From: "Daniel H. Luecking" Subject: Re: extension of probability measure Date: Tue, 11 Jan 2000 15:21:29 -0600 Newsgroups: sci.math.research Summary: [missing] On 11 Jan 2000, Robert Israel wrote: > In article <85au7d$sn3$1@usenet01.srv.cis.pitt.edu>, > Alexander R Pruss wrote: > >Let > > X=\bigtimes_{n=1}^\infty R > >be an infinite product of real lines R. Let F be all borel sets in X. > >Let F_n be all borel sets of the form > > B \times \bigtimes_{j=n+1}^\infty R > >where B is a borel set in R^n. > > >For any positive integer n, I have a borel probability measure \mu_n on > >F_n, such that \mu_n agrees with \mu_{n-1} on F_{n-1}. Can I extend > >these measures to a probability measure \mu_\infty on all of F, which > >agrees with \mu_n on F_n, for all n? > > Yes. This is a theorem of Kolmogorov. See e.g. B. Simon, Functional > Integration and Quantum Physics, Academic Press 1979, Theorem 2.1. Or see any book on probability (at the appropriate level). Look for the "Kolmogorov Extension Theorem". For example: R. B. Ash, Real Analysis and Probability, Academic Press, 1972. Kolmogorov himself (as translated by N. Morrison) in his book (Foundations of Probability, Second English Edition, Chelsea, 1956) divided this into two theorems: the Fundamental Theorem, which produces a measure on the field (not sigma-field) generated by the F_n, and the Extension Theorem, which extends that measure to the generated sigma-field. The Ash book (among others) has a stronger version allowing uncountable products, but requires something of the factor spaces (they are separable metric spaces and Borel probability measures). For countable products, essentially arbitrary probability spaces are permitted. -- Dan Luecking Dept. of Mathematical Sciences luecking@comp.uark.edu University of Arkansas http://comp.uark.edu/~luecking/ Fayetteville, AR 72101