Open Access
December, 1958 On a Problem in Measure-Spaces
V. S. Varadarajan
Ann. Math. Statist. 29(4): 1275-1278 (December, 1958). DOI: 10.1214/aoms/1177706461


Let $\mathcal{F}$ be the family of all random variables on a probability space $\Omega$ taking values from a separable and complete metric space $X$. In this paper we prove that $\mathcal{F}$ is in a certain sense a closed family. More precisely, if $\{\xi_n\}$ is a sequence of $X$-valued random variables such that their probability distributions converge weakly to a probability distribution $P$ on $X$, then there exists an $X$-valued random variable on $\Omega$ with distribution $P$. An example is also given which shows that the assumption of completeness of $X$ cannot in general be dropped.


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V. S. Varadarajan. "On a Problem in Measure-Spaces." Ann. Math. Statist. 29 (4) 1275 - 1278, December, 1958.


Published: December, 1958
First available in Project Euclid: 27 April 2007

zbMATH: 0093.14401
MathSciNet: MR100291
Digital Object Identifier: 10.1214/aoms/1177706461

Rights: Copyright © 1958 Institute of Mathematical Statistics

Vol.29 • No. 4 • December, 1958
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