 Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is.The theorem is used in one of the standard proofs of existence of a Brownian motion, by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above.As in most of the definitions of Brownian motion it is required that the sample paths are continuous almost surely, and one then uses the Kolmogorov continuity theorem to construct a continuous modification of the process constructed by the Kolmogorov extension theorem.