Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 2, Pages 137-146
doi:10.1155/S1048953300000150

Fatou's Lemma and Lebesgue's convergence theorem for measures

Onésimo Hernández-Lerma1 and Jean B. Lasserre2

1CINVESTA V-IPN, Departamento de Matemáticas, Apdo. Postal 14-740, México D.F. 07000, Mexico
2LAAS-CNRS, 7, Avenue Du Colonel Roche, Toulouse Cédex 31077, France

Received 1 December 1998; Revised 1 November 1999

Copyright © 2000 Onésimo Hernández-Lerma and Jean B. Lasserre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Analogues of Fatou's Lemma and Lebesgue's convergence theorems are established for ∫fdμn when {μn} is a sequence of measures. A “generalized” Dominated Convergence Theorem is also proved for the asymptotic behavior of ∫fndμn and the latter is shown to be a special case of a more general result established in vector lattices and related to the Dunford-Pettis property in Banach spaces.