Geodesic
Convergence of probability measures and Markov decision models with incomplete information
Informatics and Automation, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 103-124

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This paper deals with three major types of convergence of probability measures on metric spaces: weak convergence, setwise convergence, and convergence in total variation. First, it describes and compares necessary and sufficient conditions for these types of convergence, some of which are well-known, in terms of convergence of probabilities of open and closed sets and, for the probabilities on the real line, in terms of convergence of distribution functions. Second, it provides criteria for weak and setwise convergence of probability measures and continuity of stochastic kernels in terms of convergence of probabilities defined on the base of the topology generated by the metric. Third, it provides applications to control of partially observable Markov decision processes and, in particular, to Markov decision models with incomplete information. 
@article{TRSPY_2014_287_a5,
     author = {Eugene A. Feinberg and Pavlo O. Kasyanov and Michael Z. Zgurovsky},
     title = {Convergence of probability measures and {Markov} decision models with incomplete information},
     journal = {Informatics and Automation},
     pages = {103--124},
     publisher = {mathdoc},
     volume = {287},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2014_287_a5/}
}
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Eugene A. Feinberg; Pavlo O. Kasyanov; Michael Z. Zgurovsky. Convergence of probability measures and Markov decision models with incomplete information. Informatics and Automation, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 103-124. http://geodesic.mathdoc.fr/item/TRSPY_2014_287_a5/