Kolmogorov extension theorem: Difference between revisions

Revision as of 11:34, 19 September 2023

Revision as of 11:36, 19 September 2023

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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 product [[Σ-algebra#Product_σ-algebra|product σ-algebra]] of <math>(\mathbb{R}^n)^T</math>, which is not very rich.

==Explanation of the conditions==