 4 9 truncation truncation NOUN ejpam-4528 4 10 . . PUNCT ejpam-4528 5 1 we we PRON ejpam-4528 5 2 focus focus VERB ejpam-4528 5 3 our our PRON ejpam-4528 5 4 study study NOUN ejpam-4528 5 5 particularly particularly ADV ejpam-4528 5 6 on on ADP ejpam-4528 5 7 space space NOUN ejpam-4528 5 8 l1 l1 PROPN ejpam-4528 5 9 e′ e′ PROPN ejpam-4528 6 1 [ [ X ejpam-4528 6 2 e e X ejpam-4528 6 3 ] ] PUNCT ejpam-4528 6 4 . . PUNCT ejpam-4528 7 1 2020 2020 NUM ejpam-4528 7 2 mathematics mathematic NOUN ejpam-4528 7 3 subject subject NOUN ejpam-4528 7 4 classifications classification NOUN ejpam-4528 7 5 : : PUNCT ejpam-4528 7 6 28a20 28a20 NUM ejpam-4528 7 7 , , PUNCT ejpam-4528 7 8 28a25 28a25 NUM ejpam-4528 7 9 , , PUNCT ejpam-4528 7 10 40a05 40a05 NUM ejpam-4528 7 11 , , PUNCT ejpam-4528 7 12 46g10 46g10 NUM ejpam-4528 7 13 key key ADJ ejpam-4528 7 14 words word NOUN ejpam-4528 7 15 and and CCONJ ejpam-4528 7 16 phrases phrase NOUN ejpam-4528 7 17 : : PUNCT ejpam-4528 7 18 convergence convergence NOUN ejpam-4528 7 19 , , PUNCT ejpam-4528 7 20 dual dual ADJ ejpam-4528 7 21 space space NOUN ejpam-4528 7 22 , , PUNCT ejpam-4528 7 23 gelfand gelfand PROPN ejpam-4528 7 24 integral integral ADJ ejpam-4528 7 25 , , PUNCT ejpam-4528 7 26 pettis pettis PROPN ejpam-4528 7 27 integral integral ADJ ejpam-4528 7 28 , , PUNCT ejpam-4528 7 29 truncation truncation NOUN ejpam-4528 7 30 1 1 NUM ejpam-4528 7 31 . . PUNCT ejpam-4528 7 32 introduction introduction NOUN ejpam-4528 7 33 several several ADJ ejpam-4528 7 34 authors author NOUN ejpam-4528 7 35 studied study VERB ejpam-4528 7 36 the the DET ejpam-4528 7 37 pettis pettis NOUN ejpam-4528 7 38 integrability integrability NOUN ejpam-4528 7 39 of of ADP ejpam-4528 7 40 banach banach NOUN ejpam-4528 7 41 space space NOUN ejpam-4528 7 42 valued value VERB ejpam-4528 7 43 functions function NOUN ejpam-4528 7 44 ( ( PUNCT ejpam-4528 7 45 see see VERB ejpam-4528 7 46 for for ADP ejpam-4528 7 47 example example NOUN ejpam-4528 7 48 [ [ X ejpam-4528 7 49 1],[10],[11],[13],[14],[12],[18 1],[10],[11],[13],[14],[12],[18 NUM ejpam-4528 7 50 ] ] PUNCT ejpam-4528 7 51 and and CCONJ ejpam-4528 7 52 references reference NOUN ejpam-4528 7 53 therein therein ADV ejpam-4528 7 54 ) ) PUNCT ejpam-4528 7 55 and and CCONJ ejpam-4528 7 56 especially especially ADV ejpam-4528 7 57 of of ADP ejpam-4528 7 58 dual dual ADJ ejpam-4528 7 59 banach banach NOUN ejpam-4528 7 60 space space NOUN ejpam-4528 7 61 valued value VERB ejpam-4528 7 62 functions function NOUN ejpam-4528 7 63 ( ( PUNCT ejpam-4528 7 64 [ [ X ejpam-4528 7 65 2],[17],[19 2],[17],[19 NUM ejpam-4528 7 66 ] ] SYM ejpam-4528 7 67 ) ) PUNCT ejpam-4528 7 68 . . PUNCT ejpam-4528 8 1 similarly similarly ADV ejpam-4528 8 2 , , PUNCT ejpam-4528 8 3 the the DET ejpam-4528 8 4 study study NOUN ejpam-4528 8 5 of of ADP ejpam-4528 8 6 pettis pettis PROPN ejpam-4528 8 7 integrability integrability NOUN ejpam-4528 8 8 for for ADP ejpam-4528 8 9 multifunctions multifunction NOUN ejpam-4528 8 10 has have AUX ejpam-4528 8 11 been be AUX ejpam-4528 8 12 the the DET ejpam-4528 8 13 focus focus NOUN ejpam-4528 8 14 of of ADP ejpam-4528 8 15 various various ADJ ejpam-4528 8 16 papers paper NOUN ejpam-4528 8 17 ( ( PUNCT ejpam-4528 8 18 for for ADP ejpam-4528 8 19 example example NOUN ejpam-4528 8 20 [ [ X ejpam-4528 8 21 9],[15 9],[15 X ejpam-4528 8 22 ] ] PUNCT ejpam-4528 8 23 and and CCONJ ejpam-4528 8 24 [ [ X ejpam-4528 8 25 22 22 NUM ejpam-4528 8 26 ] ] PUNCT ejpam-4528 8 27 ) ) PUNCT ejpam-4528 8 28 . . PUNCT ejpam-4528 9 1 in in ADP ejpam-4528 9 2 this this DET ejpam-4528 9 3 note note NOUN ejpam-4528 9 4 , , PUNCT ejpam-4528 9 5 we we PRON ejpam-4528 9 6 are be AUX ejpam-4528 9 7 interested interested ADJ ejpam-4528 9 8 in in ADP ejpam-4528 9 9 pettis pettis NOUN ejpam-4528 9 10 integrability integrability NOUN ejpam-4528 9 11 for for ADP ejpam-4528 9 12 scalarly scalarly ADV ejpam-4528 9 13 integrable integrable ADJ ejpam-4528 9 14 functions function NOUN ejpam-4528 9 15 of of ADP ejpam-4528 9 16 l1 l1 PROPN ejpam-4528 9 17 e′ e′ PUNCT ejpam-4528 10 1 [ [ X ejpam-4528 10 2 e e X ejpam-4528 10 3 ] ] X ejpam-4528 10 4 . . PUNCT ejpam-4528 11 1 our our PRON ejpam-4528 11 2 study study NOUN ejpam-4528 11 3 is be AUX ejpam-4528 11 4 based base VERB ejpam-4528 11 5 on on ADP ejpam-4528 11 6 the the DET ejpam-4528 11 7 truncation truncation NOUN ejpam-4528 11 8 technique technique NOUN ejpam-4528 11 9 that that PRON ejpam-4528 11 10 has have AUX ejpam-4528 11 11 been be AUX ejpam-4528 11 12 adopted adopt VERB ejpam-4528 11 13 in in ADP ejpam-4528 11 14 ( ( PUNCT ejpam-4528 11 15 [ [ X ejpam-4528 11 16 5],[6 5],[6 NUM ejpam-4528 11 17 ] ] PUNCT ejpam-4528 11 18 ) ) PUNCT ejpam-4528 11 19 to to PART ejpam-4528 11 20 state state VERB ejpam-4528 11 21 some some DET ejpam-4528 11 22 komlós komlós NOUN ejpam-4528 11 23 type type NOUN ejpam-4528 11 24 theorems theorem NOUN ejpam-4528 11 25 for for ADP ejpam-4528 11 26 bochner bochner NOUN ejpam-4528 11 27 integrable integrable ADJ ejpam-4528 11 28 functions function NOUN ejpam-4528 11 29 and and CCONJ ejpam-4528 11 30 in in ADP ejpam-4528 11 31 [ [ X ejpam-4528 11 32 16 16 NUM ejpam-4528 11 33 ] ] PUNCT ejpam-4528 11 34 to to PART ejpam-4528 11 35 provide provide VERB ejpam-4528 11 36 a a DET ejpam-4528 11 37 komlós komlós NOUN ejpam-4528 11 38 type type NOUN ejpam-4528 11 39 theorem theorem NOUN ejpam-4528 11 40 in in ADP ejpam-4528 11 41 l1 l1 PROPN ejpam-4528 11 42 e′ e′ PUNCT ejpam-4528 12 1 [ [ X ejpam-4528 12 2 e e X ejpam-4528 12 3 ] ] PUNCT ejpam-4528 12 4 . . PUNCT ejpam-4528 13 1 it it PRON ejpam-4528 13 2 is be AUX ejpam-4528 13 3 well well ADV ejpam-4528 13 4 known know VERB ejpam-4528 13 5 that that SCONJ ejpam-4528 13 6 a a DET ejpam-4528 13 7 strongly strongly ADV ejpam-4528 13 8 measurable measurable ADJ ejpam-4528 13 9 and and CCONJ ejpam-4528 13 10 scalarly scalarly ADV ejpam-4528 13 11 integrable integrable ADJ ejpam-4528 13 12 function function NOUN ejpam-4528 13 13 f f PROPN ejpam-4528 13 14 : : PUNCT ejpam-4528 13 15 ω ω PROPN ejpam-4528 13 16 → → SYM ejpam-4528 13 17 e e PROPN ejpam-4528 13 18 is be AUX ejpam-4528 13 19 pettis pettis NOUN ejpam-4528 13 20 integrable integrable ADJ ejpam-4528 13 21 if if SCONJ ejpam-4528 13 22 and and CCONJ ejpam-4528 13 23 only only ADV ejpam-4528 13 24 if if SCONJ ejpam-4528 13 25 the the DET ejpam-4528 13 26 set set NOUN ejpam-4528 13 27 { { PUNCT ejpam-4528 13 28 ⟨x′ ⟨x′ PROPN ejpam-4528 13 29 , , PUNCT ejpam-4528 13 30 f⟩ f⟩ NOUN ejpam-4528 13 31 : : PUNCT ejpam-4528 13 32 ∥x′∥ ∥x′∥ NOUN ejpam-4528 13 33 ≤ ≤ ADV ejpam-4528 13 34 1 1 NUM ejpam-4528 13 35 } } PUNCT ejpam-4528 13 36 is be AUX ejpam-4528 13 37 uniformly uniformly ADV ejpam-4528 13 38 integrable integrable ADJ ejpam-4528 13 39 in in ADP ejpam-4528 13 40 l1 l1 PROPN ejpam-4528 13 41 r(µ r(µ PROPN ejpam-4528 13 42 ) ) PUNCT ejpam-4528 13 43 ( ( PUNCT ejpam-4528 13 44 [ [ X ejpam-4528 13 45 14 14 NUM ejpam-4528 13 46 ] ] PUNCT ejpam-4528 13 47 theorem theorem VERB ejpam-4528 13 48 5.2 5.2 NUM ejpam-4528 13 49 ) ) PUNCT ejpam-4528 13 50 . . PUNCT ejpam-4528 14 1 we we PRON ejpam-4528 14 2 give give VERB ejpam-4528 14 3 a a DET ejpam-4528 14 4 characterization characterization NOUN ejpam-4528 14 5 of of ADP ejpam-4528 14 6 pettis pettis PROPN ejpam-4528 14 7 integrability integrability NOUN ejpam-4528 14 8 for for ADP ejpam-4528 14 9 scalarly scalarly ADV ejpam-4528 14 10 integrable integrable ADJ ejpam-4528 14 11 function function NOUN ejpam-4528 14 12 ( ( PUNCT ejpam-4528 14 13 non non ADJ ejpam-4528 14 14 - - ADJ ejpam-4528 14 15 necessary necessary ADJ ejpam-4528 14 16 strongly strongly ADV ejpam-4528 14 17 measurable measurable ADJ ejpam-4528 14 18 ) ) PUNCT ejpam-4528 14 19 with with ADP ejpam-4528 14 20 norm norm NOUN ejpam-4528 14 21 measurable measurable ADJ ejpam-4528 14 22 function function NOUN ejpam-4528 14 23 ( ( PUNCT ejpam-4528 14 24 proposition proposition NOUN ejpam-4528 14 25 1 1 NUM ejpam-4528 14 26 ) ) PUNCT ejpam-4528 14 27 and and CCONJ ejpam-4528 14 28 , , PUNCT ejpam-4528 14 29 when when SCONJ ejpam-4528 14 30 e e PROPN ejpam-4528 14 31 is be AUX ejpam-4528 14 32 a a DET ejpam-4528 14 33 separable separable ADJ ejpam-4528 14 34 banach banach NOUN ejpam-4528 14 35 space space NOUN ejpam-4528 14 36 , , PUNCT ejpam-4528 14 37 we we PRON ejpam-4528 14 38 establish establish VERB ejpam-4528 14 39 that that SCONJ ejpam-4528 14 40 a a DET ejpam-4528 14 41 function function NOUN ejpam-4528 14 42 f f PROPN ejpam-4528 14 43 ∈ ∈ PROPN ejpam-4528 14 44 l1 l1 PROPN ejpam-4528 14 45 e′ e′ PUNCT ejpam-4528 15 1 [ [ X ejpam-4528 15 2 e e X ejpam-4528 15 3 ] ] X ejpam-4528 15 4 is be AUX ejpam-4528 15 5 pettis pettis NOUN ejpam-4528 15 6 integrable integrable ADJ ejpam-4528 15 7 if if SCONJ ejpam-4528 15 8 and and CCONJ ejpam-4528 15 9 only only ADV ejpam-4528 15 10 if if SCONJ ejpam-4528 15 11 its its PRON ejpam-4528 15 12 truncated truncated ADJ ejpam-4528 15 13 function function NOUN