Kolmogorov extension theorem

Remarks

The measure $\mu$ is first defined on cylinder sets, where consistency makes it well defined and finitely additive. The key step is countable additivity on the algebra of cylinder sets. It uses inner regularity: each $\mu_J$ is approximated from inside by compact sets, and a decreasing sequence of nonempty compact cylinders has nonempty intersection. Carathéodory's extension theorem then extends $\mu$ to the product σ-algebra, and uniqueness holds because cylinder sets form a π-system.

Example: let $T = [0, \infty)$, $E_t = \R$, and let $\mu_J$ be the centred Gaussian measure on $\R^J$ with covariance $\min(s, t)$ for $s, t \in J$. These measures are consistent, so there is a stochastic process $(X_t)_{t \ge 0}$ with these finite-dimensional laws. This is the first step in the construction of Brownian motion; the continuity of paths needs a further argument.

A topological assumption on the factors is needed. For arbitrary measurable spaces in place of Polish spaces, consistent families exist that have no extension. No such assumption is needed when the $\mu_J$ are products, as for the infinite product measure. It is not needed either when they are built from kernels, as in the Ionescu-Tulcea theorem.

Going deeper

For a student

The theorem is the standard existence result for stochastic processes with prescribed finite-dimensional distributions: Gaussian processes, Markov processes from transition functions, and sequences of independent variables. The resulting σ-algebra only contains events that depend on countably many coordinates. So for uncountable $T$, sets such as "the path is continuous" are not measurable, and one has to pass to a modification, for instance with the Kolmogorov continuity theorem.