Mean convergence and compact subsets of 𝐿₁

Halpern, Benjamin

doi:10.1090/S0002-9939-1971-0276432-3

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Mean convergence and compact subsets of $L_{1}$
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by Benjamin Halpern PDF

Proc. Amer. Math. Soc. 28 (1971), 122-126 Request permission

Abstract:

It is shown that in the usual criterion for $w$ compactness of a set $K$ in ${L_1}(\mu )$ the explicit assumption that $K$ is bounded follows from the other assumptions which are usually made if $\mu$ is nonatomic.

References

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2