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Fatou's identity and Lebesgue's convergence theorem |
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| Authors: | Heinz-Albrecht Klei |
| Affiliation: | Département de Mathématiques et Informatique, Université de Reims, Moulin de la Housse, B.P. 1039, 51687 Reims Cedex 2, France |
| 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. |
| Keywords: | Fatou's lemma Fatou's identity Lebesgue's theorem uniform integrability measure convergent sequence norm convergent sequence. |
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