Geodesic
A generalization of McMillan's theorem on the case of countable alphabet
Trudy Instituta matematiki, Tome 23 (2015) no. 1, pp. 115-122.

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In the paper we introduce a fine topology on the set of probability measures. It is proved that the entropy of a measure on a countable set is continuous with respect to the fine topology. Then with the help of this continuity we extend McMillan's theorem to the case of a countable alphabet. 
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E. E. Sokol. A generalization of McMillan's theorem on the case of countable alphabet. Trudy Instituta matematiki, Tome 23 (2015) no. 1, pp. 115-122. http://geodesic.mathdoc.fr/item/TIMB_2015_23_1_a8/

[1] Shiryaev A. N., Veroyatnost, v. 1, M., 2004

[2] McMillan B., “The Basic Theorems of Information Theory”, Ann. Math. Stat., 24 (1953), 196–219 | DOI | MR | Zbl