ejpam-4528 15 14 1{∥f∥≤n}f 1{∥f∥≤n}f NUM ejpam-4528 15 15 is be AUX ejpam-4528 15 16 pettis pettis NOUN ejpam-4528 15 17 integrable integrable ADJ ejpam-4528 15 18 for for ADP ejpam-4528 15 19 all all DET ejpam-4528 15 20 n n PRON ejpam-4528 15 21 ≥ ≥ NOUN ejpam-4528 15 22 1 1 NUM ejpam-4528 15 23 ( ( PUNCT ejpam-4528 15 24 corollary corollary ADJ ejpam-4528 15 25 1 1 NUM ejpam-4528 15 26 ) ) PUNCT ejpam-4528 15 27 . . PUNCT ejpam-4528 16 1 we we PRON ejpam-4528 16 2 also also ADV ejpam-4528 16 3 give give VERB ejpam-4528 16 4 some some DET ejpam-4528 16 5 criteria criterion NOUN ejpam-4528 16 6 that that PRON ejpam-4528 16 7 guarantee guarantee VERB ejpam-4528 16 8 the the DET ejpam-4528 16 9 pettis pettis NOUN ejpam-4528 16 10 integrability integrability NOUN ejpam-4528 16 11 of of ADP ejpam-4528 16 12 the the DET ejpam-4528 16 13 limit limit NOUN ejpam-4528 16 14 of of ADP ejpam-4528 16 15 a a DET ejpam-4528 16 16 pettis pettis PROPN ejpam-4528 16 17 integrable integrable ADJ ejpam-4528 16 18 l1 l1 PROPN ejpam-4528 16 19 e′ e′ PUNCT ejpam-4528 17 1 [ [ X ejpam-4528 17 2 e]-convergent e]-convergent NOUN ejpam-4528 17 3 sequence sequence NOUN ejpam-4528 17 4 . . PUNCT ejpam-4528 18 1 more more ADV ejpam-4528 18 2 precisely precisely ADV ejpam-4528 18 3 , , PUNCT ejpam-4528 18 4 we we PRON ejpam-4528 18 5 show show VERB ejpam-4528 18 6 that that SCONJ ejpam-4528 18 7 if if SCONJ ejpam-4528 18 8 a a DET ejpam-4528 18 9 sequence sequence NOUN ejpam-4528 18 10 of of ADP ejpam-4528 18 11 pettis pettis PROPN ejpam-4528 18 12 integrable integrable ADJ ejpam-4528 18 13 functions function NOUN ejpam-4528 18 14 bounded bound VERB ejpam-4528 18 15 in in ADP ejpam-4528 18 16 l1 l1 PROPN ejpam-4528 18 17 e′ e′ PUNCT ejpam-4528 19 1 [ [ X ejpam-4528 19 2 e e X ejpam-4528 19 3 ] ] X ejpam-4528 19 4 converges converge VERB ejpam-4528 19 5 weakly weakly ADJ ejpam-4528 19 6 a.e a.e PROPN ejpam-4528 19 7 . . PROPN ejpam-4528 20 1 in in ADP ejpam-4528 20 2 e′ e′ PROPN ejpam-4528 20 3 ( ( PUNCT ejpam-4528 20 4 resp resp NOUN ejpam-4528 20 5 . . PUNCT ejpam-4528 21 1 converges converge VERB ejpam-4528 21 2 pointwise pointwise VERB ejpam-4528 21 3 in in ADP ejpam-4528 21 4 l∞ l∞ NOUN ejpam-4528 21 5 r r NOUN ejpam-4528 21 6 ( ( PUNCT ejpam-4528 21 7 µ µ NOUN ejpam-4528 21 8 ) ) PUNCT ejpam-4528 21 9 ⊗ ⊗ PROPN ejpam-4528 21 10 e′′ e′′ PROPN ejpam-4528 21 11 ) ) PUNCT ejpam-4528 21 12 to to ADP ejpam-4528 21 13 a a DET ejpam-4528 21 14 scalarly scalarly ADV ejpam-4528 21 15 integrable integrable ADJ ejpam-4528 21 16 function function NOUN ejpam-4528 21 17 f f PROPN ejpam-4528 21 18 , , PUNCT ejpam-4528 21 19 then then ADV ejpam-4528 21 20 f f PROPN ejpam-4528 21 21 is be AUX ejpam-4528 21 22 pettis pettis PROPN ejpam-4528 21 23 integrable integrable ADJ ejpam-4528 21 24 theorem theorem ADJ ejpam-4528 21 25 2 2 NUM ejpam-4528 21 26 ( ( PUNCT ejpam-4528 21 27 resp resp NOUN ejpam-4528 21 28 . . PUNCT ejpam-4528 22 1 theorem theorem ADJ ejpam-4528 22 2 4 4 NUM ejpam-4528 22 3 ) ) PUNCT ejpam-4528 22 4 . . PUNCT ejpam-4528 23 1 it it PRON ejpam-4528 23 2 is be AUX ejpam-4528 23 3 important important ADJ ejpam-4528 23 4 to to PART ejpam-4528 23 5 note note VERB ejpam-4528 23 6 that that SCONJ ejpam-4528 23 7 a a DET ejpam-4528 23 8 bounded bound VERB ejpam-4528 23 9 scalarly scalarly ADV ejpam-4528 23 10 integrable integrable ADJ ejpam-4528 23 11 function function NOUN ejpam-4528 23 12 is be AUX ejpam-4528 23 13 not not PART ejpam-4528 23 14 in in ADP ejpam-4528 23 15 general general ADJ ejpam-4528 23 16 pettis pettis PROPN ejpam-4528 23 17 integrable integrable ADJ ejpam-4528 23 18 , , PUNCT ejpam-4528 23 19 one one PRON ejpam-4528 23 20 can can AUX ejpam-4528 23 21 find find VERB ejpam-4528 23 22 some some DET ejpam-4528 23 23 examples example NOUN ejpam-4528 23 24 in in ADP ejpam-4528 23 25 [ [ X ejpam-4528 23 26 2],[19 2],[19 NOUN ejpam-4528 23 27 ] ] PUNCT ejpam-4528 23 28 . . PUNCT ejpam-4528 24 1 we we PRON ejpam-4528 24 2 note note VERB ejpam-4528 24 3 that that SCONJ ejpam-4528 24 4 the the DET ejpam-4528 24 5 results result NOUN ejpam-4528 24 6 in in ADP ejpam-4528 24 7 [ [ X ejpam-4528 24 8 16 16 NUM ejpam-4528 24 9 ] ] PUNCT ejpam-4528 24 10 will will AUX ejpam-4528 24 11 play play VERB ejpam-4528 24 12 an an DET ejpam-4528 24 13 important important ADJ ejpam-4528 24 14 role role NOUN ejpam-4528 24 15 for for ADP ejpam-4528 24 16 the the DET ejpam-4528 24 17 development development NOUN ejpam-4528 24 18 of of ADP ejpam-4528 24 19 this this DET ejpam-4528 24 20 work work NOUN ejpam-4528 24 21 and and CCONJ ejpam-4528 24 22 a a DET ejpam-4528 24 23 version version NOUN ejpam-4528 24 24 of of ADP ejpam-4528 24 25 theorem theorem NOUN ejpam-4528 24 26 4 4 NUM ejpam-4528 24 27 in in ADP ejpam-4528 24 28 [ [ X ejpam-4528 24 29 16 16 NUM ejpam-4528 24 30 ] ] PUNCT ejpam-4528 24 31 with with ADP ejpam-4528 24 32 pettis pettis PROPN ejpam-4528 24 33 integrable integrable ADJ ejpam-4528 24 34 functions function NOUN ejpam-4528 24 35 is be AUX ejpam-4528 24 36 given give VERB ejpam-4528 24 37 ( ( PUNCT ejpam-4528 24 38 theorem theorem NOUN ejpam-4528 24 39 6 6 NUM ejpam-4528 24 40 ) ) PUNCT ejpam-4528 24 41 . . PUNCT ejpam-4528 25 1 ∗corresponding ∗corresponde VERB ejpam-4528 25 2 author author NOUN ejpam-4528 25 3 . . PUNCT ejpam-4528 26 1 doi doi NOUN ejpam-4528 26 2 : : PUNCT ejpam-4528 26 3 https://doi.org/10.29020/nybg.ejpam.v15i4.4528 https://doi.org/10.29020/nybg.ejpam.v15i4.4528 PUNCT ejpam-4528 26 4 email email NOUN ejpam-4528 26 5 addresses address NOUN ejpam-4528 26 6 : : PUNCT ejpam-4528 26 7 sabiri.noureddine@gmail.com sabiri.noureddine@gmail.com X ejpam-4528 26 8 ( ( PUNCT ejpam-4528 26 9 n. n. PROPN ejpam-4528 26 10 sabiri sabiri PROPN ejpam-4528 26 11 ) ) PUNCT ejpam-4528 26 12 , , PUNCT ejpam-4528 26 13 guessousjssous@yahoo.fr guessousjssous@yahoo.fr PROPN ejpam-4528 26 14 ( ( PUNCT ejpam-4528 26 15 m. m. NOUN ejpam-4528 26 16 guessous guessous ADJ ejpam-4528 26 17 ) ) PUNCT ejpam-4528 26 18 https://www.ejpam.com https://www.ejpam.com NOUN ejpam-4528 26 19 1512 1512 NUM ejpam-4528 27 1 © © PROPN ejpam-4528 27 2 2022 2022 NUM ejpam-4528 27 3 ejpam ejpam VERB ejpam-4528 27 4 all all DET ejpam-4528 27 5 rights right NOUN ejpam-4528 27 6 reserved reserve VERB ejpam-4528 27 7 . . PUNCT ejpam-4528 28 1 n. n. PROPN ejpam-4528 28 2 sabiri sabiri PROPN ejpam-4528 28 3 , , PUNCT ejpam-4528 28 4 m. m. NOUN ejpam-4528 28 5 guessous guessous ADJ ejpam-4528 28 6 / / SYM ejpam-4528 28 7 eur eur PROPN ejpam-4528 28 8 . . PUNCT ejpam-4528 29 1 j. j. PROPN ejpam-4528 29 2 pure pure PROPN ejpam-4528 29 3 appl appl PROPN ejpam-4528 29 4 . . PROPN ejpam-4528 29 5 math math PROPN ejpam-4528 29 6 , , PUNCT ejpam-4528 29 7 15 15 NUM ejpam-4528 29 8 ( ( PUNCT ejpam-4528 29 9 4 4 NUM ejpam-4528 29 10 ) ) PUNCT ejpam-4528 29 11 ( ( PUNCT ejpam-4528 29 12 2022 2022 NUM ejpam-4528 29 13 ) ) PUNCT ejpam-4528 29 14 , , PUNCT ejpam-4528 29 15 1512 1512 NUM ejpam-4528 29 16 - - SYM ejpam-4528 29 17 1520 1520 NUM ejpam-4528 29 18 1513 1513 NUM ejpam-4528 29 19 2 2 NUM ejpam-4528 29 20 . . PUNCT ejpam-4528 29 21 notations notation NOUN ejpam-4528 29 22 and and CCONJ ejpam-4528 29 23 preliminaries preliminary NOUN ejpam-4528 29 24 let let VERB ejpam-4528 29 25 ( ( PUNCT ejpam-4528 29 26 ω ω PROPN ejpam-4528 29 27 , , PUNCT ejpam-4528 29 28 f f PROPN ejpam-4528 29 29 , , PUNCT ejpam-4528 29 30 µ µ X ejpam-4528 29 31 ) ) PUNCT ejpam-4528 29 32 be be AUX ejpam-4528 29 33 a a DET ejpam-4528 29 34 complete complete ADJ ejpam-4528 29 35 probability probability NOUN ejpam-4528 29 36 space space NOUN ejpam-4528 29 37 , , PUNCT ejpam-4528 29 38 e e X ejpam-4528 29 39 a a DET ejpam-4528 29 40 banach banach NOUN ejpam-4528 29 41 space space NOUN ejpam-4528 29 42 and and CCONJ ejpam-4528 29 43 e′ e′ VERB ejpam-4528 29 44 its its PRON ejpam-4528 29 45 topological topological ADJ ejpam-4528 29 46 dual dual NOUN ejpam-4528 29 47 . . PUNCT ejpam-4528 30 1 the the DET ejpam-4528 30 2 weak weak ADJ ejpam-4528 30 3 topology topology NOUN ejpam-4528 30 4 σ(e σ(e PROPN ejpam-4528 30 5 , , PUNCT ejpam-4528 30 6 e′ e′ NOUN ejpam-4528 30 7 ) ) PUNCT ejpam-4528 30 8 on on ADP ejpam-4528 30 9 e e X ejpam-4528 30 10 ( ( PUNCT ejpam-4528 30 11 resp resp NOUN ejpam-4528 30 12 . . PUNCT ejpam-4528 31 1 the the DET ejpam-4528 31 2 weak weak ADJ ejpam-4528 31 3 * * X ejpam-4528 31 4 topology topology NOUN ejpam-4528 31 5 σ(e′ σ(e′ PROPN ejpam-4528 31 6 , , PUNCT ejpam-4528 31 7 e e NOUN ejpam-4528 31 8 ) ) PUNCT ejpam-4528 31 9 on on ADP ejpam-4528 31 10 e′ e′ PROPN ejpam-4528 31 11 ) ) PUNCT ejpam-4528 31 12 will will AUX ejpam-4528 31 13 be be AUX ejpam-4528 31 14 referred refer VERB ejpam-4528 31 15 to to ADP ejpam-4528 31 16 by by ADP ejpam-4528 31 17 the the DET ejpam-4528 31 18 symbol symbol NOUN ejpam-4528 31 19 ” " PUNCT ejpam-4528 31 20 w w PROPN ejpam-4528 31 21 ” " PUNCT ejpam-4528 31 22 ( ( PUNCT ejpam-4528 31 23 resp resp NOUN ejpam-4528 31 24 . . PUNCT ejpam-4528 31 25 ” " PUNCT ejpam-4528 32 1 w w X ejpam-4528 32 2 * * PUNCT ejpam-4528 32 3 ” " PUNCT ejpam-4528 32 4 ) ) PUNCT ejpam-4528 32 5 . . PUNCT ejpam-4528 33 1 a a DET ejpam-4528 33 2 function function NOUN ejpam-4528 33 3 f f NOUN ejpam-4528 33 4 : : PUNCT ejpam-4528 33 5 ω ω PROPN ejpam-4528 33 6 → → SYM ejpam-4528 33 7 e e X ejpam-4528 33 8 ( ( PUNCT ejpam-4528 33 9 resp resp NOUN ejpam-4528 33 10 f f X ejpam-4528 33 11 : : PUNCT ejpam-4528 33 12 ω ω PROPN ejpam-4528 33 13 → → SYM ejpam-4528 33 14 e′ e′ NUM ejpam-4528 33 15 ) ) PUNCT ejpam-4528 33 16 is be AUX ejpam-4528 33 17 w w NOUN ejpam-4528 33 18 - - ADJ ejpam-4528 33 19 measurable measurable ADJ ejpam-4528 33 20 ( ( PUNCT ejpam-4528 33 21 resp resp NOUN ejpam-4528 33 22 w*-measurable w*-measurable ADJ ejpam-4528 33 23 ) ) PUNCT ejpam-4528 33 24 , , PUNCT ejpam-4528 33 25 if if SCONJ ejpam-4528 33 26 for for ADP ejpam-4528 33 27 any any DET ejpam-4528 33 28 x′ x′ PROPN ejpam-4528 33 29 ∈ ∈ PROPN ejpam-4528 33 30 e′ e′ PROPN ejpam-4528 33 31 , , PUNCT ejpam-4528 33 32 ( ( PUNCT ejpam-4528 33 33 resp resp NOUN ejpam-4528 33 34 x x SYM ejpam-4528 33 35 ∈ ∈ PROPN ejpam-4528 33 36 e e X ejpam-4528 33 37 ) ) PUNCT ejpam-4528 33 38 the the DET ejpam-4528 33 39 function function NOUN ejpam-4528 33 40 ⟨f ⟨f ADV ejpam-4528 33 41 , , PUNCT ejpam-4528 33 42 x′⟩ x′⟩ PROPN ejpam-4528 33 43 : : PUNCT ejpam-4528 34 1 ω ω NUM ejpam-4528 34 2 7→ 7→ PROPN ejpam-4528 34 3 ⟨f(ω ⟨f(ω NOUN ejpam-4528 34 4 ) ) PUNCT ejpam-4528 34 5 , , PUNCT ejpam-4528 34 6 x′⟩ x′⟩ PROPN ejpam-4528 34 7 ( ( PUNCT ejpam-4528 34 8 resp resp NOUN ejpam-4528 34 9 ⟨f ⟨f PROPN ejpam-4528 34 10 , , PUNCT ejpam-4528 34 11 x⟩ x⟩ PUNCT ejpam-4528 34 12 : : PUNCT ejpam-4528 35 1 ω ω X ejpam-4528 35 2 7→ 7→ NUM ejpam-4528 35 3 ⟨f(ω ⟨f(ω NOUN ejpam-4528 35 4 ) ) PUNCT ejpam-4528 35 5 , , PUNCT ejpam-4528 35 6 x⟩ x⟩ PUNCT ejpam-4528 35 7 ) ) PUNCT ejpam-4528 35 8 is be AUX ejpam-4528 35 9 measurable measurable ADJ ejpam-4528 35 10 . . PUNCT ejpam-4528 36 1 two two NUM ejpam-4528 36 2 functions function NOUN ejpam-4528 36 3 f f NOUN ejpam-4528 36 4 , , PUNCT ejpam-4528 36 5 g g PROPN ejpam-4528 36 6 : : PUNCT ejpam-4528 36 7 ω ω PROPN ejpam-4528 36 8 → → SYM ejpam-4528 36 9 e e X ejpam-4528 36 10 ( ( PUNCT ejpam-4528 36 11 resp resp NOUN ejpam-4528 36 12 f f X ejpam-4528 36 13 , , PUNCT ejpam-4528 36 14 g g PROPN ejpam-4528 36 15 : : PUNCT ejpam-4528 36 16 ω ω PROPN ejpam-4528 36 17 → → SYM ejpam-4528 36 18 e′ e′ NUM ejpam-4528 36 19 ) ) PUNCT ejpam-4528 36 20 are be AUX ejpam-4528 36 21 w w NOUN ejpam-4528 36 22 - - PUNCT ejpam-4528 36 23 equivalent equivalent ADJ ejpam-4528 36 24 ( ( PUNCT ejpam-4528 36 25 resp resp NOUN ejpam-4528 36 26 w*-equivalent w*-equivalent PROPN ejpam-4528 36 27 ) ) PUNCT ejpam-4528 36 28 , , PUNCT ejpam-4528 36 29 if if SCONJ ejpam-4528 36 30 ⟨f ⟨f X ejpam-4528 36 31 , , PUNCT ejpam-4528 36 32 x′⟩ x′⟩ PROPN ejpam-4528 36 33 = = SYM ejpam-4528 36 34 ⟨g ⟨g PROPN ejpam-4528 36 35 , , PUNCT ejpam-4528 36 36 x′⟩ x′⟩ PROPN ejpam-4528 36 37 µ µ PROPN ejpam-4528 36 38 − − PROPN ejpam-4528 36 39 a.e a.e PROPN ejpam-4528 36 40 . . PROPN ejpam-4528 36 41 for for ADP ejpam-4528 36 42 every every DET ejpam-4528 36 43 x′ x′ PROPN ejpam-4528 36 44 ∈ ∈ PROPN ejpam-4528 36 45 e′ e′ PROPN ejpam-4528 36 46 , , PUNCT ejpam-4528 36 47 ( ( PUNCT ejpam-4528 36 48 resp resp NOUN ejpam-4528 36 49 ⟨f ⟨f PROPN ejpam-4528 36 50 , , PUNCT ejpam-4528 36 51 x⟩ x⟩ PUNCT ejpam-4528 37 1 = = SYM ejpam-4528 37 2 ⟨g ⟨g PROPN ejpam-4528 37 3 , , PUNCT ejpam-4528 37 4 x⟩ x⟩ PUNCT ejpam-4528 38 1 µ− µ− PROPN ejpam-4528 38 2 a.e a.e PROPN ejpam-4528 38 3 . . PROPN ejpam-4528 39 1 for for ADP ejpam-4528 39 2 every every DET ejpam-4528 39 3 x x SYM ejpam-4528 39 4 ∈ ∈ PROPN ejpam-4528 39 5 e e NOUN ejpam-4528 39 6 ) ) PUNCT ejpam-4528 39 7 . . PUNCT ejpam-4528 40 1 a a DET ejpam-4528 40 2 function function NOUN ejpam-4528 40 3 f f NOUN ejpam-4528 40 4 : : PUNCT ejpam-4528 40 5 ω ω PROPN ejpam-4528 40 6 → → SYM ejpam-4528 40 7 e e X ejpam-4528 40 8 ( ( PUNCT ejpam-4528 40 9 resp resp NOUN ejpam-4528 40 10 f f X ejpam-4528 40 11 : : PUNCT ejpam-4528 40 12 ω ω PROPN ejpam-4528 40 13 → → SYM ejpam-4528 40 14 e′ e′ NUM ejpam-4528 40 15 ) ) PUNCT ejpam-4528 40 16 is be AUX ejpam-4528 40 17 scalarly scalarly ADV ejpam-4528 40 18 integrable integrable ADJ ejpam-4528 40 19 ( ( PUNCT ejpam-4528 40 20 resp resp NOUN ejpam-4528 40 21 w*-scalarly w*-scalarly ADV ejpam-4528 40 22 integrable integrable ADJ ejpam-4528 40 23 ) ) PUNCT ejpam-4528 40 24 if if SCONJ ejpam-4528 40 25 for for ADP ejpam-4528 40 26 every every DET ejpam-4528 40 27 x′ x′ PROPN ejpam-4528 40 28 ∈ ∈ PROPN ejpam-4528 40 29 e′ e′ PROPN ejpam-4528 40 30 the the DET ejpam-4528 40 31 function function NOUN ejpam-4528 40 32 ⟨f ⟨f ADV ejpam-4528 40 33 , , PUNCT ejpam-4528 40 34 x′⟩ x′⟩ PROPN ejpam-4528 40 35 ( ( PUNCT ejpam-4528 40 36 resp resp VERB ejpam-4528 40 37 for for ADP ejpam-4528 40 38 every every DET ejpam-4528 40 39 x x SYM ejpam-4528 40 40 ∈ ∈ PROPN ejpam-4528 40 41 e e NOUN ejpam-4528 40 42 the the DET ejpam-4528 40 43 function function NOUN ejpam-4528 40 44 ⟨f ⟨f X ejpam-4528 40 45 , , PUNCT ejpam-4528 40 46 x⟩ x⟩ NUM ejpam-4528 40 47 ) ) PUNCT ejpam-4528 40 48 is be AUX ejpam-4528 40 49 µ-integrable µ-integrable ADJ ejpam-4528 40 50 . . PUNCT ejpam-4528 41 1 if if SCONJ ejpam-4528 41 2 f f PROPN ejpam-4528 41 3 : : PUNCT ejpam-4528 41 4 ω ω PROPN ejpam-4528 41 5 → → SYM ejpam-4528 41 6 e e X ejpam-4528 41 7 is be AUX ejpam-4528 41 8 scalarly scalarly ADV ejpam-4528 41 9 integrable integrable ADJ ejpam-4528 41 10 , , PUNCT ejpam-4528 41 11 then then ADV ejpam-4528 41 12 ( ( PUNCT ejpam-4528 41 13 [ [ X ejpam-4528 41 14 7 7 X ejpam-4528 41 15 ] ] X ejpam-4528 41 16 lemma lemma PROPN ejpam-4528 41 17 1 1 NUM ejpam-4528 41 18 . . PUNCT ejpam-4528 42 1 p. p. NOUN ejpam-4528 42 2 52 52 NUM ejpam-4528 42 3 ) ) PUNCT ejpam-4528 42 4 for for ADP ejpam-4528 42 5 every every DET ejpam-4528 42 6 a a DET ejpam-4528 42 7 ∈ ∈ PROPN ejpam-4528 42 8 f f NOUN ejpam-4528 42 9 there there PRON ejpam-4528 42 10 exists exist VERB ejpam-4528 42 11 x′′f x′′f PROPN ejpam-4528 42 12 ( ( PUNCT ejpam-4528 42 13 a a NOUN ejpam-4528 42 14 ) ) PUNCT ejpam-4528 42 15 in in ADP ejpam-4528 42 16 e′′ e′′ PROPN ejpam-4528 42 17 such such ADJ ejpam-4528 42 18 that that SCONJ ejpam-4528 42 19 , , PUNCT ejpam-4528 42 20 for for ADP ejpam-4528 42 21 every every DET ejpam-4528 42 22 x′ x′ PROPN ejpam-4528 42 23 ∈ ∈ PROPN ejpam-4528 42 24 e′ e′ PROPN ejpam-4528 42 25 ⟨x′′f ⟨x′′f PROPN ejpam-4528 42 26 ( ( PUNCT ejpam-4528 42 27 a a NOUN ejpam-4528 42 28 ) ) PUNCT ejpam-4528 42 29 , , PUNCT ejpam-4528 42 30 x′⟩ x′⟩ PROPN ejpam-4528 43 1 = = PUNCT ejpam-4528 43 2 ∫ ∫ PROPN ejpam-4528 43 3 a a DET ejpam-4528 43 4 ⟨f ⟨f NOUN ejpam-4528 43 5 , , PUNCT ejpam-4528 43 6 x′⟩ x′⟩ PROPN ejpam-4528 43 7 dµ dµ PROPN ejpam-4528 43 8 , , PUNCT ejpam-4528 43 9 the the DET ejpam-4528 43 10 element element NOUN ejpam-4528 43 11 x′′f x′′f PROPN ejpam-4528 43 12 ( ( PUNCT ejpam-4528 43 13 a a X ejpam-4528 43 14 ) ) PUNCT ejpam-4528 43 15 is be AUX ejpam-4528 43 16 called call VERB ejpam-4528 43 17 the the DET ejpam-4528 43 18 dunford dunford NOUN ejpam-4528 43 19 integral integral ADJ ejpam-4528 43 20 of of ADP ejpam-4528 43 21 f f PROPN ejpam-4528 43 22 over over ADP ejpam-4528 43 23 a a PRON ejpam-4528 43 24 and and CCONJ ejpam-4528 43 25 denoted denote VERB ejpam-4528 43 26 by by ADP ejpam-4528 43 27 ( ( PUNCT ejpam-4528 43 28 d)− d)− PROPN ejpam-4528 43 29 ∫ ∫ PROPN ejpam-4528 43 30 a a DET ejpam-4528 43 31 fdµ. fdµ. NOUN ejpam-4528 43 32 by by ADP ejpam-4528 43 33 definition definition NOUN ejpam-4528 43 34 , , PUNCT ejpam-4528 43 35 f f PROPN ejpam-4528 43 36 is be AUX ejpam-4528 43 37 pettis pettis PROPN ejpam-4528 43 38 integrable integrable ADJ ejpam-4528 43 39 if if SCONJ ejpam-4528 43 40 ( ( PUNCT ejpam-4528 43 41 d d X ejpam-4528 43 42 ) ) PUNCT ejpam-4528 44 1 − − ADP ejpam-4528 44 2 ∫ ∫ PROPN ejpam-4528 44 3 a a DET ejpam-4528 44 4 fdµ fdµ NOUN ejpam-4528 44 5 ∈ ∈ PROPN ejpam-4528 44 6 e e NOUN ejpam-4528 44 7 for for ADP ejpam-4528 44 8 all all DET ejpam-4528 44 9 a a DET ejpam-4528 44 10 ∈ ∈ NOUN ejpam-4528 44 11 f f NOUN ejpam-4528 45 1 and and CCONJ ejpam-4528 45 2 we we PRON ejpam-4528 45 3 write write VERB ejpam-4528 45 4 ( ( PUNCT ejpam-4528 45 5 p p NOUN ejpam-4528 45 6 ) ) PUNCT ejpam-4528 45 7 − − ADP ejpam-4528 45 8 ∫ ∫ PROPN ejpam-4528 46 1 a a DET ejpam-4528 46 2 fdµ fdµ NOUN ejpam-4528 46 3 instead instead ADV ejpam-4528 46 4 of of ADP ejpam-4528 46 5 ( ( PUNCT ejpam-4528 46 6 d d NOUN ejpam-4528 46 7 ) ) PUNCT ejpam-4528 46 8 − − ADP ejpam-4528 46 9 ∫ ∫ PROPN ejpam-4528 46 10 a a DET ejpam-4528 46 11 fdµ. fdµ. NOUN ejpam-4528 46 12 also also ADV ejpam-4528 46 13 , , PUNCT ejpam-4528 46 14 ( ( PUNCT ejpam-4528 46 15 [ [ X ejpam-4528 46 16 7 7 X ejpam-4528 46 17 ] ] PUNCT ejpam-4528 46 18 p. p. NOUN ejpam-4528 46 19 53 53 NUM ejpam-4528 46 20 ) ) PUNCT ejpam-4528 46 21 if if SCONJ ejpam-4528 