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Proceedings of the American Mathematical Society

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



Mean convergence and compact subsets of $L_{1}$

Author: Benjamin Halpern
Journal: Proc. Amer. Math. Soc. 28 (1971), 122-126
MSC: Primary 28.20
MathSciNet review: 0276432
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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 [Enhancements On Off] (What's this?)

  • 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 Company, Inc., New York, N. Y., 1950. MR 0033869

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Keywords: Mean convergenc, <IMG WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${L_1}$">, equicontinuity, absolute continuity
Article copyright: © Copyright 1971 American Mathematical Society