# On Pettis integrals with separable range

Colloquium Mathematicae (1993)

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

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topPlebanek, 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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- [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
- [P1] G. Plebanek, On the space of continuous functions on a dyadic set, Mathematika 38 (1991), 42-49. Zbl0776.46019
- [SW] F. D. Sentilles and R. F. Wheeler, Pettis integration via the Stonian transform, Pacific J. Math. 107 (1983), 473-496. Zbl0488.46038
- [T] M. Talagrand, Sur les mesures vectorielles définies par une application Pettis- intégrable, Bull. Soc. Math. France 108 (1980), 475-483. Zbl0459.46029
- [Ta] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 307 (1984). Zbl0582.46049
- [Wh] R. F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 1 (1983), 97-190. Zbl0522.28009

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