46 22 f f PROPN ejpam-4528 46 23 : : PUNCT ejpam-4528 46 24 ω ω PROPN ejpam-4528 46 25 → → SYM ejpam-4528 46 26 e′ e′ X ejpam-4528 46 27 is be AUX ejpam-4528 46 28 w*-scalarly w*-scalarly ADV ejpam-4528 46 29 integrable integrable ADJ ejpam-4528 46 30 then then ADV ejpam-4528 46 31 for for ADP ejpam-4528 46 32 every every DET ejpam-4528 46 33 a a DET ejpam-4528 46 34 ∈ ∈ ADJ ejpam-4528 46 35 f f NOUN ejpam-4528 46 36 there there PRON ejpam-4528 46 37 exists exist VERB ejpam-4528 46 38 x′f x′f PUNCT ejpam-4528 47 1 ( ( PUNCT ejpam-4528 47 2 a a X ejpam-4528 47 3 ) ) PUNCT ejpam-4528 47 4 in in ADP ejpam-4528 47 5 e′ e′ PROPN ejpam-4528 47 6 such such ADJ ejpam-4528 47 7 that that SCONJ ejpam-4528 47 8 , , PUNCT ejpam-4528 47 9 for for ADP ejpam-4528 47 10 every every DET ejpam-4528 47 11 x x SYM ejpam-4528 47 12 ∈ ∈ PROPN ejpam-4528 47 13 e e X ejpam-4528 47 14 ⟨x′f ⟨x′f X ejpam-4528 47 15 ( ( PUNCT ejpam-4528 47 16 a a NOUN ejpam-4528 47 17 ) ) PUNCT ejpam-4528 47 18 , , PUNCT ejpam-4528 47 19 x⟩ x⟩ PUNCT ejpam-4528 48 1 = = PRON ejpam-4528 48 2 ∫ ∫ PROPN ejpam-4528 48 3 a a DET ejpam-4528 48 4 ⟨f ⟨f NOUN ejpam-4528 48 5 , , PUNCT ejpam-4528 48 6 x⟩ x⟩ PROPN ejpam-4528 49 1 dµ dµ PROPN ejpam-4528 49 2 , , PUNCT ejpam-4528 49 3 the the DET ejpam-4528 49 4 element element NOUN ejpam-4528 49 5 x′f x′f PROPN ejpam-4528 49 6 ( ( PUNCT ejpam-4528 49 7 a a X ejpam-4528 49 8 ) ) PUNCT ejpam-4528 49 9 is be AUX ejpam-4528 49 10 called call VERB ejpam-4528 49 11 the the DET ejpam-4528 49 12 weak weak ADJ ejpam-4528 49 13 * * ADJ ejpam-4528 49 14 integral integral ADJ ejpam-4528 49 15 ( ( PUNCT ejpam-4528 49 16 or or CCONJ ejpam-4528 49 17 gelfand gelfand ADJ ejpam-4528 49 18 integral integral ADJ ejpam-4528 49 19 ) ) PUNCT ejpam-4528 49 20 of of ADP ejpam-4528 49 21 f f PROPN ejpam-4528 49 22 over over ADP ejpam-4528 49 23 a a PRON ejpam-4528 49 24 and and CCONJ ejpam-4528 49 25 denoted denote VERB ejpam-4528 49 26 by by ADP ejpam-4528 49 27 ( ( PUNCT ejpam-4528 49 28 w∗ w∗ NOUN ejpam-4528 49 29 ) ) PUNCT ejpam-4528 49 30 − − ADP ejpam-4528 49 31 ∫ ∫ PROPN ejpam-4528 49 32 a a DET ejpam-4528 49 33 fdµ. fdµ. NOUN ejpam-4528 49 34 a a DET ejpam-4528 49 35 sequence sequence NOUN ejpam-4528 49 36 ( ( PUNCT ejpam-4528 49 37 fn fn NOUN ejpam-4528 49 38 ) ) PUNCT ejpam-4528 49 39 of of ADP ejpam-4528 49 40 e e NOUN ejpam-4528 49 41 - - VERB ejpam-4528 49 42 valued value VERB ejpam-4528 49 43 scalarly scalarly ADV ejpam-4528 49 44 integrable integrable ADJ ejpam-4528 49 45 functions function NOUN ejpam-4528 49 46 converges converge VERB ejpam-4528 49 47 pointwise pointwise VERB ejpam-4528 49 48 on on ADP ejpam-4528 49 49 l∞ l∞ NOUN ejpam-4528 49 50 r r NOUN ejpam-4528 49 51 ( ( PUNCT ejpam-4528 49 52 µ µ NOUN ejpam-4528 49 53 ) ) PUNCT ejpam-4528 49 54 ⊗ ⊗ NOUN ejpam-4528 49 55 e′ e′ PROPN ejpam-4528 49 56 to to ADP ejpam-4528 49 57 an an DET ejpam-4528 49 58 e e NOUN ejpam-4528 49 59 - - VERB ejpam-4528 49 60 valued value VERB ejpam-4528 49 61 scalarly scalarly ADV ejpam-4528 49 62 integrable integrable ADJ ejpam-4528 49 63 function function NOUN ejpam-4528 49 64 f f PROPN ejpam-4528 49 65 if if SCONJ ejpam-4528 49 66 ∀h ∀h PROPN ejpam-4528 49 67 ∈ ∈ PROPN ejpam-4528 49 68 l∞ l∞ NOUN ejpam-4528 49 69 r r NOUN ejpam-4528 49 70 ( ( PUNCT ejpam-4528 49 71 µ),∀x′ µ),∀x′ X ejpam-4528 49 72 ∈ ∈ PROPN ejpam-4528 49 73 e′ e′ PROPN ejpam-4528 49 74 , , PUNCT ejpam-4528 49 75 ∫ ∫ PROPN ejpam-4528 49 76 ω ω NUM ejpam-4528 49 77 h⟨fn h⟨fn PROPN ejpam-4528 49 78 , , PUNCT ejpam-4528 49 79 x′⟩ x′⟩ PROPN ejpam-4528 49 80 dµ dµ PROPN ejpam-4528 49 81 → → SYM ejpam-4528 49 82 ∫ ∫ PROPN ejpam-4528 49 83 ω ω PROPN ejpam-4528 49 84 h⟨f h⟨f PROPN ejpam-4528 49 85 , , PUNCT ejpam-4528 49 86 x′⟩ x′⟩ PROPN ejpam-4528 49 87 dµ dµ PROPN ejpam-4528 49 88 , , PUNCT ejpam-4528 49 89 or or CCONJ ejpam-4528 49 90 equivalently equivalently ADV ejpam-4528 49 91 ( ( PUNCT ejpam-4528 49 92 [ [ X ejpam-4528 49 93 8 8 NUM ejpam-4528 49 94 ] ] PUNCT ejpam-4528 49 95 theorem theorem NOUN ejpam-4528 49 96 7 7 NUM ejpam-4528 49 97 . . PUNCT ejpam-4528 50 1 p. p. NOUN ejpam-4528 50 2 291 291 NUM ejpam-4528 50 3 ) ) PUNCT ejpam-4528 50 4 for for ADP ejpam-4528 50 5 every every DET ejpam-4528 50 6 x′ x′ PROPN ejpam-4528 50 7 ∈ ∈ PROPN ejpam-4528 50 8 e′ e′ PROPN ejpam-4528 50 9 , , PUNCT ejpam-4528 50 10 the the DET ejpam-4528 50 11 sequence sequence NOUN ejpam-4528 50 12 ( ( PUNCT ejpam-4528 50 13 ⟨fn ⟨fn NUM ejpam-4528 50 14 , , PUNCT ejpam-4528 50 15 x′⟩)n x′⟩)n PROPN ejpam-4528 50 16 is be AUX ejpam-4528 50 17 bounded bound VERB ejpam-4528 50 18 in in ADP ejpam-4528 50 19 l1 l1 PROPN ejpam-4528 50 20 r(µ r(µ PROPN ejpam-4528 50 21 ) ) PUNCT ejpam-4528 50 22 and and CCONJ ejpam-4528 50 23 ∀a ∀a NOUN ejpam-4528 50 24 ∈ ∈ PROPN ejpam-4528 50 25 f f PROPN ejpam-4528 50 26 , , PUNCT ejpam-4528 50 27 ∫ ∫ PROPN ejpam-4528 50 28 a a DET ejpam-4528 50 29 ⟨fn ⟨fn PROPN ejpam-4528 50 30 , , PUNCT ejpam-4528 50 31 x′⟩ x′⟩ PROPN ejpam-4528 50 32 dµ dµ PROPN ejpam-4528 50 33 → → SYM ejpam-4528 50 34 ∫ ∫ PROPN ejpam-4528 50 35 a a DET ejpam-4528 50 36 ⟨f ⟨f NOUN ejpam-4528 50 37 , , PUNCT ejpam-4528 50 38 x′⟩ x′⟩ PROPN ejpam-4528 50 39 dµ. dµ. PROPN ejpam-4528 50 40 let let VERB ejpam-4528 50 41 p p NOUN ejpam-4528 50 42 1 1 NUM ejpam-4528 50 43 e(µ e(µ NOUN ejpam-4528 50 44 ) ) PUNCT ejpam-4528 50 45 denote denote VERB ejpam-4528 50 46 the the DET ejpam-4528 50 47 ( ( PUNCT ejpam-4528 50 48 quotient quotient NOUN ejpam-4528 50 49 ) ) PUNCT ejpam-4528 50 50 space space NOUN ejpam-4528 50 51 of of ADP ejpam-4528 50 52 pettis pettis PROPN ejpam-4528 50 53 integrable integrable ADJ ejpam-4528 50 54 e e ADJ ejpam-4528 50 55 - - ADJ ejpam-4528 50 56 valued value VERB ejpam-4528 50 57 functions function NOUN ejpam-4528 50 58 . . PUNCT ejpam-4528 51 1 the the DET ejpam-4528 51 2 weak weak ADJ ejpam-4528 51 3 topology topology NOUN ejpam-4528 51 4 on on ADP ejpam-4528 51 5 p p NOUN ejpam-4528 51 6 1 1 NUM ejpam-4528 51 7 e(µ e(µ NOUN ejpam-4528 51 8 ) ) PUNCT ejpam-4528 51 9 is be AUX ejpam-4528 51 10 the the DET ejpam-4528 51 11 weak weak ADJ ejpam-4528 51 12 topology topology NOUN ejpam-4528 51 13 induced induce VERB ejpam-4528 51 14 by by ADP ejpam-4528 51 15 the the DET ejpam-4528 51 16 duality duality NOUN ejpam-4528 51 17 ( ( PUNCT ejpam-4528 51 18 p p NOUN ejpam-4528 51 19 1 1 NUM ejpam-4528 51 20 e(µ e(µ NOUN ejpam-4528 51 21 ) ) PUNCT ejpam-4528 51 22 , , PUNCT ejpam-4528 51 23 l l NOUN ejpam-4528 51 24 ∞ ∞ PROPN ejpam-4528 51 25 r r NOUN ejpam-4528 51 26 ( ( PUNCT ejpam-4528 51 27 µ µ NOUN ejpam-4528 51 28 ) ) PUNCT ejpam-4528 51 29 ⊗ ⊗ NOUN ejpam-4528 51 30 e′ e′ NUM ejpam-4528 51 31 ) ) PUNCT ejpam-4528 51 32 . . PUNCT ejpam-4528 52 1 if if SCONJ ejpam-4528 52 2 e e PROPN ejpam-4528 52 3 is be AUX ejpam-4528 52 4 separable separable ADJ ejpam-4528 52 5 and and CCONJ ejpam-4528 52 6 f f PROPN ejpam-4528 52 7 : : PUNCT ejpam-4528 52 8 ω ω PROPN ejpam-4528 52 9 → → SYM ejpam-4528 52 10 e′ e′ X ejpam-4528 52 11 is be AUX ejpam-4528 52 12 w*-measurable w*-measurable ADJ ejpam-4528 52 13 , , PUNCT ejpam-4528 52 14 the the DET ejpam-4528 52 15 function function NOUN ejpam-4528 52 16 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 52 17 is be AUX ejpam-4528 52 18 measurable measurable ADJ ejpam-4528 52 19 [ [ X ejpam-4528 52 20 20 20 NUM ejpam-4528 52 21 ] ] PUNCT ejpam-4528 52 22 however however ADV ejpam-4528 52 23 , , PUNCT ejpam-4528 52 24 this this PRON ejpam-4528 52 25 is be AUX ejpam-4528 52 26 not not PART ejpam-4528 52 27 always always ADV ejpam-4528 52 28 the the DET ejpam-4528 52 29 case case NOUN ejpam-4528 52 30 if if SCONJ ejpam-4528 52 31 e e PRON ejpam-4528 52 32 is be AUX ejpam-4528 52 33 a a DET ejpam-4528 52 34 general general ADJ ejpam-4528 52 35 banach banach NOUN ejpam-4528 52 36 space space NOUN ejpam-4528 52 37 ( ( PUNCT ejpam-4528 52 38 [ [ X ejpam-4528 52 39 14 14 NUM ejpam-4528 52 40 ] ] PUNCT ejpam-4528 52 41 example example NOUN ejpam-4528 52 42 3.3 3.3 NUM ejpam-4528 52 43 ) ) PUNCT ejpam-4528 52 44 . . PUNCT ejpam-4528 53 1 with with ADP ejpam-4528 53 2 e e X ejpam-4528 53 3 being be AUX ejpam-4528 53 4 separable separable ADJ ejpam-4528 53 5 , , PUNCT ejpam-4528 53 6 the the DET ejpam-4528 53 7 banach banach NOUN ejpam-4528 53 8 space space NOUN ejpam-4528 53 9 ( ( PUNCT ejpam-4528 53 10 l1 l1 PROPN ejpam-4528 53 11 e′ e′ PROPN ejpam-4528 53 12 [ [ X ejpam-4528 53 13 e e X ejpam-4528 53 14 ] ] X ejpam-4528 53 15 , , PUNCT ejpam-4528 53 16 n1 n1 PROPN ejpam-4528 53 17 ) ) PUNCT ejpam-4528 53 18 ( ( PUNCT ejpam-4528 53 19 [ [ X ejpam-4528 53 20 3],[21],[16 3],[21],[16 NUM ejpam-4528 53 21 ] ] PUNCT ejpam-4528 53 22 ) ) PUNCT ejpam-4528 53 23 is be AUX ejpam-4528 53 24 simply simply ADV ejpam-4528 53 25 the the DET ejpam-4528 53 26 ( ( PUNCT ejpam-4528 53 27 quotient quotient NOUN ejpam-4528 53 28 ) ) PUNCT ejpam-4528 53 29 space space NOUN ejpam-4528 53 30 of of ADP ejpam-4528 53 31 w*-scalarly w*-scalarly ADV ejpam-4528 53 32 integrable