Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

The $\mu$ -PIP and integrability of a single function

Author(s): Gunnar F. Stefánsson
Journal: Proc. Amer. Math. Soc. 124 (1996), 539-542.
MSC (1991): Primary 46G10, 28B05
MathSciNet review: 1301529
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Abstract: Two examples are given that answer in the negative the following question asked by E. M. Bator: If $f:\Omega\to X^*$ is bounded and weakly measurable and for each $x^{**}$ in $X^{**}$ there is a bounded sequence $(x_n)$ in $X$ such that $x^{**}f=\lim_nfx_n$ a.e., does it follow that $f$ is Pettis integrable?

References:

1: E. M. Bator, Pettis integrability and equality of the norms of the weak* integrable and the Dunford integral, Proc. Amer. Math. Soc. 95 (1985), 265--270. MR 87a:46074
2: K. Musial and G. Plebanek, Pettis integrability and equality of the norms of the weak* integral and the Dunford integral, Hiroshima Math. J. 19 (1989), 329--332. MR 91c:46064
3: R. Phillips, Integrability in a convex linear topological space, Trans. Amer. Math. Soc. 47 (1940), 114--145. MR 2:103c
4: G. F. Stefánsson, Pettis integrability, Ph.D. thesis, Pennsylvania State University, 1989.
5: M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc., no. 307, 1984. MR 86j:46042

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