Geodesic
On almost sure convergence in linear spaces of random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 2, pp. 351-357

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In this paper, almost sure convergence in a linear space $L$ of random variables with values in a normed space $\mathfrak X$ is considered. In case when $\mathfrak X=R$, it is shown that the necessary and sufficient condition for convergence in measure and a.s. convergence to coincide is nuclearity of $L$. 
@article{TVP_1970_15_2_a16,
     author = {D. Kh. Mushtari},
     title = {On almost sure convergence in linear spaces of random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {351--357},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1970_15_2_a16/}
}
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D. Kh. Mushtari. On almost sure convergence in linear spaces of random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 2, pp. 351-357. http://geodesic.mathdoc.fr/item/TVP_1970_15_2_a16/