integrable ADJ ejpam-4528 53 33 functions function NOUN ejpam-4528 53 34 f f X ejpam-4528 53 35 : : PUNCT ejpam-4528 53 36 ω ω PROPN ejpam-4528 53 37 → → SYM ejpam-4528 53 38 e′ e′ X ejpam-4528 53 39 such such ADJ ejpam-4528 53 40 that that SCONJ ejpam-4528 53 41 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 53 42 is be AUX ejpam-4528 53 43 µ-integrable µ-integrable ADJ ejpam-4528 53 44 , , PUNCT ejpam-4528 53 45 and and CCONJ ejpam-4528 53 46 n1(f n1(f X ejpam-4528 53 47 ) ) PUNCT ejpam-4528 53 48 = = SYM ejpam-4528 54 1 ∫ ∫ PROPN ejpam-4528 54 2 ω ω NUM ejpam-4528 54 3 ∥f(ω)∥ ∥f(ω)∥ PROPN ejpam-4528 54 4 dµ(ω dµ(ω PUNCT ejpam-4528 54 5 ) ) PUNCT ejpam-4528 54 6 , , PUNCT ejpam-4528 54 7 f f PROPN ejpam-4528 54 8 ∈ ∈ PROPN ejpam-4528 54 9 l1 l1 PROPN ejpam-4528 54 10 e′ e′ PUNCT ejpam-4528 55 1 [ [ X ejpam-4528 55 2 e e X ejpam-4528 55 3 ] ] PUNCT ejpam-4528 55 4 . . PUNCT ejpam-4528 56 1 n. n. PROPN ejpam-4528 56 2 sabiri sabiri PROPN ejpam-4528 56 3 , , PUNCT ejpam-4528 56 4 m. m. NOUN ejpam-4528 56 5 guessous guessous ADJ ejpam-4528 56 6 / / SYM ejpam-4528 56 7 eur eur PROPN ejpam-4528 56 8 . . PUNCT ejpam-4528 57 1 j. j. PROPN ejpam-4528 57 2 pure pure PROPN ejpam-4528 57 3 appl appl PROPN ejpam-4528 57 4 . . PROPN ejpam-4528 57 5 math math PROPN ejpam-4528 57 6 , , PUNCT ejpam-4528 57 7 15 15 NUM ejpam-4528 57 8 ( ( PUNCT ejpam-4528 57 9 4 4 NUM ejpam-4528 57 10 ) ) PUNCT ejpam-4528 57 11 ( ( PUNCT ejpam-4528 57 12 2022 2022 NUM ejpam-4528 57 13 ) ) PUNCT ejpam-4528 57 14 , , PUNCT ejpam-4528 57 15 1512 1512 NUM ejpam-4528 57 16 - - SYM ejpam-4528 57 17 1520 1520 NUM ejpam-4528 57 18 1514 1514 NUM ejpam-4528 57 19 finally finally ADV ejpam-4528 57 20 , , PUNCT ejpam-4528 57 21 we we PRON ejpam-4528 57 22 recall recall VERB ejpam-4528 57 23 that that SCONJ ejpam-4528 57 24 a a DET ejpam-4528 57 25 set set ADJ ejpam-4528 57 26 h h NOUN ejpam-4528 57 27 of of ADP ejpam-4528 57 28 l1 l1 PROPN ejpam-4528 57 29 r(µ r(µ PROPN ejpam-4528 57 30 ) ) PUNCT ejpam-4528 57 31 is be AUX ejpam-4528 57 32 uniformly uniformly ADV ejpam-4528 57 33 integrable integrable ADJ ejpam-4528 57 34 ( ( PUNCT ejpam-4528 57 35 briefly briefly NOUN ejpam-4528 57 36 ui ui PROPN ejpam-4528 57 37 ) ) PUNCT ejpam-4528 57 38 if if SCONJ ejpam-4528 57 39 it it PRON ejpam-4528 57 40 is be AUX ejpam-4528 57 41 bounded bound VERB ejpam-4528 57 42 and and CCONJ ejpam-4528 58 1 lim lim PROPN ejpam-4528 58 2 µ(a)→0 µ(a)→0 NOUN ejpam-4528 58 3 sup sup PROPN ejpam-4528 58 4 f∈h f∈h VERB ejpam-4528 58 5 ∫ ∫ PROPN ejpam-4528 58 6 a a DET ejpam-4528 58 7 |f |f PROPN ejpam-4528 58 8 | | ADV ejpam-4528 58 9 dµ dµ ADJ ejpam-4528 58 10 = = SYM ejpam-4528 58 11 0 0 X ejpam-4528 58 12 . . PUNCT ejpam-4528 59 1 a a DET ejpam-4528 59 2 set set NOUN ejpam-4528 59 3 k k PROPN ejpam-4528 59 4 of of ADP ejpam-4528 59 5 l1 l1 PROPN ejpam-4528 59 6 e′ e′ PUNCT ejpam-4528 60 1 [ [ X ejpam-4528 60 2 e e X ejpam-4528 60 3 ] ] X ejpam-4528 60 4 is be AUX ejpam-4528 60 5 ui ui PROPN ejpam-4528 61 1 [ [ X ejpam-4528 61 2 16 16 NUM ejpam-4528 61 3 ] ] PUNCT ejpam-4528 61 4 if if SCONJ ejpam-4528 61 5 the the DET ejpam-4528 61 6 set set NOUN ejpam-4528 61 7 { { PUNCT ejpam-4528 61 8 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 61 9 : : PUNCT ejpam-4528 62 1 f f PROPN ejpam-4528 62 2 ∈ ∈ PROPN ejpam-4528 62 3 k k X ejpam-4528 62 4 } } PUNCT ejpam-4528 62 5 is be AUX ejpam-4528 62 6 ui ui PROPN ejpam-4528 62 7 in in ADP ejpam-4528 62 8 l1 l1 PROPN ejpam-4528 62 9 r(µ r(µ PROPN ejpam-4528 62 10 ) ) PUNCT ejpam-4528 62 11 , , PUNCT ejpam-4528 62 12 and and CCONJ ejpam-4528 62 13 we we PRON ejpam-4528 62 14 say say VERB ejpam-4528 62 15 that that SCONJ ejpam-4528 62 16 a a DET ejpam-4528 62 17 set set ADJ ejpam-4528 62 18 h h NOUN ejpam-4528 62 19 of of ADP ejpam-4528 62 20 e e NOUN ejpam-4528 62 21 - - VERB ejpam-4528 62 22 valued value VERB ejpam-4528 62 23 scalarly scalarly ADV ejpam-4528 62 24 integrable integrable ADJ ejpam-4528 62 25 functions function NOUN ejpam-4528 62 26 is be AUX ejpam-4528 62 27 scalarly scalarly ADV ejpam-4528 62 28 uniformly uniformly ADV ejpam-4528 62 29 integrable integrable ADJ ejpam-4528 62 30 briefly briefly NOUN ejpam-4528 62 31 sui sui PROPN ejpam-4528 62 32 ( ( PUNCT ejpam-4528 62 33 resp resp PROPN ejpam-4528 62 34 w w ADV ejpam-4528 62 35 - - PUNCT ejpam-4528 62 36 scalarly scalarly ADV ejpam-4528 62 37 uniformly uniformly ADV ejpam-4528 62 38 integrable integrable ADJ ejpam-4528 62 39 briefly briefly NOUN ejpam-4528 62 40 wsui wsui NOUN ejpam-4528 62 41 ) ) PUNCT ejpam-4528 62 42 , , PUNCT ejpam-4528 62 43 if if SCONJ ejpam-4528 62 44 the the DET ejpam-4528 62 45 set set NOUN ejpam-4528 62 46 { { PUNCT ejpam-4528 62 47 ⟨x′ ⟨x′ PROPN ejpam-4528 62 48 , , PUNCT ejpam-4528 62 49 f⟩ f⟩ NOUN ejpam-4528 62 50 : : PUNCT ejpam-4528 62 51 ∥x′∥ ∥x′∥ NOUN ejpam-4528 62 52 ≤ ≤ ADV ejpam-4528 62 53 1 1 NUM ejpam-4528 62 54 , , PUNCT ejpam-4528 62 55 f f PROPN ejpam-4528 62 56 ∈ ∈ PROPN ejpam-4528 62 57 h h NOUN ejpam-4528 62 58 } } PUNCT ejpam-4528 62 59 ( ( PUNCT ejpam-4528 62 60 resp resp VERB ejpam-4528 62 61 for for ADP ejpam-4528 62 62 each each DET ejpam-4528 62 63 x′ x′ PROPN ejpam-4528 62 64 ∈ ∈ PROPN ejpam-4528 62 65 e′ e′ PROPN ejpam-4528 62 66 , , PUNCT ejpam-4528 62 67 the the DET ejpam-4528 62 68 set set NOUN ejpam-4528 62 69 { { PUNCT ejpam-4528 62 70 ⟨x′ ⟨x′ PROPN ejpam-4528 62 71 , , PUNCT ejpam-4528 62 72 f⟩ f⟩ NOUN ejpam-4528 62 73 : : PUNCT ejpam-4528 62 74 f f PROPN ejpam-4528 62 75 ∈ ∈ PROPN ejpam-4528 62 76 h h NOUN ejpam-4528 62 77 } } PUNCT ejpam-4528 62 78 ) ) PUNCT ejpam-4528 62 79 is be AUX ejpam-4528 62 80 ui ui PROPN ejpam-4528 62 81 in in ADP ejpam-4528 62 82 l1 l1 PROPN ejpam-4528 62 83 r(µ r(µ PROPN ejpam-4528 62 84 ) ) PUNCT ejpam-4528 62 85 . . PUNCT ejpam-4528 63 1 3 3 X ejpam-4528 63 2 . . X ejpam-4528 63 3 pettis pettis PROPN ejpam-4528 63 4 integrability integrability NOUN ejpam-4528 63 5 and and CCONJ ejpam-4528 63 6 truncation truncation NOUN ejpam-4528 63 7 by by ADP ejpam-4528 63 8 ( ( PUNCT ejpam-4528 63 9 [ [ X ejpam-4528 63 10 10 10 NUM ejpam-4528 63 11 ] ] PUNCT ejpam-4528 63 12 p.82 p.82 NOUN ejpam-4528 63 13 ) ) PUNCT ejpam-4528 63 14 , , PUNCT ejpam-4528 63 15 if if SCONJ ejpam-4528 63 16 f f PROPN ejpam-4528 63 17 : : PUNCT ejpam-4528 63 18 ω ω PROPN ejpam-4528 63 19 → → SYM ejpam-4528 63 20 e e PROPN ejpam-4528 63 21 is be AUX ejpam-4528 63 22 pettis pettis PROPN ejpam-4528 63 23 integrable integrable ADJ ejpam-4528 63 24 then then ADV ejpam-4528 63 25 { { PUNCT ejpam-4528 63 26 f f X ejpam-4528 63 27 } } PUNCT ejpam-4528 63 28 is be AUX ejpam-4528 63 29 sui sui PROPN ejpam-4528 63 30 and and CCONJ ejpam-4528 63 31 the the DET ejpam-4528 63 32 converse converse NOUN ejpam-4528 63 33 remains remain VERB ejpam-4528 63 34 true true ADJ ejpam-4528 63 35 if if SCONJ ejpam-4528 63 36 f f PROPN ejpam-4528 63 37 is be AUX ejpam-4528 63 38 strongly strongly ADV ejpam-4528 63 39 measurable measurable ADJ ejpam-4528 63 40 ( ( PUNCT ejpam-4528 63 41 [ [ X ejpam-4528 63 42 14 14 NUM ejpam-4528 63 43 ] ] PUNCT ejpam-4528 63 44 theorem theorem VERB ejpam-4528 63 45 5.2 5.2 NUM ejpam-4528 63 46 ) ) PUNCT ejpam-4528 63 47 . . PUNCT ejpam-4528 64 1 for for ADP ejpam-4528 64 2 the the DET ejpam-4528 64 3 instance instance NOUN ejpam-4528 64 4 of of ADP ejpam-4528 64 5 l1 l1 PROPN ejpam-4528 64 6 e′ e′ PUNCT ejpam-4528 65 1 [ [ X ejpam-4528 65 2 e e X ejpam-4528 65 3 ] ] X ejpam-4528 65 4 , , PUNCT ejpam-4528 65 5 we we PRON ejpam-4528 65 6 give give VERB ejpam-4528 65 7 some some DET ejpam-4528 65 8 characterizations characterization NOUN ejpam-4528 65 9 of of ADP ejpam-4528 65 10 the the DET ejpam-4528 65 11 pettis pettis NOUN ejpam-4528 65 12 integrability integrability NOUN ejpam-4528 65 13 by by ADP ejpam-4528 65 14 the the DET ejpam-4528 65 15 mean mean NOUN ejpam-4528 65 16 of of ADP ejpam-4528 65 17 the the DET ejpam-4528 65 18 associated associated ADJ ejpam-4528 65 19 truncated truncate VERB ejpam-4528 65 20 functions function NOUN ejpam-4528 65 21 . . PUNCT ejpam-4528 66 1 our our PRON ejpam-4528 66 2 work work NOUN ejpam-4528 66 3 build build VERB ejpam-4528 66 4 on on ADP ejpam-4528 66 5 the the DET ejpam-4528 66 6 following follow VERB ejpam-4528 66 7 ( ( PUNCT ejpam-4528 66 8 [ [ X ejpam-4528 66 9 4 4 NUM ejpam-4528 66 10 ] ] PUNCT ejpam-4528 66 11 , , PUNCT ejpam-4528 66 12 theorem theorem VERB ejpam-4528 66 13 3.1 3.1 NUM ejpam-4528 66 14 ): ): PUNCT ejpam-4528 66 15 theorem theorem NOUN ejpam-4528 66 16 1 1 NUM ejpam-4528 66 17 . . PUNCT ejpam-4528 67 1 let let VERB ejpam-4528 67 2 e e PRON ejpam-4528 67 3 be be AUX ejpam-4528 67 4 a a DET ejpam-4528 67 5 banach banach NOUN ejpam-4528 67 6 space space NOUN ejpam-4528 67 7 , , PUNCT ejpam-4528 67 8 ( ( PUNCT ejpam-4528 67 9 fn fn NOUN ejpam-4528 67 10 ) ) PUNCT ejpam-4528 67 11 a a DET ejpam-4528 67 12 sequence sequence NOUN ejpam-4528 67 13 of of ADP ejpam-4528 67 14 e e NOUN ejpam-4528 67 15 - - VERB ejpam-4528 67 16 valued value VERB