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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Fatou's lemma in several dimensions


Author: David Schmeidler
Journal: Proc. Amer. Math. Soc. 24 (1970), 300-306
MSC: Primary 28.25
MathSciNet review: 0248316
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Abstract: In this note the following generalization of Fatou's lemma is proved:

Lemma. Let $ ({f_n})_{n - 1}^\infty $ be a sequence of integrable functions on a measure space $ S$ with values in $ R_ + ^d$, the nonnegative orthant of a $ d$-dimensional Euclidean space, for which $ \int {{f_n} \to a \in R_ + ^d} $. Then there exists an integrable function $ f$, from $ S$ to $ R_ + ^d$, such that a.e. $ f(s)$ is a limit point of $ ({f_n}(s))_{n - 1}^\infty $ and $ \int {f \leqq a} $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0248316-7
PII: S 0002-9939(1970)0248316-7
Keywords: Fatou lemma, vector valued integrals
Article copyright: © Copyright 1970 American Mathematical Society