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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. These measures may sometimes be extended appropriately to larger sigma algebras, 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>. 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: *Brownian motion, i.e, the [[Wiener process]],
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