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

 

Mean convergence in $ L\sp{p}$ spaces


Author: W. P. Novinger
Journal: Proc. Amer. Math. Soc. 34 (1972), 627-628
MSC: Primary 28A20; Secondary 46E30
MathSciNet review: 0294595
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Abstract: Let $ (X,\mathcal{B},\mu )$ be a measure space and $ \{ {f_n}\} $ be a sequence in $ {L^p}(X,\mathcal{B},\mu ),0 < p < \infty $. The author presents a very short proof of the familiar fact that if $ {f_n} \to f\mu {\text{-a}}.{\text{e}}.$ and $ {\left\Vert {{f_n}} \right\Vert _p} \to {\left\Vert f \right\Vert _p} < \infty $, then $ {\left\Vert {{f_n} - f} \right\Vert _p} \to 0$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0294595-1
PII: S 0002-9939(1972)0294595-1
Keywords: Convergence in mean, types of convergence in $ {L^p}$ spaces
Article copyright: © Copyright 1972 American Mathematical Society