Kolmogorov extension theorem: Difference between revisions

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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(induced by the <math>\Omega_F tau_t</math>) (induced by theon <math>\tau_tOmega_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})_* \mu_G</math>