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


Author: Gunnar F. Stefánsson
Journal: Proc. Amer. Math. Soc. 124 (1996), 539-542
MSC (1991): Primary 46G10, 28B05
DOI: https://doi.org/10.1090/S0002-9939-96-03105-X
MathSciNet review: 1301529
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Abstract | References | Similar Articles | Additional Information

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

  • 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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Additional Information

Gunnar F. Stefánsson
Affiliation: Department of Mathematics, Pennsylvania State University, Altoona Campus, Altoona, Pennsylvania 16601
Email: gfs@math.psu.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03105-X
Keywords: $\mathrm{weak}^*$ integral, Pettis integral
Received by editor(s): July 14, 1993
Received by editor(s) in revised form: September 7, 1994
Communicated by: Dale Alspach
Article copyright: © Copyright 1996 American Mathematical Society

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