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→General form of the theorem: Necessary that subset J be finite. |
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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>{{cite book |first=T. |last=Tao |authorlink=Terence Tao |title=An Introduction to Measure Theory |series=[[Graduate Studies in Mathematics]] |volume=126 |___location=Providence |publisher=American Mathematical Society |year=2011 |isbn=978-0-8218-6919-2 |page=195 |url=https://books.google.com/books?id=HoGDAwAAQBAJ&pg=PA195 }}</ref>
Let <math>T</math> be any set. Let <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 space|Hausdorff topology]] on <math> \Omega_t</math>. For each finite subset <math>J \subset T</math>, define
:<math>\Omega_J := \prod_{t\in J} \Omega_t</math>.
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