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The Kolmogorov extention theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some <math>\mathbb{R}^{n}</math>-valued stochastic process, but the assumption that the state space be <math>\mathbb{R}^{n}</math> is unnecessary. In fact, any collection of measurable spaces together with a collection of [[inner regular measure]]s 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.<ref>T. Tao, ''An Introduction to Measure Theory'', Graduate Studies in Mathematics, Vol. 126, 2011, p. 195 </ref>
Let <math>T</math> be any set. Let <math> \{ (\Omega_t, \mathcal{F}_t) \}_{t \in T} </math> be some collection of measurable spaces, and for each <math> t \in T </math>, let <math> \tau_t</math> be a Hausdorff topology on <math> \Omega_t</math>. For each subset <math>J \subset T</math>,
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For subsets <math>I \subset J \subset T</math>, let <math>\pi_{I \leftarrow J}: \Omega_J \to \Omega_I</math> denote the canonical projection map <math> \omega \mapsto \omega|_I </math>.
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