Geodesic
Limit Approach under the Signs of Information and Entropy
Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 29-37

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The main result of this paper amounts to the following statement: If a sequence of pairs of random variables $(\xi_n,\eta_n)$ is given and this sequence converges in variation to a pair of random variables $(\xi,\eta)$, then $\lim _{n\to\infty}I(\xi_n,\eta_n)=I(\xi,\eta)(I(\xi,\eta)$ is the information of the pair $(\xi,\eta)$ if and only if the sequence of corresponding information densities is uniformly integrable. A similar result is proved for entropies and for a new concept in information within a probability $E$ of events. Conditions are found for the convergence of these quantities. 
@article{TVP_1960_5_1_a2,
     author = {R. L. Dobrushin},
     title = {Limit {Approach} under the {Signs} of {Information} and {Entropy}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {29--37},
     publisher = {mathdoc},
     volume = {5},
     number = {1},
     year = {1960},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1960_5_1_a2/}
}
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R. L. Dobrushin. Limit Approach under the Signs of Information and Entropy. Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 29-37. http://geodesic.mathdoc.fr/item/TVP_1960_5_1_a2/