ejpam-4528 67 17 pettis pettis PROPN ejpam-4528 67 18 integrable integrable ADJ ejpam-4528 67 19 functions function NOUN ejpam-4528 67 20 and and CCONJ ejpam-4528 67 21 f f NOUN ejpam-4528 67 22 : : PUNCT ejpam-4528 67 23 ω ω X ejpam-4528 67 24 → → SYM ejpam-4528 67 25 e e X ejpam-4528 67 26 a a DET ejpam-4528 67 27 scalarly scalarly ADV ejpam-4528 67 28 integrable integrable ADJ ejpam-4528 67 29 function function NOUN ejpam-4528 67 30 satisfying satisfying ADJ ejpam-4528 67 31 : : PUNCT ejpam-4528 67 32 ( ( PUNCT ejpam-4528 67 33 i i NOUN ejpam-4528 67 34 ) ) PUNCT ejpam-4528 67 35 { { PUNCT ejpam-4528 67 36 f f X ejpam-4528 67 37 } } PUNCT ejpam-4528 67 38 is be AUX ejpam-4528 67 39 sui sui PROPN ejpam-4528 67 40 , , PUNCT ejpam-4528 67 41 ( ( PUNCT ejpam-4528 67 42 ii ii NOUN ejpam-4528 67 43 ) ) PUNCT ejpam-4528 67 44 ( ( PUNCT ejpam-4528 67 45 fn fn NOUN ejpam-4528 67 46 ) ) PUNCT ejpam-4528 67 47 converges converge VERB ejpam-4528 67 48 pointwise pointwise VERB ejpam-4528 67 49 on on ADP ejpam-4528 67 50 l∞ l∞ NOUN ejpam-4528 67 51 r r NOUN ejpam-4528 67 52 ( ( PUNCT ejpam-4528 67 53 µ µ NOUN ejpam-4528 67 54 ) ) PUNCT ejpam-4528 67 55 ⊗ ⊗ NOUN ejpam-4528 67 56 e′ e′ PROPN ejpam-4528 67 57 to to ADP ejpam-4528 67 58 f f PROPN ejpam-4528 67 59 . . PUNCT ejpam-4528 68 1 then then ADV ejpam-4528 68 2 f f PROPN ejpam-4528 68 3 is be AUX ejpam-4528 68 4 pettis pettis PROPN ejpam-4528 68 5 integrable integrable ADJ ejpam-4528 68 6 . . PUNCT ejpam-4528 69 1 the the DET ejpam-4528 69 2 next next ADJ ejpam-4528 69 3 lemma lemma PROPN ejpam-4528 69 4 is be AUX ejpam-4528 69 5 useful useful ADJ ejpam-4528 69 6 . . PUNCT ejpam-4528 70 1 lemma lemma PROPN ejpam-4528 70 2 1 1 NUM ejpam-4528 70 3 . . PUNCT ejpam-4528 71 1 if if SCONJ ejpam-4528 71 2 f f PROPN ejpam-4528 71 3 : : PUNCT ejpam-4528 71 4 ω ω PROPN ejpam-4528 71 5 → → SYM ejpam-4528 71 6 e e NOUN ejpam-4528 71 7 is be AUX ejpam-4528 71 8 scalarly scalarly ADV ejpam-4528 71 9 integrable integrable ADJ ejpam-4528 71 10 and and CCONJ ejpam-4528 71 11 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 71 12 is be AUX ejpam-4528 71 13 measurable measurable ADJ ejpam-4528 71 14 , , PUNCT ejpam-4528 71 15 then then ADV ejpam-4528 71 16 the the DET ejpam-4528 71 17 sequence sequence NOUN ejpam-4528 71 18 ( ( PUNCT ejpam-4528 71 19 1{∥f∥≤n}f)n 1{∥f∥≤n}f)n NUM ejpam-4528 71 20 converges converge VERB ejpam-4528 71 21 pointwise pointwise VERB ejpam-4528 71 22 on on ADP ejpam-4528 71 23 l∞ l∞ NOUN ejpam-4528 71 24 r r NOUN ejpam-4528 71 25 ( ( PUNCT ejpam-4528 71 26 µ µ NOUN ejpam-4528 71 27 ) ) PUNCT ejpam-4528 71 28 ⊗ ⊗ NOUN ejpam-4528 71 29 e′ e′ PROPN ejpam-4528 71 30 to to ADP ejpam-4528 71 31 f f PROPN ejpam-4528 71 32 . . PUNCT ejpam-4528 72 1 proof proof NOUN ejpam-4528 72 2 . . PUNCT ejpam-4528 73 1 let let VERB ejpam-4528 73 2 h h PROPN ejpam-4528 73 3 ∈ ∈ PROPN ejpam-4528 73 4 l∞ l∞ NOUN ejpam-4528 73 5 r r NOUN ejpam-4528 73 6 ( ( PUNCT ejpam-4528 73 7 µ µ NOUN ejpam-4528 73 8 ) ) PUNCT ejpam-4528 73 9 and and CCONJ ejpam-4528 73 10 x′ x′ PROPN ejpam-4528 73 11 ∈ ∈ PROPN ejpam-4528 73 12 e′. e′. NOUN ejpam-4528 73 13 we we PRON ejpam-4528 73 14 have have AUX ejpam-4528 73 15 h(ω)⟨1{∥f∥≤n}f(ω h(ω)⟨1{∥f∥≤n}f(ω VERB ejpam-4528 73 16 ) ) PUNCT ejpam-4528 73 17 , , PUNCT ejpam-4528 73 18 x x X ejpam-4528 73 19 ′⟩ ′⟩ PROPN ejpam-4528 73 20 → → SYM ejpam-4528 73 21 h(ω)⟨f(ω h(ω)⟨f(ω PROPN ejpam-4528 73 22 ) ) PUNCT ejpam-4528 73 23 , , PUNCT ejpam-4528 73 24 x′⟩ x′⟩ PROPN ejpam-4528 73 25 ∀ω ∀ω PROPN ejpam-4528 73 26 ∈ ∈ PROPN ejpam-4528 73 27 ω ω PROPN ejpam-4528 73 28 , , PUNCT ejpam-4528 73 29 and and CCONJ ejpam-4528 73 30 |h(ω)⟨1{∥f∥≤n}f(ω |h(ω)⟨1{∥f∥≤n}f(ω NOUN ejpam-4528 73 31 ) ) PUNCT ejpam-4528 73 32 , , PUNCT ejpam-4528 73 33 x x PROPN ejpam-4528 73 34 ′⟩| ′⟩| NOUN ejpam-4528 73 35 ≤ ≤ NUM ejpam-4528 73 36 ∥h∥∞|⟨f(ω ∥h∥∞|⟨f(ω NOUN ejpam-4528 73 37 ) ) PUNCT ejpam-4528 73 38 , , PUNCT ejpam-4528 74 1 x′⟩| x′⟩| PROPN ejpam-4528 74 2 a.e a.e PROPN ejpam-4528 74 3 . . PROPN ejpam-4528 74 4 , , PUNCT ejpam-4528 74 5 then then ADV ejpam-4528 74 6 by by ADP ejpam-4528 74 7 the the DET ejpam-4528 74 8 lebesgue lebesgue NOUN ejpam-4528 74 9 dominated dominate VERB ejpam-4528 74 10 convergence convergence NOUN ejpam-4528 74 11 theorem∫ theorem∫ VERB ejpam-4528 74 12 ω ω PROPN ejpam-4528 74 13 |⟨h(ω)1{∥f∥≤n}f(ω)− |⟨h(ω)1{∥f∥≤n}f(ω)− PROPN ejpam-4528 74 14 f(ω f(ω PROPN ejpam-4528 74 15 ) ) PUNCT ejpam-4528 74 16 , , PUNCT ejpam-4528 74 17 x′⟩|dµ(ω x′⟩|dµ(ω PROPN ejpam-4528 74 18 ) ) PUNCT ejpam-4528 75 1 → → SYM ejpam-4528 75 2 0 0 NUM ejpam-4528 75 3 , , PUNCT ejpam-4528 75 4 and and CCONJ ejpam-4528 75 5 therefore therefore ADV ejpam-4528 75 6 ∫ ∫ PROPN ejpam-4528 75 7 ω ω NUM ejpam-4528 75 8 h(ω)⟨1{∥f∥≤n}f(ω h(ω)⟨1{∥f∥≤n}f(ω PROPN ejpam-4528 75 9 ) ) PUNCT ejpam-4528 75 10 , , PUNCT ejpam-4528 75 11 x x X ejpam-4528 75 12 ′⟩dµ(ω ′⟩dµ(ω X ejpam-4528 75 13 ) ) PUNCT ejpam-4528 75 14 → → SYM ejpam-4528 75 15 ∫ ∫ PROPN ejpam-4528 75 16 ω ω PROPN ejpam-4528 75 17 h(ω)⟨f(ω h(ω)⟨f(ω PROPN ejpam-4528 75 18 ) ) PUNCT ejpam-4528 75 19 , , PUNCT ejpam-4528 75 20 x′⟩dµ(ω x′⟩dµ(ω NUM ejpam-4528 75 21 ) ) PUNCT ejpam-4528 75 22 . . PUNCT ejpam-4528 76 1 n. n. PROPN ejpam-4528 76 2 sabiri sabiri PROPN ejpam-4528 76 3 , , PUNCT ejpam-4528 76 4 m. m. NOUN ejpam-4528 76 5 guessous guessous ADJ ejpam-4528 76 6 / / SYM ejpam-4528 76 7 eur eur PROPN ejpam-4528 76 8 . . PUNCT ejpam-4528 77 1 j. j. PROPN ejpam-4528 77 2 pure pure PROPN ejpam-4528 77 3 appl appl PROPN ejpam-4528 77 4 . . PROPN ejpam-4528 77 5 math math PROPN ejpam-4528 77 6 , , PUNCT ejpam-4528 77 7 15 15 NUM ejpam-4528 77 8 ( ( PUNCT ejpam-4528 77 9 4 4 NUM ejpam-4528 77 10 ) ) PUNCT ejpam-4528 77 11 ( ( PUNCT ejpam-4528 77 12 2022 2022 NUM ejpam-4528 77 13 ) ) PUNCT ejpam-4528 77 14 , , PUNCT ejpam-4528 77 15 1512 1512 NUM ejpam-4528 77 16 - - SYM ejpam-4528 77 17 1520 1520 NUM ejpam-4528 77 18 1515 1515 NUM ejpam-4528 77 19 proposition proposition NOUN ejpam-4528 77 20 1 1 NUM ejpam-4528 77 21 . . PUNCT ejpam-4528 78 1 if if SCONJ ejpam-4528 78 2 f f PROPN ejpam-4528 78 3 : : PUNCT ejpam-4528 78 4 ω ω PROPN ejpam-4528 78 5 → → SYM ejpam-4528 78 6 e e NOUN ejpam-4528 78 7 is be AUX ejpam-4528 78 8 scalarly scalarly ADV ejpam-4528 78 9 integrable integrable ADJ ejpam-4528 78 10 and and CCONJ ejpam-4528 78 11 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 78 12 is be AUX ejpam-4528 78 13 measurable measurable ADJ ejpam-4528 78 14 , , PUNCT ejpam-4528 78 15 then then ADV ejpam-4528 78 16 f f PROPN ejpam-4528 78 17 is be AUX ejpam-4528 78 18 pettis pettis PROPN ejpam-4528 78 19 integrable integrable ADJ ejpam-4528 78 20 if if SCONJ ejpam-4528 78 21 and and CCONJ ejpam-4528 78 22 only only ADV ejpam-4528 78 23 if if SCONJ ejpam-4528 78 24 ( ( PUNCT ejpam-4528 78 25 i i NOUN ejpam-4528 78 26 ) ) PUNCT ejpam-4528 78 27 { { PUNCT ejpam-4528 78 28 f f X ejpam-4528 78 29 } } PUNCT ejpam-4528 78 30 is be AUX ejpam-4528 78 31 sui sui PROPN ejpam-4528 78 32 , , PUNCT ejpam-4528 78 33 and and CCONJ ejpam-4528 78 34 ( ( PUNCT ejpam-4528 78 35 ii ii NOUN ejpam-4528 78 36 ) ) PUNCT ejpam-4528 78 37 1{∥f∥≤n}f 1{∥f∥≤n}f NUM ejpam-4528 78 38 is be AUX ejpam-4528 78 39 pettis pettis NOUN ejpam-4528 78 40 integrable integrable ADJ ejpam-4528 78 41 for for ADP ejpam-4528 78 42 all all DET ejpam-4528 78 43 n n PRON ejpam-4528 78 44 ≥ ≥ NOUN ejpam-4528 78 45 1 1 NUM ejpam-4528 78 46 . . PUNCT ejpam-4528 79 1 proof proof NOUN ejpam-4528 79 2 . . PUNCT ejpam-4528 80 1 if if SCONJ ejpam-4528 80 2 f f PROPN ejpam-4528 80 3 is be AUX ejpam-4528 80 4 pettis pettis PROPN ejpam-4528 80 5 integrable integrable ADJ ejpam-4528 80 6 then then ADV ejpam-4528 80 7 { { PUNCT ejpam-4528 80 8 f f X ejpam-4528 80 9 } } PUNCT ejpam-4528 80 10 is be AUX ejpam-4528 80 11 sui sui PROPN ejpam-4528 80 12 and and CCONJ ejpam-4528 80 13 1{∥f∥≤n}f 1{∥f∥≤n}f NUM ejpam-4528 80 14 is be AUX ejpam-4528 80 15 pettis pettis PROPN ejpam-4528 80 16 integrable integrable ADJ ejpam-4528 80 17 ∀n ∀n NUM ejpam-4528 80 18 ≥ ≥ NOUN ejpam-4528 80 19 1 1 NUM ejpam-4528 80 20 . . PUNCT ejpam-4528 81 1 the the DET ejpam-4528 81 2 converse converse NOUN ejpam-4528 81 3 follows follow VERB ejpam-4528 81 4 from from ADP ejpam-4528 81 5 theorem theorem ADJ ejpam-4528 81 6 1 1 NUM ejpam-4528 81 7 and and CCONJ ejpam-4528 81 8 lemma lemma PROPN ejpam-4528 81 9 1 1 NUM ejpam-4528 81 10 . . PUNCT ejpam-4528 82 1 the the DET ejpam-4528 82 2 above above ADJ ejpam-4528 82 3 result result NOUN ejpam-4528 82 4 gives give VERB ejpam-4528 82 5 a a DET ejpam-4528 82 6 characterization characterization NOUN ejpam-4528 82 7 of of ADP ejpam-4528 82 8 pettis pettis PROPN ejpam-4528 82 9 integrability integrability NOUN ejpam-4528 82 10 for for ADP ejpam-4528 82 11 scalarly scalarly ADV ejpam-4528 82 12 integrable integrable ADJ ejpam-4528 82 13 function function NOUN ejpam-4528 82 14 with with ADP ejpam-4528 82 15 measurable measurable ADJ ejpam-4528 82 16 norm norm NOUN ejpam-4528 82 17 