Geodesic
The existence of a~stationary $\varepsilon$-optimal policy for a~finite Markov chain
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 313-330

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The existence of a stationary average reward $\varepsilon$-optimal policy is proved for discrete time Markov decision chains with finitely many states, compact sets of actions, continuous transition functions and upper semicontinuous reward functions. 
@article{TVP_1978_23_2_a5,
     author = {E. A. Faǐnberg},
     title = {The existence of a~stationary $\varepsilon$-optimal policy for a~finite {Markov} chain},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {313--330},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a5/}
}
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E. A. Faǐnberg. The existence of a~stationary $\varepsilon$-optimal policy for a~finite Markov chain. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 313-330. http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a5/