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SillyBunnies (talk | contribs) →A more general form of the theorem: The previous few dozen edits were devoted to adding new material relating to a more general form of the theorem. See the Tao book (reference 3) for the proof. |
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As a remark, all of the measures <math>\mu_F,\mu</math> are defined on the product [[sigma algebra]] on their respective spaces, which (as mentioned before) is rather coarse. The measure <math>\mu</math> may sometimes be extended appropriately to a larger sigma algebra, if there is additional structure involved.
Note that the original statement of the theorem is just a special case of this theorem with <math>\Omega_t = \mathbb{R}^n </math> for all <math>t \in T</math>, and <math> \mu_{\{t_1,...,t_k\}}=\nu_{t_1 \dots t_k}</math> for <math> t_1,...,t_k \in T</math>. The stochastic process would simply be the canonical process <math> (\pi_t)_{t \in T}</math>, defined on <math>\Omega=(\mathbb{R}^n)^T</math> with probability measure <math>P=\mu</math>.
This theorem has many far-reaching consequences; for example it can be used to prove the existence of the following, among others:
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