Fatou’s Lemma for Weakly Converging Probabilities∗

S f(s)μ(ds) = 1, ∫ S f(s)μn(ds) = 0, and (1.3) does not hold. Theorem 1.1 presents Fatou’s lemma for weakly converging measures μn and nonnegative functions fn. This fact is useful for the analysis of Markov decision processes and stochastic games. Serfozo [7, Lemma 3.2] establishes inequality (1.4) for a vaguely convergent sequence of measures on a locally compact metric space S and for nonnegative functions fn. In its current form, Theorem 1.1 is formulated in [6, Lemma 2.3(ii)] without proof, in [4, Lemma 3.2] with short explanations on how the proof from [7, Lemma 3.2] can be adapted to weak convergence on metric spaces, and in [3, Lemma 4] with a proof. To make this paper logically complete, we provide the proof of Theorem 1.1 in section 3. The provided proof is shorter and simpler than the proof in [3]. Theorem 4.2 below extends Theorem 1.1 to functions fn