A Kolmogorov Consistency Theorem in the Multiple Probabilities Setting

TL;DR

This paper establishes a Kolmogorov consistency theorem for systems of finite-dimensional distributions that are weak* closed, enabling the construction of a compatible system of random variables under a set of measures.

Contribution

It extends the classical Kolmogorov theorem to multiple probability measures with weak* closed finite-dimensional distributions, broadening its applicability.

Findings

Defines conditions for constructing random variables from multiple measures.

Proves the existence of a probability space compatible with the given distribution sets.

Shows the measure is determined up to a set of measures.

Abstract

We consider a system of weak* closed sets of finite-dimensional distributions. We show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures, provided that the sets of finite-dimensional distributions are consistent.