The $\mu$-PIP and integrability of a single function
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- by Gunnar F. Stefánsson PDF
- Proc. Amer. Math. Soc. 124 (1996), 539-542 Request permission
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
- Elizabeth M. Bator, Pettis integrability and the equality of the norms of the weak$^\ast$ integral and the Dunford integral, Proc. Amer. Math. Soc. 95 (1985), no. 2, 265–270. MR 801336, DOI 10.1090/S0002-9939-1985-0801336-4
- Kazimierz Musiał and Grzegorz Plebanek, Pettis integrability and the equality of the norms of the weak$^*$ integral and the Dunford integral, Hiroshima Math. J. 19 (1989), no. 2, 329–332. MR 1027936
- Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
- G. F. Stefánsson, Pettis integrability, Ph.D. thesis, Pennsylvania State University, 1989.
- Michel Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224. MR 756174, DOI 10.1090/memo/0307
Additional Information
- Gunnar F. Stefánsson
- Affiliation: Department of Mathematics, Pennsylvania State University, Altoona Campus, Altoona, Pennsylvania 16601
- Email: gfs@math.psu.edu
- Received by editor(s): July 14, 1993
- Received by editor(s) in revised form: September 7, 1994
- Communicated by: Dale Alspach
- © Copyright 1996 American Mathematical Society
- 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