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{{short description|
{{About|a theorem on stochastic processes|a theorem on extension of pre-measure|Hahn–Kolmogorov theorem}}
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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>
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>,
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In fact, it is always possible to take as the underlying probability space <math>\Omega = (\mathbb{R}^n)^T</math> and to take for <math>X</math> the canonical process <math>X\colon (t,Y) \mapsto Y_t</math>. Therefore, an alternative way of stating Kolmogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure <math>\nu</math> on <math>(\mathbb{R}^n)^T</math> with marginals <math>\nu_{t_{1} \dots t_{k}}</math> for any finite collection of times <math>t_{1} \dots t_{k}</math>. Kolmogorov's extension theorem applies when <math>T</math> is uncountable, but the price to pay
for this level of generality is that the measure <math>\nu</math> is only defined on the
==Explanation of the conditions==
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* Aldrich, J. (2007) [http://www.emis.de/journals/JEHPS/Decembre2007/Aldrich.pdf "But you have to remember P.J.Daniell of Sheffield"] [http://www.emis.de/journals/JEHPS/indexang.html Electronic Journ@l for History of Probability and Statistics] December 2007.
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