  This note describes Fatou's lemma and Lebesgue's dominated convergence
theorem for a sequence of measures converging weakly to a finite measure and
for a sequence of functions whose negative parts are uniformly integrable with
respect to these measures. The note also provides new formulations of uniform
Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis
theorem, and the fundamental theorem for Young measures based on the
equivalence of uniform integrability and the apparently weaker property of
asymptotic uniform integrability for sequences of functions and finite
measures.
