Geodesic
An $\varepsilon$-optimal control of finite Markov chain with average
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 71-82

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Discrete time Markov decition chain with average reward criterion is considered. It is proved that if the state space is finite and the sets of actions are measurable subsets of Polish space, then there exist non-randomized Markov $\varepsilon$-optimal policies. An example showing that there exists a Markov decition chain with countable state space and finite sets of actions such that randomized Markov $\varepsilon$-optimal policies for this chain don't exist is constructed. 
@article{TVP_1980_25_1_a5,
     author = {E. A. Feǐnberg},
     title = {An $\varepsilon$-optimal control of finite {Markov} chain with average},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {71--82},
     publisher = {mathdoc},
     volume = {25},
     number = {1},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_1_a5/}
}
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E. A. Feǐnberg. An $\varepsilon$-optimal control of finite Markov chain with average. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 71-82. http://geodesic.mathdoc.fr/item/TVP_1980_25_1_a5/