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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniform $ \sigma $-additivity in spaces of Bochner or Pettis integrable functions over a locally compact group


Author: Nicolae Dinculeanu
Journal: Proc. Amer. Math. Soc. 87 (1983), 627-633
MSC: Primary 28B05; Secondary 43A20, 46G10
MathSciNet review: 687630
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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\vert f \right\vert 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_{E'}^\infty )$ of convolution and translation operators. In case $ E = R$, this gives a new characterization of relatively weakly compact sets $ K \subset {L^1}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0687630-7
PII: S 0002-9939(1983)0687630-7
Keywords: Locally compact group, Harr measure, Bochner integral, Pettis integral, convolution, translation, approximate unit, uniform $ \sigma $-additivity, uniform convergence
Article copyright: © Copyright 1983 American Mathematical Society