Pettis integrability

Author:
Gunnar F. Stefánsson

Journal:
Trans. Amer. Math. Soc. **330** (1992), 401-418

MSC:
Primary 46G10; Secondary 28B05, 46B20

DOI:
https://doi.org/10.1090/S0002-9947-1992-1070352-0

MathSciNet review:
1070352

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Abstract: A weakly measurable function is said to be determined by a subspace of if for each , implies that a.e. For a given Dunford integrable function with a countably additive indefinite integral we show that is Pettis integrable if and only if is determined by a weakly compactly generated subspace of if and only if is determined by a subspace which has Mazur's property.

We show that if is Pettis integrable then there exists a sequence ( ) of valued simple functions such that for all , a.e. if and only if is determined by a separable subspace of .

For a bounded weakly measurable function into a dual of a weakly compactly generated space, we show that is Pettis integrable if and only if is determined by a separable subspace of if and only if is weakly equivalent to a Pettis integrable function that takes its range in .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1070352-0

Keywords:
Banach space,
Pettis integral,
weak measurability,
integral

Article copyright:
© Copyright 1992
American Mathematical Society