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Transactions of the American Mathematical Society

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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 $ f:\Omega \to X$ is said to be determined by a subspace $ D$ of $ X$ if for each $ {x^{\ast} } \in {X^{\ast} }$, $ {x^{\ast}}{\vert _D} = 0$ implies that $ {x^{\ast}}\;f= 0$ a.e. For a given Dunford integrable function $ f:\Omega \to X$ with a countably additive indefinite integral we show that $ f$ is Pettis integrable if and only if $ f$ is determined by a weakly compactly generated subspace of $ X$ if and only if $ f$ is determined by a subspace which has Mazur's property.

We show that if $ f:\Omega \to X$ is Pettis integrable then there exists a sequence ( $ {\varphi _n}$) of $ X$ valued simple functions such that for all $ {x^{\ast}} \in {X^{\ast}}$, $ {x^{\ast}}f= {\lim _n}{x^{\ast}}\,{\varphi _n}$ a.e. if and only if $ f$ is determined by a separable subspace of $ X$.

For a bounded weakly measurable function $ f:\Omega \to {X^{\ast} }$ into a dual of a weakly compactly generated space, we show that $ f$ is Pettis integrable if and only if $ f$ is determined by a separable subspace of $ {X^{\ast}}$ if and only if $ f$ is weakly equivalent to a Pettis integrable function that takes its range in $ {\text{cor}}_f^{\ast} (\Omega)$.


References [Enhancements On Off] (What's this?)

  • [1] D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. 80 (1968), 34-46. MR 0228983 (37:4562)
  • [2] K. T. Andrews, Universal Pettis integrability, Canad. J. Math. 37 (1985), 141-159. MR 777045 (86j:46040)
  • [3] W. J. Davis, T. Figiel, W. B. Johnson, and A. Pelczynski, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327. MR 0355536 (50:8010)
  • [4] J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York, 1984. MR 737004 (85i:46020)
  • [5] J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R.I., 1977. MR 0453964 (56:12216)
  • [6] D. H. Fremlin and M. Talagrand, A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means, Math. Z. 168 (1979), 117-142. MR 544700 (80k:28004)
  • [7] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
  • [8] R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1981), 81-86. MR 603606 (82c:28018)
  • [9] R. E. Huff, Some remarks on the Pettis integral, Proc. Amer. Math. Soc. 96 (1986), 402-404. MR 822428 (87g:46074)
  • [10] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. No. 307, 1984, reprint 1986. MR 756174 (86j:46042)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1070352-0
Keywords: Banach space, Pettis integral, weak measurability, $ {\text{weak}}^{\ast} $ integral
Article copyright: © Copyright 1992 American Mathematical Society

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