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The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions f : X → Y exists, then it can be used to specify the probability distribution of finite-dimensional random variables [f(x1),...,f(xn)]. Now, from this n-dimensional probability distribution we can deduce an (n-1)-dimensional marginal probability distribution for [f(x1),...,f(xn-1)]. There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation.

The Kolmogorov extension theorem guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman-Kolmogorov compatibility condition.

Separability, or what the Kolmogorov extension does not provide

Recall that, in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer.

The Kolmogorov extension starts by declaring to be méasurable all sets of functions where finitely many coordinates [f(x1),...,f(xn)] are restricted to lie in méasurable subsets of Yn. In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.

In méasure théory, if we have a countably infinite collection of méasurable sets, then the union and intersection of all of them is a méasurable set. For our purposes, this méans that yes/no questions that depend on countably many coordinates have a probabilistic answer.

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the réally bad news is that virtually all concepts of calculus are of this sort. For example: