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Algebras of random variables and their representations. (English) Zbl 0693.60003
In different parts of probability theory, Kolmogorov’s representation theorem gives a basic tool showing us the connection of a compatible family of finite dimensional probability distributions with a probability space corresponding to it. In this paper the problem of describing the probability spaces and measures wich represent a given algebra of random variables (ARV) is presented. It is shown that among all representations of an ARV there is a fundamental one, the homomorphism of the ARV into the algebra of continuous functions on the space of real characters of the ARV. It is shown that the space of any representation may be transformed by canonical mapping into a part of the space of characters of the ARV.
Reviewer: I.G.Kalmár
60A05 Axioms; other general questions in probability
60G07 General theory of stochastic processes