function function NOUN ejpam-4528 82 18 ( ( PUNCT ejpam-4528 82 19 compare compare VERB ejpam-4528 82 20 with with ADP ejpam-4528 82 21 theorem theorem ADJ ejpam-4528 82 22 5.2 5.2 NUM ejpam-4528 82 23 in in ADP ejpam-4528 82 24 [ [ X ejpam-4528 82 25 14 14 NUM ejpam-4528 82 26 ] ] PUNCT ejpam-4528 82 27 ) ) PUNCT ejpam-4528 82 28 and and CCONJ ejpam-4528 82 29 it it PRON ejpam-4528 82 30 can can AUX ejpam-4528 82 31 be be AUX ejpam-4528 82 32 seen see VERB ejpam-4528 82 33 as as ADP ejpam-4528 82 34 a a DET ejpam-4528 82 35 generalization generalization NOUN ejpam-4528 82 36 for for ADP ejpam-4528 82 37 the the DET ejpam-4528 82 38 case case NOUN ejpam-4528 82 39 of of ADP ejpam-4528 82 40 strongly strongly ADV ejpam-4528 82 41 measurable measurable ADJ ejpam-4528 82 42 functions function NOUN ejpam-4528 82 43 since since SCONJ ejpam-4528 82 44 , , PUNCT ejpam-4528 82 45 if if SCONJ ejpam-4528 82 46 f f PROPN ejpam-4528 82 47 : : PUNCT ejpam-4528 82 48 ω ω PROPN ejpam-4528 82 49 → → SYM ejpam-4528 82 50 e e X ejpam-4528 82 51 is be AUX ejpam-4528 82 52 strongly strongly ADV ejpam-4528 82 53 measurable measurable ADJ ejpam-4528 82 54 then then ADV ejpam-4528 82 55 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 82 56 is be AUX ejpam-4528 82 57 measurable measurable ADJ ejpam-4528 82 58 and and CCONJ ejpam-4528 82 59 hence hence ADV ejpam-4528 82 60 1{∥f∥≤n}f 1{∥f∥≤n}f NUM ejpam-4528 82 61 is be AUX ejpam-4528 82 62 bochner bochn ADJ ejpam-4528 82 63 then then ADV ejpam-4528 82 64 pettis pettis PROPN ejpam-4528 82 65 integrable integrable ADJ ejpam-4528 82 66 . . PUNCT ejpam-4528 83 1 we we PRON ejpam-4528 83 2 obtain obtain VERB ejpam-4528 83 3 the the DET ejpam-4528 83 4 following follow VERB ejpam-4528 83 5 characterization characterization NOUN ejpam-4528 83 6 of of ADP ejpam-4528 83 7 pettis pettis PROPN ejpam-4528 83 8 integrability integrability NOUN ejpam-4528 83 9 in in ADP ejpam-4528 83 10 l1 l1 PROPN ejpam-4528 83 11 e′ e′ PUNCT ejpam-4528 84 1 [ [ X ejpam-4528 84 2 e e X ejpam-4528 84 3 ] ] PUNCT ejpam-4528 84 4 . . PUNCT ejpam-4528 85 1 corollary corollary ADJ ejpam-4528 85 2 1 1 NUM ejpam-4528 85 3 . . PUNCT ejpam-4528 86 1 let let VERB ejpam-4528 86 2 e e PRON ejpam-4528 86 3 be be AUX ejpam-4528 86 4 a a DET ejpam-4528 86 5 separable separable ADJ ejpam-4528 86 6 banach banach NOUN ejpam-4528 86 7 space space NOUN ejpam-4528 86 8 and and CCONJ ejpam-4528 86 9 f f PROPN ejpam-4528 86 10 ∈ ∈ PROPN ejpam-4528 86 11 l1 l1 PROPN ejpam-4528 86 12 e′ e′ PUNCT ejpam-4528 87 1 [ [ X ejpam-4528 87 2 e e X ejpam-4528 87 3 ] ] PUNCT ejpam-4528 87 4 . . PUNCT ejpam-4528 88 1 then then ADV ejpam-4528 88 2 f f PROPN ejpam-4528 88 3 is be AUX ejpam-4528 88 4 pettis pettis PROPN ejpam-4528 88 5 integrable integrable PROPN ejpam-4528 88 6 iff iff PROPN ejpam-4528 88 7 1{∥f∥≤n}f 1{∥f∥≤n}f PROPN ejpam-4528 88 8 is be AUX ejpam-4528 88 9 pettis pettis NOUN ejpam-4528 88 10 integrable integrable ADJ ejpam-4528 88 11 for for ADP ejpam-4528 88 12 all all DET ejpam-4528 88 13 n n PRON ejpam-4528 88 14 ≥ ≥ NOUN ejpam-4528 88 15 1 1 NUM ejpam-4528 88 16 . . PUNCT ejpam-4528 89 1 proof proof NOUN ejpam-4528 89 2 . . PUNCT ejpam-4528 90 1 as as SCONJ ejpam-4528 90 2 e e PROPN ejpam-4528 90 3 is be AUX ejpam-4528 90 4 separable separable ADJ ejpam-4528 90 5 then then ADV ejpam-4528 90 6 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 90 7 is be AUX ejpam-4528 90 8 measurable measurable ADJ ejpam-4528 90 9 . . PUNCT ejpam-4528 91 1 the the DET ejpam-4528 91 2 direct direct ADJ ejpam-4528 91 3 implication implication NOUN ejpam-4528 91 4 is be AUX ejpam-4528 91 5 immediate immediate ADJ ejpam-4528 91 6 we we PRON ejpam-4528 91 7 show show VERB ejpam-4528 91 8 the the DET ejpam-4528 91 9 converse converse NOUN ejpam-4528 91 10 . . PUNCT ejpam-4528 92 1 for for ADP ejpam-4528 92 2 every every DET ejpam-4528 92 3 x′′ x′′ PROPN ejpam-4528 92 4 ∈ ∈ PROPN ejpam-4528 92 5 e′′ e′′ VERB ejpam-4528 92 6 the the DET ejpam-4528 92 7 function function NOUN ejpam-4528 92 8 ⟨f ⟨f X ejpam-4528 92 9 ( ( PUNCT ejpam-4528 92 10 . . PUNCT ejpam-4528 92 11 ) ) PUNCT ejpam-4528 92 12 , , PUNCT ejpam-4528 92 13 x′′⟩ x′′⟩ PROPN ejpam-4528 92 14 is be AUX ejpam-4528 92 15 measurable measurable ADJ ejpam-4528 92 16 a a DET ejpam-4528 92 17 simple simple ADJ ejpam-4528 92 18 limit limit NOUN ejpam-4528 92 19 of of ADP ejpam-4528 92 20 ( ( PUNCT ejpam-4528 92 21 ⟨1{∥f∥≤n}f ⟨1{∥f∥≤n}f X ejpam-4528 92 22 ( ( PUNCT ejpam-4528 92 23 . . PUNCT ejpam-4528 92 24 ) ) PUNCT ejpam-4528 92 25 , , PUNCT ejpam-4528 92 26 x x PUNCT ejpam-4528 93 1 ′⟩)n ′⟩)n PROPN ejpam-4528 93 2 . . NOUN ejpam-4528 93 3 for for ADP ejpam-4528 93 4 all all DET ejpam-4528 93 5 ω ω NUM ejpam-4528 93 6 ∈ ∈ PROPN ejpam-4528 93 7 ω ω NOUN ejpam-4528 93 8 and and CCONJ ejpam-4528 93 9 x′′ x′′ PROPN ejpam-4528 93 10 ∈ ∈ PROPN ejpam-4528 93 11 be′′ be′′ PROPN ejpam-4528 93 12 , , PUNCT ejpam-4528 93 13 we we PRON ejpam-4528 93 14 have have VERB ejpam-4528 93 15 |⟨f(ω |⟨f(ω PROPN ejpam-4528 93 16 ) ) PUNCT ejpam-4528 93 17 , , PUNCT ejpam-4528 93 18 x′′⟩| x′′⟩| VERB ejpam-4528 93 19 ≤ ≤ NUM ejpam-4528 93 20 ∥f(ω)∥. ∥f(ω)∥. PRON ejpam-4528 93 21 as as ADP ejpam-4528 93 22 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 93 23 ∈ ∈ PROPN ejpam-4528 93 24 l1 l1 PROPN ejpam-4528 93 25 r(µ r(µ PROPN ejpam-4528 93 26 ) ) PUNCT ejpam-4528 93 27 then then ADV ejpam-4528 93 28 { { PUNCT ejpam-4528 93 29 f f X ejpam-4528 93 30 } } PUNCT ejpam-4528 93 31 is be AUX ejpam-4528 93 32 sui sui PROPN ejpam-4528 93 33 . . PUNCT ejpam-4528 94 1 therefore therefore ADV ejpam-4528 94 2 we we PRON ejpam-4528 94 3 apply apply VERB ejpam-4528 94 4 proposition proposition NOUN ejpam-4528 94 5 1 1 NUM ejpam-4528 94 6 . . PUNCT ejpam-4528 94 7 from from ADP ejpam-4528 94 8 now now ADV ejpam-4528 94 9 , , PUNCT ejpam-4528 94 10 we we PRON ejpam-4528 94 11 suppose suppose VERB ejpam-4528 94 12 that that SCONJ ejpam-4528 94 13 e e PROPN ejpam-4528 94 14 is be AUX ejpam-4528 94 15 separable separable ADJ ejpam-4528 94 16 . . PUNCT ejpam-4528 95 1 if if SCONJ ejpam-4528 95 2 ( ( PUNCT ejpam-4528 95 3 fn fn NOUN ejpam-4528 95 4 ) ) PUNCT ejpam-4528 95 5 is be AUX ejpam-4528 95 6 a a DET ejpam-4528 95 7 convergent convergent ADJ ejpam-4528 95 8 sequence sequence NOUN ejpam-4528 95 9 of of ADP ejpam-4528 95 10 pettis pettis PROPN ejpam-4528 95 11 integrable integrable ADJ ejpam-4528 95 12 functions function NOUN ejpam-4528 95 13 of of ADP ejpam-4528 95 14 l1 l1 PROPN ejpam-4528 95 15 e′ e′ PUNCT ejpam-4528 96 1 [ [ X ejpam-4528 96 2 e e X ejpam-4528 96 3 ] ] X ejpam-4528 96 4 , , PUNCT ejpam-4528 96 5 when when SCONJ ejpam-4528 96 6 does do AUX ejpam-4528 96 7 ( ( PUNCT ejpam-4528 96 8 fn fn NOUN ejpam-4528 96 9 ) ) PUNCT ejpam-4528 96 10 have have VERB ejpam-4528 96 11 a a DET ejpam-4528 96 12 pettis pettis NOUN ejpam-4528 96 13 integrable integrable ADJ ejpam-4528 96 14 limit limit NOUN ejpam-4528 96 15 ? ? PUNCT ejpam-4528 97 1 here here ADV ejpam-4528 97 2 the the DET ejpam-4528 97 3 convergence convergence NOUN ejpam-4528 97 4 is be AUX ejpam-4528 97 5 taken take VERB ejpam-4528 97 6 in in ADP ejpam-4528 97 7 the the DET ejpam-4528 97 8 sense sense NOUN ejpam-4528 97 9 of of ADP ejpam-4528 97 10 weak weak ADJ ejpam-4528 97 11 convergence convergence NOUN ejpam-4528 97 12 a.e a.e PROPN ejpam-4528 97 13 . . PROPN ejpam-4528 97 14 or or CCONJ ejpam-4528 97 15 the the DET ejpam-4528 97 16 pointwise pointwise ADJ ejpam-4528 97 17 convergence convergence NOUN ejpam-4528 97 18 on on ADP ejpam-4528 97 19 l∞ l∞ NOUN ejpam-4528 97 20 r r NOUN ejpam-4528 97 21 ( ( PUNCT ejpam-4528 97 22 µ µ NOUN ejpam-4528 97 23 ) ) PUNCT ejpam-4528 97 24 ⊗ ⊗ PROPN ejpam-4528 97 25 e′′. e′′. PROPN ejpam-4528 98 1 the the DET ejpam-4528 98 2 following follow VERB ejpam-4528 98 3 result result NOUN ejpam-4528 98 4 is be AUX ejpam-4528 98 5 an an DET ejpam-4528 98 6 analogue analogue NOUN ejpam-4528 98 7 of of ADP ejpam-4528 98 8 vitali vitali PROPN ejpam-4528 98 9 ’s ’s PART ejpam-4528 98 10 convergence convergence NOUN ejpam-4528 98 11 theorem theorem NOUN ejpam-4528 98 12 for for ADP ejpam-4528 98 13 pettis pettis PROPN ejpam-4528 98 14 integrable integrable ADJ ejpam-4528 98 15 functions function NOUN ejpam-4528 98 16 . . PUNCT ejpam-4528 99 1 lemma lemma PROPN ejpam-4528 99 2 2 2 X ejpam-4528 99 3 . . PUNCT ejpam-4528 100 1 let let VERB ejpam-4528 100 2 f f PROPN ejpam-4528 100 3 ∈ ∈ PROPN ejpam-4528 100 4 l1 l1 PROPN ejpam-4528 100 5 e′ e′ PUNCT ejpam-4528 101 1 [ [ X ejpam-4528 101 2 e e X ejpam-4528 101 3 ] ] X ejpam-4528 101 4 be be AUX ejpam-4528 101 5 a a DET ejpam-4528 101 6 scalarly scalarly ADV ejpam-4528 101 7 integrable integrable ADJ ejpam-4528 101 8 function function NOUN ejpam-4528 101 9 . . PUNCT ejpam-4528 102 1 suppose suppose VERB ejpam-4528 102 2 that that SCONJ ejpam-4528 102 3 there there PRON ejpam-4528 102 4 exists exist VERB ejpam-4528 102 5 a a DET ejpam-4528 102 6 sequence sequence NOUN ejpam-4528 102 