Uniform $\sigma$-additivity in spaces of Bochner or Pettis integrable functions over a locally compact group
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- by Nicolae Dinculeanu PDF
- Proc. Amer. Math. Soc. 87 (1983), 627-633 Request permission
Abstract:
If $G$ is an abelian locally compact group with Haar measure $\mu$, $E$ is a Banach space and $K \subset L_E^1(\mu )$, we give necessary and sufficient conditions for the set $\left \{ {{f_{( \cdot )}}\left | f \right |d\mu ;f \in K} \right \}$ to be uniformly $\sigma$-additive in terms of uniform convergence on $K$, for the topology $\sigma (L_E^1,L_{Eā}^\infty )$ of convolution and translation operators. In case $E = R$, this gives a new characterization of relatively weakly compact sets $K \subset {L^1}$.References
- N. Dinculeanu, Conditional expectations for general measure spaces, J. Multivariate Anal. 1 (1971), 347ā364. MR 301770, DOI 10.1016/0047-259X(71)90014-5 ā, Integration on locally compact spaces, Noordhoff, Leyden, 1974.
- Nicolae Dinculeanu, Uniform $\sigma$-additivity and uniform convergence of conditional expectations in the space of Bochner or Pettis integrable functions, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980) Academic Press, New York-London, 1981, pp.Ā 391ā397. MR 619065
- Nicolae Dinculeanu, On Kolmogorov-Tamarkin and M. Riesz compactness criteria in function spaces over a locally compact group, J. Math. Anal. Appl. 89 (1982), no.Ā 1, 67ā85. MR 672189, DOI 10.1016/0022-247X(82)90091-9
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 627-633
- MSC: Primary 28B05; Secondary 43A20, 46G10
- DOI: https://doi.org/10.1090/S0002-9939-1983-0687630-7
- MathSciNet review: 687630