## Fatou's Lemma in Its Classic Form and Lebesgue's Convergence Theorems for Varying Measures with Applications to MDPs. (arXiv:1902.01525v1 [math.PR])

The classic Fatou lemma states that the lower limit of a sequence of
integrals of functions is greater or equal than the integral of the lower
limit. It is known that Fatou's lemma for a sequence of weakly converging
measures states a weaker inequality because the integral of the lower limit is
replaced with integral of the lower limit in two parameters, where the second
parameter is the argument of the functions. This paper provides sufficient
conditions when Fatou's lemma holds in its classic form for a sequence of
weakly converging measures. The functions can take both positive and negative
values. The paper also provides similar results for sequences of setwise
converging measures. It also provides Lebesgue's and monotone convergence
theorem for sequences of weakly and setwise converging measures. The obtained
results are used to prove broad sufficient conditions for the validity of
optimality equations for average-costs Markov decision processes.查看全文