Kolmogorov extension theorem: Difference between revisions

Revision as of 18:41, 9 July 2016

Revision as of 21:42, 3 September 2016

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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 subset <math>J \subset T</math>, define

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

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For subsets <math>I \subset J \subset T</math>, let <math>\pi^J_I: \Omega_J \to \Omega_I</math> denote the canonical projection map <math> \omega \mapsto \omega|_I </math>.

For each finite subset <math> F \subset T</math>, suppose we have a probability measure <math> \mu_F </math> on <math> \Omega_F </math> which is [[inner regular]] with respect to the [[product topology]] (induced by the <math>\tau_t</math>) on <math>\Omega_F </math>. Suppose also that this collection <math>\{\mu_F\}</math> of measures satisfies the following compatibility relation: for finite subsets <math>F \subset G \subset T</math>, we have that

:<math>\mu_F = (\pi^G_F)_* \mu_G</math>

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Then there exists a unique probability measure <math>\mu</math> on <math>\Omega_T </math> such that <math>\mu_F=(\pi^T_F)_* \mu</math> for every finite subset <math>F \subset T</math>.

As a remark, all of the measures <math>\mu_F,\mu</math> are defined on the product [[sigma algebra|product sigma algebra]] on their respective spaces, which (as mentioned before) is rather coarse. The measure <math>\mu</math> may sometimes be extended appropriately to a larger sigma algebra, if there is additional structure involved.

Note that the original statement of the theorem is just a special case of this theorem with <math>\Omega_t = \mathbb{R}^n </math> for all <math>t \in T</math>, and <math> \mu_{\{t_1,...,t_k\}}=\nu_{t_1 \dots t_k}</math> for <math> t_1,...,t_k \in T</math>. The stochastic process would simply be the canonical process <math> (\pi_t)_{t \in T}</math>, defined on <math>\Omega=(\mathbb{R}^n)^T</math> with probability measure <math>P=\mu</math>. The reason that the original statement of the theorem does not mention inner regularity of the measures <math>\nu_{t_1\dots t_k}</math> is that this would automatically follow, since Borel probability measures on [[Polish space]]s are automatically [[Radon measure|Radon]].