Real Analysis Exchange

Another Proof That Lp-Bounded Pointwise Convergence Implies Weak Convergence

Marian Jakszto

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Abstract

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

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

Dates
First available in Project Euclid: 11 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.rae/1321020515

Mathematical Reviews number (MathSciNet)
MR3016731

Zentralblatt MATH identifier
1245.46023

Subjects
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

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

Citation

Jakszto, Marian. Another Proof That L p -Bounded Pointwise Convergence Implies Weak Convergence. Real Anal. Exchange 36 (2010), no. 2, 479--482. https://projecteuclid.org/euclid.rae/1321020515


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References

  • S. Banach and S. Saks, Sur la convergence forte dans le espaces ${L}^{p}$, Studia Math., 2 (1930), 51–57.
  • E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, 1975.
  • E. H. Lieb and M. Loss, Analysis, American Mathematical Society, 2001.
  • F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, 1990.