Proceedings of the American Mathematical Society

Author(s): Heinz-Albrecht Klei
Journal: Proc. Amer. Math. Soc. 127 (1999), 2297-2302.
MSC (1991): Primary 26D15, 28A20, 28A25
Posted: April 9, 1999
Retrieve article in: PDF DVI PostScript
This article is available free of charge

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.

1.: Z. Artstein, A note on Fatou's lemma in several dimensions, J. Math. Econom. 6 (1979), 277-282. MR 82b:28008
2.: R. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. MR 32:2543
3.: E. J. Balder, A unifying note on Fatou's lemma in several dimensions, Math. Oper. Res. 9 (1984), 267-275. MR 85f:28003
4.: E. J. Balder and C. Hess, Fatou's lemma for multifunctions with unbounded values, Math. Oper. Res. 20 (1995), 175-188. MR 96a:28020
5.: L. Cesari and M. B. Suryanarayana, An existence theorem for Pareto problems, Nonlinear Anal. 2 (1978), 225-233. MR 81g:49008
6.: W. Hildenbrand and J. F. Mertens, On Fatou's lemma in several dimensions, Z. Wahrsch. Verw. Gebiete. 17 (1971), 151-155. MR 43:3414
7.: H.-A. Klei, Convergences faible, en mesure et au sens de Cesaro dans $L^1(\mathbf R)$ , C. R. Acad. Sci. Paris, Sér. I, 315 (1992), 9-12. MR 93e:28005
8.: -, Measure convergent sequences in Lebesgue spaces and Fatou's lemma, Bull. Austral. Math. Soc. 54 (1996), 197-202. MR 97k:28008
9.: -, Convergence and extraction of bounded sequences in $L^1(\mathbf R)$ , J. Math. Anal. Appl. 198 (1996), 636-642. MR 96m:46047
10.: H.-A. Klei and M. Miyara, Une extension du lemme de Fatou, Bull. Sci. Math. (2) 115 (1991), 211-222. MR 92b:28004
11.: J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217-229. MR 35:1071
12.: C. Olech, On $n$ -dimensional extensions of Fatou's lemma, Z. Angew. Math. Phys. 38 (1987), 266-271. MR 88g:28009
13.: F. H. Page, Komlós limits and Fatou's lemma in several dimensions, Canad. Math. Bull. 34 (1991), 109-112. MR 92m:28007
14.: H. Richter, Verallgemeinerung eines in der Statistik benötigten Satzes der Masstheorie, Math. Ann. 150, (1963), 85-90 and 440-441. MR 26:3851; MR 27:2025
15.: H. P. Rosenthal, Sous-espaces de , lectures, University Paris VI, 1979.
16.: D. Schmeidler, Fatou's lemma in several dimensions, Proc. Amer. Math. Soc. 24, (1970), 300-306. MR 40:1568

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 26D15, 28A20, 28A25

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: 10.1090/S0002-9939-99-05099-6
PII: S 0002-9939(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
Posted: April 9, 1999
Communicated by: Frederick W. Gehring
Copyright of article: Copyright 1999, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS