On Pettis integrals with separable range

Grzegorz Plebanek

Colloquium Mathematicae (1993)

  • Volume: 64, Issue: 1, page 71-78
  • ISSN: 0010-1354

Abstract

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Several techniques have been developed to study Pettis integrability of weakly measurable functions with values in Banach spaces. As shown by M. Talagrand [Ta], it is fruitful to regard a weakly measurable mapping as a pointwise compact set of measurable functions - its Pettis integrability is then a purely measure-theoretic question of an appropriate continuity of a measure. On the other hand, properties of weakly measurable functions can be translated into the language of topological measure theory by means of weak Baire measures on Banach spaces. This approach, originated by G. A. Edgar [E1, E2], was remarkably developed by M. Talagrand. Following this idea, we show that the Pettis Integral Property of a Banach space E, together with the requirement of separability of E-valued Pettis integrals, is equivalent to the fact that every weak Baire measure on E is, in a certain weak sense, concentrated on a separable subspace. We base on a lemma which is a version of Talagrand's Lemma 5-1-2 from [Ta]. Our lemma easily yields a sequential completeness of the spaces of Grothendieck measures, a related result proved by Pallarés-Vera [PV]. We also present two results on Pettis integrability in the spaces of continuous functions.

How to cite

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Plebanek, Grzegorz. "On Pettis integrals with separable range." Colloquium Mathematicae 64.1 (1993): 71-78. <http://eudml.org/doc/210175>.

@article{Plebanek1993,
abstract = {Several techniques have been developed to study Pettis integrability of weakly measurable functions with values in Banach spaces. As shown by M. Talagrand [Ta], it is fruitful to regard a weakly measurable mapping as a pointwise compact set of measurable functions - its Pettis integrability is then a purely measure-theoretic question of an appropriate continuity of a measure. On the other hand, properties of weakly measurable functions can be translated into the language of topological measure theory by means of weak Baire measures on Banach spaces. This approach, originated by G. A. Edgar [E1, E2], was remarkably developed by M. Talagrand. Following this idea, we show that the Pettis Integral Property of a Banach space E, together with the requirement of separability of E-valued Pettis integrals, is equivalent to the fact that every weak Baire measure on E is, in a certain weak sense, concentrated on a separable subspace. We base on a lemma which is a version of Talagrand's Lemma 5-1-2 from [Ta]. Our lemma easily yields a sequential completeness of the spaces of Grothendieck measures, a related result proved by Pallarés-Vera [PV]. We also present two results on Pettis integrability in the spaces of continuous functions.},
author = {Plebanek, Grzegorz},
journal = {Colloquium Mathematicae},
keywords = {Pettis integrals; separable range; scalarly bounded function; weak Baire measure; Grothendieck measures; completely regular Hausdorff spaces},
language = {eng},
number = {1},
pages = {71-78},
title = {On Pettis integrals with separable range},
url = {http://eudml.org/doc/210175},
volume = {64},
year = {1993},
}

TY - JOUR
AU - Plebanek, Grzegorz
TI - On Pettis integrals with separable range
JO - Colloquium Mathematicae
PY - 1993
VL - 64
IS - 1
SP - 71
EP - 78
AB - Several techniques have been developed to study Pettis integrability of weakly measurable functions with values in Banach spaces. As shown by M. Talagrand [Ta], it is fruitful to regard a weakly measurable mapping as a pointwise compact set of measurable functions - its Pettis integrability is then a purely measure-theoretic question of an appropriate continuity of a measure. On the other hand, properties of weakly measurable functions can be translated into the language of topological measure theory by means of weak Baire measures on Banach spaces. This approach, originated by G. A. Edgar [E1, E2], was remarkably developed by M. Talagrand. Following this idea, we show that the Pettis Integral Property of a Banach space E, together with the requirement of separability of E-valued Pettis integrals, is equivalent to the fact that every weak Baire measure on E is, in a certain weak sense, concentrated on a separable subspace. We base on a lemma which is a version of Talagrand's Lemma 5-1-2 from [Ta]. Our lemma easily yields a sequential completeness of the spaces of Grothendieck measures, a related result proved by Pallarés-Vera [PV]. We also present two results on Pettis integrability in the spaces of continuous functions.
LA - eng
KW - Pettis integrals; separable range; scalarly bounded function; weak Baire measure; Grothendieck measures; completely regular Hausdorff spaces
UR - http://eudml.org/doc/210175
ER -

References

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  1. [AC] M. Ya. Antonovskiĭ and D. Chudnovsky, Some questions of general topology and Tikhonov semifields. II, Russian Math. Surveys 31 (3) (1976), 69-128. 
  2. [E1] G. A. Edgar, Measurability in a Banach space, Indiana Univ. Math. J. 26 (1977), 663-677. Zbl0361.46017
  3. [E2] G. A. Edgar, Measurability in a Banach space, II, ibid. 28 (1979), 559-579. Zbl0418.46034
  4. [En] R. Engelking, General Topology, PWN, Warszawa 1977. 
  5. [FT] D. H. Fremlin and M. Talagrand, A decomposition theorem for additive set functions, with application to Pettis integrals and ergodic means, Math. Z. 168 (1979), 117-142. Zbl0393.28005
  6. [Ma] S. Mazur, On continuous mappings on Cartesian products, Fund. Math. 39 (1952), 229-238. Zbl0050.16802
  7. [M1] K. Musiał, Martingales of Pettis integrable functions, in: Measure Theory, Oberwolfach 1979, Lecture Notes in Math. 794, Springer, 1980, 324-339. 
  8. [M2] K. Musiał, Pettis integration, in: Proc. 13th Winter School on Abstract Analysis, Suppl. Rend. Circ. Mat. Palermo 10 (1985), 133-142. Zbl0649.46040
  9. [PV] A. J. Pallarés and G. Vera, Pettis integrability of weakly continuous functions and Baire measures, J. London Math. Soc. 32 (1985), 479-487. Zbl0591.28004
  10. [P1] G. Plebanek, On the space of continuous functions on a dyadic set, Mathematika 38 (1991), 42-49. Zbl0776.46019
  11. [SW] F. D. Sentilles and R. F. Wheeler, Pettis integration via the Stonian transform, Pacific J. Math. 107 (1983), 473-496. Zbl0488.46038
  12. [T] M. Talagrand, Sur les mesures vectorielles définies par une application Pettis- intégrable, Bull. Soc. Math. France 108 (1980), 475-483. Zbl0459.46029
  13. [Ta] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 307 (1984). Zbl0582.46049
  14. [Wh] R. F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 1 (1983), 97-190. Zbl0522.28009

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