7 of of ADP ejpam-4528 102 8 pettis pettis PROPN ejpam-4528 102 9 integrable integrable ADJ ejpam-4528 102 10 functions function NOUN ejpam-4528 102 11 ( ( PUNCT ejpam-4528 102 12 fn fn NOUN ejpam-4528 102 13 ) ) PUNCT ejpam-4528 102 14 such such ADJ ejpam-4528 102 15 that that SCONJ ejpam-4528 102 16 ( ( PUNCT ejpam-4528 102 17 i i NOUN ejpam-4528 102 18 ) ) PUNCT ejpam-4528 102 19 ( ( PUNCT ejpam-4528 102 20 fn fn NOUN ejpam-4528 102 21 ) ) PUNCT ejpam-4528 102 22 is be AUX ejpam-4528 102 23 wsui wsui VERB ejpam-4528 102 24 , , PUNCT ejpam-4528 102 25 and and CCONJ ejpam-4528 102 26 ( ( PUNCT ejpam-4528 102 27 ii ii NOUN ejpam-4528 102 28 ) ) PUNCT ejpam-4528 102 29 for for ADP ejpam-4528 102 30 each each DET ejpam-4528 102 31 x′′ x′′ PROPN ejpam-4528 102 32 ∈ ∈ PROPN ejpam-4528 102 33 e′′ e′′ PROPN ejpam-4528 102 34 , , PUNCT ejpam-4528 102 35 limn→∞⟨fn limn→∞⟨fn PROPN ejpam-4528 102 36 , , PUNCT ejpam-4528 102 37 x′′⟩ x′′⟩ PUNCT ejpam-4528 102 38 = = PUNCT ejpam-4528 103 1 ⟨f ⟨f X ejpam-4528 103 2 , , PUNCT ejpam-4528 103 3 x′′⟩ x′′⟩ AUX ejpam-4528 104 1 a.e a.e PROPN ejpam-4528 104 2 . . PROPN ejpam-4528 105 1 then then ADV ejpam-4528 105 2 f f PROPN ejpam-4528 105 3 is be AUX ejpam-4528 105 4 pettis pettis PROPN ejpam-4528 105 5 integrable integrable ADJ ejpam-4528 106 1 and and CCONJ ejpam-4528 106 2 ( ( PUNCT ejpam-4528 106 3 fn fn NOUN ejpam-4528 106 4 ) ) PUNCT ejpam-4528 106 5 converges converge VERB ejpam-4528 106 6 weakly weakly ADJ ejpam-4528 106 7 to to ADP ejpam-4528 106 8 f f PROPN ejpam-4528 106 9 in in ADP ejpam-4528 106 10 p p PROPN ejpam-4528 106 11 1 1 NUM ejpam-4528 106 12 e′(µ e′(µ PROPN ejpam-4528 106 13 ) ) PUNCT ejpam-4528 106 14 . . PUNCT ejpam-4528 107 1 proof proof NOUN ejpam-4528 107 2 . . PUNCT ejpam-4528 108 1 as as SCONJ ejpam-4528 108 2 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 108 3 is be AUX ejpam-4528 108 4 integrable integrable ADJ ejpam-4528 108 5 then then ADV ejpam-4528 108 6 { { PUNCT ejpam-4528 108 7 f f X ejpam-4528 108 8 } } PUNCT ejpam-4528 108 9 is be AUX ejpam-4528 108 10 sui sui PROPN ejpam-4528 108 11 . . PUNCT ejpam-4528 109 1 we we PRON ejpam-4528 109 2 apply apply VERB ejpam-4528 109 3 theorem theorem ADJ ejpam-4528 109 4 1 1 NUM ejpam-4528 109 5 and and CCONJ ejpam-4528 109 6 vitali vitali PROPN ejpam-4528 109 7 ’s ’s PART ejpam-4528 109 8 theorem theorem NOUN ejpam-4528 109 9 in in ADP ejpam-4528 109 10 l1 l1 PROPN ejpam-4528 109 11 r(µ r(µ PROPN ejpam-4528 109 12 ) ) PUNCT ejpam-4528 109 13 . . PUNCT ejpam-4528 110 1 theorem theorem NOUN ejpam-4528 110 2 2 2 NUM ejpam-4528 110 3 . . PUNCT ejpam-4528 111 1 let let VERB ejpam-4528 111 2 ( ( PUNCT ejpam-4528 111 3 fn)n∈n fn)n∈n NUM ejpam-4528 111 4 a a DET ejpam-4528 111 5 bounded bounded ADJ ejpam-4528 111 6 sequence sequence NOUN ejpam-4528 111 7 in in ADP ejpam-4528 111 8 l1 l1 PROPN ejpam-4528 111 9 e′ e′ PUNCT ejpam-4528 112 1 [ [ X ejpam-4528 112 2 e e X ejpam-4528 112 3 ] ] X ejpam-4528 112 4 . . PUNCT ejpam-4528 113 1 if if SCONJ ejpam-4528 113 2 ( ( PUNCT ejpam-4528 113 3 fn fn NOUN ejpam-4528 113 4 ) ) PUNCT ejpam-4528 113 5 w*-converges w*-converge NOUN ejpam-4528 113 6 a.e a.e PROPN ejpam-4528 113 7 . . PROPN ejpam-4528 113 8 to to ADP ejpam-4528 113 9 a a DET ejpam-4528 113 10 function function NOUN ejpam-4528 113 11 f f NOUN ejpam-4528 113 12 : : PUNCT ejpam-4528 113 13 ω ω PROPN ejpam-4528 113 14 → → SYM ejpam-4528 113 15 e′ e′ PROPN ejpam-4528 113 16 then then ADV ejpam-4528 113 17 f f PROPN ejpam-4528 113 18 ∈ ∈ PROPN ejpam-4528 113 19 l1 l1 PROPN ejpam-4528 113 20 e′ e′ PUNCT ejpam-4528 114 1 [ [ X ejpam-4528 114 2 e e X ejpam-4528 114 3 ] ] X ejpam-4528 114 4 . . PUNCT ejpam-4528 115 1 if if SCONJ ejpam-4528 115 2 fn fn PROPN ejpam-4528 115 3 is be AUX ejpam-4528 115 4 pettis pettis NOUN ejpam-4528 115 5 integrable integrable ADJ ejpam-4528 115 6 for for SCONJ ejpam-4528 115 7 all all DET ejpam-4528 115 8 n n PRON ejpam-4528 115 9 and and CCONJ ejpam-4528 115 10 ( ( PUNCT ejpam-4528 115 11 fn fn NOUN ejpam-4528 115 12 ) ) PUNCT ejpam-4528 115 13 w w NOUN ejpam-4528 115 14 - - PUNCT ejpam-4528 115 15 converges converge VERB ejpam-4528 115 16 a.e a.e PROPN ejpam-4528 115 17 . . PROPN ejpam-4528 115 18 to to ADP ejpam-4528 115 19 f f PROPN ejpam-4528 115 20 , , PUNCT ejpam-4528 115 21 then then ADV ejpam-4528 115 22 f f PROPN ejpam-4528 115 23 is be AUX ejpam-4528 115 24 pettis pettis PROPN ejpam-4528 115 25 integrable integrable ADJ ejpam-4528 115 26 . . PUNCT ejpam-4528 116 1 n. n. PROPN ejpam-4528 116 2 sabiri sabiri PROPN ejpam-4528 116 3 , , PUNCT ejpam-4528 116 4 m. m. NOUN ejpam-4528 116 5 guessous guessous ADJ ejpam-4528 116 6 / / SYM ejpam-4528 116 7 eur eur PROPN ejpam-4528 116 8 . . PUNCT ejpam-4528 117 1 j. j. PROPN ejpam-4528 117 2 pure pure PROPN ejpam-4528 117 3 appl appl PROPN ejpam-4528 117 4 . . PROPN ejpam-4528 117 5 math math PROPN ejpam-4528 117 6 , , PUNCT ejpam-4528 117 7 15 15 NUM ejpam-4528 117 8 ( ( PUNCT ejpam-4528 117 9 4 4 NUM ejpam-4528 117 10 ) ) PUNCT ejpam-4528 117 11 ( ( PUNCT ejpam-4528 117 12 2022 2022 NUM ejpam-4528 117 13 ) ) PUNCT ejpam-4528 117 14 , , PUNCT ejpam-4528 117 15 1512 1512 NUM ejpam-4528 117 16 - - SYM ejpam-4528 117 17 1520 1520 NUM ejpam-4528 117 18 1516 1516 NUM ejpam-4528 117 19 proof proof NOUN ejpam-4528 117 20 . . PUNCT ejpam-4528 118 1 as as ADP ejpam-4528 118 2 ( ( PUNCT ejpam-4528 118 3 fn(ω))n fn(ω))n ADP ejpam-4528 118 4 w*-converges w*-converge NOUN ejpam-4528 118 5 a.e a.e PROPN ejpam-4528 118 6 . . PROPN ejpam-4528 118 7 to to ADP ejpam-4528 118 8 f(ω f(ω PROPN ejpam-4528 118 9 ) ) PUNCT ejpam-4528 118 10 we we PRON ejpam-4528 118 11 have have VERB ejpam-4528 118 12 ∥f(ω)∥ ∥f(ω)∥ PROPN ejpam-4528 118 13 ≤ ≤ NUM ejpam-4528 118 14 lim lim PROPN ejpam-4528 118 15 inf inf PROPN ejpam-4528 118 16 n n CCONJ ejpam-4528 118 17 ∥fn(ω)∥ ∥fn(ω)∥ ADJ ejpam-4528 118 18 a.e a.e PROPN ejpam-4528 118 19 . . PROPN ejpam-4528 118 20 by by ADP ejpam-4528 118 21 fatou fatou PROPN ejpam-4528 118 22 ’s ’s PART ejpam-4528 118 23 lemma lemma PROPN ejpam-4528 118 24 and and CCONJ ejpam-4528 118 25 the the DET ejpam-4528 118 26 boundedness boundedness NOUN ejpam-4528 118 27 of of ADP ejpam-4528 118 28 ( ( PUNCT ejpam-4528 118 29 fn fn NOUN ejpam-4528 118 30 ) ) PUNCT ejpam-4528 118 31 in in ADP ejpam-4528 118 32 l1 l1 PROPN ejpam-4528 118 33 e′ e′ PUNCT ejpam-4528 119 1 [ [ X ejpam-4528 119 2 e e X ejpam-4528 119 3 ] ] X ejpam-4528 119 4 we we PRON ejpam-4528 119 5 get∫ get∫ PROPN ejpam-4528 119 6 ω ω PUNCT ejpam-4528 119 7 ∥f∥dµ ∥f∥dµ PUNCT ejpam-4528 119 8 ≤ ≤ NUM ejpam-4528 119 9 lim lim PROPN ejpam-4528 119 10 inf inf PROPN ejpam-4528 119 11 n n PROPN ejpam-4528 119 12 ∫ ∫ PROPN ejpam-4528 119 13 ω ω X ejpam-4528 119 14 ∥fn∥dµ ∥fn∥dµ X ejpam-4528 119 15 < < X ejpam-4528 119 16 ∞ ∞ PROPN ejpam-4528 119 17 , , PUNCT ejpam-4528 119 18 thus thus ADV ejpam-4528 119 19 f f PROPN ejpam-4528 119 20 ∈ ∈ PROPN ejpam-4528 119 21 l1 l1 PROPN ejpam-4528 119 22 e′ e′ PUNCT ejpam-4528 120 1 [ [ X ejpam-4528 120 2 e e X ejpam-4528 120 3 ] ] PUNCT ejpam-4528 120 4 . . PUNCT ejpam-4528 121 1 now now ADV ejpam-4528 121 2 suppose suppose VERB ejpam-4528 121 3 that that SCONJ ejpam-4528 121 4 fn fn PROPN ejpam-4528 121 5 is be AUX ejpam-4528 121 6 pettis pettis NOUN ejpam-4528 121 7 integrable integrable ADJ ejpam-4528 121 8 for for SCONJ ejpam-4528 121 9 all all DET ejpam-4528 121 10 n n PRON ejpam-4528 121 11 and and CCONJ ejpam-4528 121 12 ( ( PUNCT ejpam-4528 121 13 fn fn NOUN ejpam-4528 121 14 ) ) PUNCT ejpam-4528 121 15 w w NOUN ejpam-4528 121 16 - - PUNCT ejpam-4528 121 17 converges converge VERB ejpam-4528 121 18 a.e a.e PROPN ejpam-4528 121 19 . . PROPN ejpam-4528 121 20 to to ADP ejpam-4528 121 21 f f PROPN ejpam-4528 121 22 . . PUNCT ejpam-4528 122 1 then then ADV ejpam-4528 122 2 f f PROPN ejpam-4528 122 3 is be AUX ejpam-4528 122 4 w w NOUN ejpam-4528 122 5 - - NOUN ejpam-4528 122 6 measurable measurable ADJ ejpam-4528 122 7 with with ADP ejpam-4528 122 8 ∥f(.)∥ ∥f(.)∥ PROPN ejpam-4528 122 9 is be AUX ejpam-4528 122 10 integrable integrable ADJ ejpam-4528 122 11 , , PUNCT ejpam-4528 122 12 so so SCONJ ejpam-4528 122 13 that that SCONJ ejpam-4528 122 14 f f PROPN ejpam-4528 122 15 is be AUX ejpam-4528 122 16 scalarly scalarly ADV ejpam-4528 122 17 integrable integrable ADJ ejpam-4528 122 18 . . PUNCT ejpam-4528 123 1 by by ADP ejpam-4528 123 2 lemma lemma PROPN ejpam-4528 123 3 2 2 NUM ejpam-4528 123 4 in in ADP ejpam-4528 123 5 [ [ X ejpam-4528 123 6 16 16 NUM ejpam-4528 123 7 ] ] PUNCT ejpam-4528 123 8 there there PRON ejpam-4528 123 9 exists exist VERB ejpam-4528 123 10 a a DET ejpam-4528 123 11 subsequence subsequence NOUN ejpam-4528 123 12 ( ( PUNCT ejpam-4528 123 13 gn gn NOUN ejpam-4528 123 14 ) ) PUNCT ejpam-4528 123 15 of of ADP ejpam-4528 123 16 ( ( PUNCT ejpam-4528 123 17 fn fn NOUN ejpam-4528 123 18 ) ) PUNCT ejpam-4528 123 19 such such ADJ ejpam-4528 123 20 that that SCONJ ejpam-4528 123 21 ( ( PUNCT ejpam-4528 123 22 1{∥gn∥