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Fatou's identity
and Lebesgue's convergence theorem


Author: Heinz-Albrecht Klei
Journal: Proc. Amer. Math. Soc. 127 (1999), 2297-2302
MSC (1991): Primary 26D15, 28A20, 28A25
DOI: https://doi.org/10.1090/S0002-9939-99-05099-6
Published electronically: April 9, 1999
MathSciNet review: 1636974
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Abstract | References | Similar Articles | Additional Information

Abstract: The classical Fatou lemma for bounded sequences of nonnegative integrable functions is represented as an equality. A similar result is stated for measure convergent sequences. Neither result requires a uniform integrability assumption. For the latter a converse is proven. Two extensions of Lebesgue's convergence theorem are presented.


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Additional Information

Heinz-Albrecht Klei
Affiliation: Département de Mathématiques et Informatique, Université de Reims, Moulin de la Housse, B.P. 1039, 51687 Reims Cedex 2, France
Email: heinz.klei@univ-reims.fr

DOI: https://doi.org/10.1090/S0002-9939-99-05099-6
Keywords: Fatou's lemma, Fatou's identity, Lebesgue's theorem, uniform integrability, measure convergent sequence, norm convergent sequence.
Received by editor(s): October 27, 1997
Published electronically: April 9, 1999
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society

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