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.