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Proceedings of the American Mathematical Society

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Mean convergence and compact subsets of $ L\sb{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?)

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

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Keywords: Mean convergenc, $ {L_1}$, equicontinuity, absolute continuity
Article copyright: © Copyright 1971 American Mathematical Society