Fatou’s identity and Lebesgue’s convergence theorem

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.

Locations

• Proceedings of the American Mathematical Society - View - PDF