Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1971-0276432-3
MathSciNet review: 0276432
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28.20

Retrieve articles in all journals with MSC: 28.20


Additional Information

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