Mean convergence in $L^{p}$ spaces
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- by W. P. Novinger
- Proc. Amer. Math. Soc. 34 (1972), 627-628
- DOI: https://doi.org/10.1090/S0002-9939-1972-0294595-1
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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 ${\left \| {{f_n}} \right \|_p} \to {\left \| f \right \|_p} < \infty$, then ${\left \| {{f_n} - f} \right \|_p} \to 0$.References
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387
- J. E. Littlewood, Lectures on the Theory of Functions, Oxford University Press, 1944. MR 0012121
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- 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