the Kolmogorov extension theorem

Suppose you want to model an infinite sequence of coin tosses, or a process that runs forever in continuous time. You can easily say what should happen on any finite stretch — the probabilities for the first n tosses, for any n. But does a single, consistent probability measure actually exist on the infinite space of all entire sequences at once? It is not obvious that one does. The Kolmogorov extension theorem is the guarantee that it does: it builds a measure on the infinite product from a consistent family of finite-dimensional descriptions.

The input is a family of finite-dimensional distributions: for every finite set of time-indices you specify the joint distribution of the process at those times. These must satisfy two natural consistency conditions: the description should not depend on the order in which you list the indices, and the distribution for a small set of indices must be the marginal you get by ignoring (integrating out) the extra coordinates of a larger set. Given that consistency, the theorem produces a unique probability measure on the full infinite-dimensional product space whose finite-dimensional marginals are exactly the ones you prescribed. In effect it stitches countless overlapping finite snapshots into one coherent infinite picture, with no contradictions at the seams.

This is the existence theorem for stochastic processes. It is what lets you legitimately speak of an infinite sequence of independent fair coin tosses, of a Poisson process on the whole half-line, or of Brownian motion's finite-dimensional structure, as living on a genuine probability space. The one technical caveat is that it needs the value spaces to be reasonably nice (standard Borel spaces, such as the real numbers), which covers essentially every model in practice; and it pins down only the finite-dimensional distributions, so finer path properties like continuity require additional work beyond the bare theorem.