Real Analysis Exchange

Another Proof That Lp-Bounded Pointwise Convergence Implies Weak Convergence

Marian Jakszto

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This note gives another proof of the known fact that \(L^{p}\)-bounded pointwise convergence implies weak convergence in \(L^{p},\) \(p>1.\) The proof is based on Banach and Saks’ theorem. The same method applies to convergence in measure.

Article information

Real Anal. Exchange, Volume 36, Number 2 (2010), 479-482.

First available in Project Euclid: 11 November 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

\(L^p\)-spaces Banach-Saks property weak convergence pointwise convergence convergence in measure


Jakszto, Marian. Another Proof That L p -Bounded Pointwise Convergence Implies Weak Convergence. Real Anal. Exchange 36 (2010), no. 2, 479--482.

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