Kolmogorov extension theorem: Difference between revisions

Revision as of 10:50, 9 October 2015

Revision as of 00:43, 20 December 2015

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

For subsets <math>I \subset J \subset T</math>, let <math>\pi_{I \leftarrow J}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_{F \leftarrow G}pi^G_F)_* \mu_G</math>

where <math>(\pi_{F \leftarrow G}pi^G_F)_* \mu_G</math> denotes the [[pushforward measure]] of <math> \mu_G</math> induced by the canonical projection map <math>\pi_{F \leftarrow G}pi^G_F</math>.

Then there exists a unique probability measure <math>\mu</math> on <math>\Omega_T </math> such that <math>\mu_F=(\pi_{F \leftarrow T}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]] 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.