Kolmogorov extension theorem: Difference between revisions

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==Statement of the theorem==

Let <math>T</math> denote some [[Interval (mathematics)|interval]] (thought of as "[[time]]"), and let <math>n \in \mathbb{N}</math>. For each <math>k \in \mathbb{N}</math> and finite [[sequence]] of distinct times <math>t_{1}, \dots, t_{k} \in T</math>, let <math>\nu_{t_{1} \dots t_{k}}</math> be a [[probability measure]] on <math>(\mathbb{R}^{n})^{k}.</math> Suppose that these measures satisfy two consistency conditions:

1. for all [[permutation]]s <math>\pi</math> of <math>\{ 1, \dots, k \}</math> and measurable sets <math>F_{i} \subseteq \mathbb{R}^{n}</math>,