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

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.