Abstract. This paper initiates the study of probability measures corresponding to stochastic processes based on the Dinculeanu-Foias notion of algebraic models for probability measures. The main result is a general extension theorem of Kolmogorov type which can be summarized as follows: Let {(X, si¡, ¡i,), i e 1} be a directed family of probability measure spaces. Then there is an associated directed family of probability measure spaces {(G, &t, vt), i e 1} and a probability measure v on the »-algebra M generated by the 3)¡ such that (i) v(B) = v¡(B), Be &,, ie I, and (ii) for each iel the spaces (X, s/t, /it) and (G, < v¡) are conjugate. The importance of the main theorem is that under certain mild conditions there exists an embedding ip: X-* G such that the induced measures vt on G are extendable to v, although the measures n¡ on X may not be extendable. Using the algebraic model formulation, the Kolmogorov extension property and the notion of a representation of a directed family of probability measure spaces are discussed.1. Introduction. Let (í¿, sé, /x) be a probability measure space and T a subset of the real line. If (X, 3S) is some measurable space, then an jaZ-measurable function x(oS): Q -> X is called a random element in X. A random function, or stochastic process, defined on T with values in A" is a function x(t, tu): Tx £2 -> X such that x(t, of) is a random element in A" for each t e T. To fix ideas, we will restrict our attention to the classical case and take X= R (the real line) and 38 the a-algebra of Borel sets. Let S denote the set of all functions x = x(t) defined on T= [a, b], say.