Let {X,} teT be a family of real (R) random variables defined on a probability space (Q,s#,P) and having the ranges in a subset S of R, that is, X t (Q) c S for all t. Let X be the mapping of ft into the function space S T such that for any coeCl We shall write X = {X t } teT and call X the random function arising from {X t } tsT . It is well-known that any finite subfamily of {X t } leT induces a "finite joint distribution" in S T , and according to Kolmogorov (1933) these finite joint

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... tions can be simultaneously extended to a probability measure P o on the Borel class 38 0 of subsets of S T . This extension is natural in the sense that for any Be38 0 P 0 (B) turns out to be exactly P[X~1(B)]. The Kolmogorov extension P o has however a shortcoming in that its domain 38 0 is not broad enough to include many events of practical interest. Following Kakutani (1943 ), Nelson (1959 has formulated a regular Borel measure P lt which extends the Kolmogorov extension P g to the topological Borel class 38 1 containing 38 0 provided that S is a compact subset of R. The Kakutani extension P t has the following regularity property: for any where F and G respectively denote closed and open subsets of S T . The definitions of ^x and 38 O are given in section 2. The purpose of this paper is to show by a simple well-known example of Doob (1953) that the Kakutani extension is not natural in that given Be3S t 371 use, available at https://www.cambridge.org/core/terms. https://doi.