Content deleted Content added
Line 42:
For subsets <math>I \subset J \subset T</math>, let <math>\pi_{I \leftarrow J}: \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 on <math>\Omega_F </math> (induced by the <math>\tau_t</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})_* \mu_G</math>
Line 49:
Then there exists a unique probability measure <math>\mu</math> on <math>\Omega_T </math> such that <math>\mu_F=(\pi_{F \leftarrow T})_* \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. These measures may sometimes be extended appropriately to larger sigma algebras, 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>. This theorem has many far-reaching consequences; for example it can be used to prove the existence of the following, among others:
| |||