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Une extension du lemme de Fatou. (An extension of Fatou’s lemma). (French) Zbl 0734.28004

The paper establishes the following result. Let \(f_ k\) be a sequence of positive functions, \(L^ 1\)-bounded with respect to the measure P. Suppose \(\lim \int f_ k dP\) exists. Then \(\lim \int f_ k dP\geq \int \liminf f_ k dP+\eta,\) where the number \(\eta\) is the modulus of uniform integrability of the sequence \(f_ k,namely\eta =\lim \eta (\epsilon)\quad as\quad \epsilon \to 0,\) and \(\eta\) (\(\epsilon\)) is the supremum over \(\int_{A}| f_ k| dP\) with \(P(A)\leq \epsilon.\) The result generalizes Fatou’s lemma, which establishes the inequality without the term \(\eta\). It is also proved that equality holds if and only if \(f_ k\) has a subsequence which converges in measure to \(\liminf f_ k.\) The main tool is Rosenthal’s subsequence splitting lemma.

MSC:

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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