= Kolmogorov extension theorem
{c}
Consistent <probability distributions> on finite collections of coordinates in <standard Borel spaces> determine a unique <probability measure> on the <product sigma-algebra>. Consistency means that discarding coordinates by <marginalization> agrees with the law already assigned to the smaller collection, and relabeling coordinates relabels the law. The theorem turns finite recipes into a complete <stochastic process>. In particular, products of one fixed <probability distribution> construct an <independent and identically distributed> sequence; putting one common <random variable> in every coordinate constructs the perfectly shared process above. The theorem supplies a law on the coordinate space, but does not by itself ensure <continuous> <sample paths>.
Back to article page
