Kolmogorov extension theorem: Difference between revisions

Revision as of 04:57, 21 June 2015

Revision as of 04:58, 21 June 2015

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The Kolmogorov extention 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.<ref>T. Tao, ''An Introduction to Measure Theory'', Graduate Studies in Mathematics, Vol. 126, 2011, p. 195 </ref>

Let <math>T</math> be any set,. and letLet <math> \{ (\Omega_t, \mathcal{F}_t) \}_{t \in T} </math> be some collection of measurable spaces, and for each <math> t \in T </math>, let <math> \tau_t</math> be a Hausdorff topology on <math> \Omega_t</math>. For each subset <math>J \subset T</math>, let

: <math>\Omega_J := \prod_{t\in J} \Omega_t</math>.