Kolmogorov extension theorem: Difference between revisions

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The theorem is used in one of the standard proofs of existence of a [[Brownian motion]], by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above. As in most of the definitions of [[Brownian motion]] it is required that the sample paths are continuous almost surely, one then uses [[kolmogorov continuity theorem]] to construct a continuous modification of the process constructed by Kolmogorov extension theorem.

==A more generalGeneral form of the theorem==

The Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some <math>\mathbb{R}^{n}</math>-valued stochastic process, but the assumption that the state space be <math>\mathbb{R}^{n}</math> is unnecessary. In fact, any collection of measurable spaces together with a collection of [[inner regular measure]]s defined on the finite products of these spaces would suffice, provided that these measures satisfy a certain compatibility relation. The formal statement of the general theorem is as follows.<ref>T. Tao, ''An Introduction to Measure Theory'', [[Graduate Studies in Mathematics]], Vol. 126, 2011, p. 195</ref>