Geodesic
Fatou's lemma in its classical form and Lebesgue's convergence theorems for varying measures with applications to Markov decision processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 2, pp. 338-367

Voir la notice de l'article provenant de la source Math-Net.Ru

The classical Fatou lemma states that the lower limit of a sequence of integrals of functions is greater than or equal to 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 the integral of the lower limit in two parameters, where the second parameter is the argument of the functions. In the present paper, we provide sufficient conditions when Fatou's lemma holds in its classical form for a sequence of weakly converging measures. The functions can take both positive and negative values. Similar results for sequences of setwise converging measures are also proved. We also put forward analogies of Lebesgue's and the monotone convergence theorems for sequences of weakly and setwise converging measures. The results obtained are used to prove broad sufficient conditions for the validity of optimality equations for average-cost Markov decision processes.
Mots-clés : Fatou's lemma, setwise convergence
Keywords: measure, weak convergence, Markov decision process. 
@article{TVP_2020_65_2_a4,
     author = {E. A. Feinberg and P. O. Kas'yanov and Y. Liang},
     title = {Fatou's lemma in its classical form and {Lebesgue's} convergence theorems for varying measures with applications to {Markov} decision processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {338--367},
     publisher = {mathdoc},
     volume = {65},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2020_65_2_a4/}
}
TY  - JOUR
AU  - E. A. Feinberg
AU  - P. O. Kas'yanov
AU  - Y. Liang
TI  - Fatou's lemma in its classical form and Lebesgue's convergence theorems for varying measures with applications to Markov decision processes
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2020
SP  - 338
EP  - 367
VL  - 65
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TVP_2020_65_2_a4/
LA  - ru
ID  - TVP_2020_65_2_a4
ER  - 
%0 Journal Article
%A E. A. Feinberg
%A P. O. Kas'yanov
%A Y. Liang
%T Fatou's lemma in its classical form and Lebesgue's convergence theorems for varying measures with applications to Markov decision processes
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2020
%P 338-367
%V 65
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TVP_2020_65_2_a4/
%G ru
%F TVP_2020_65_2_a4
E. A. Feinberg; P. O. Kas'yanov; Y. Liang. Fatou's lemma in its classical form and Lebesgue's convergence theorems for varying measures with applications to Markov decision processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 2, pp. 338-367. http://geodesic.mathdoc.fr/item/TVP_2020_65_2_a4/