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Proceedings of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9939-1972-0294595-1
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$.


References [Enhancements On Off] (What's this?)

  • [1] P. Duren, Theory of $ {H^p}$ spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [2] E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 32 #5826. MR 0188387 (32:5826)
  • [3] J. E. Littlewood, Lectures on the theory of functions, Oxford Univ. Press, Oxford, 1944. MR 6, 261. MR 0012121 (6:261f)

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Additional Information

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

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