 The Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some formula_41-valued stochastic process, but the assumption that the state space be formula_41 is unnecessary.In fact, any collection of measurable spaces together with a collection of inner regular measures defined on the finite products of these spaces would suffice, provided that these measures satisfy a certain compatibility relation.The formal statement of the general theorem is as follows.