# Defining Brownian motion through Kolmogorov's extension theorem

In section 2.2. of Oksendal's book on Stochasic differential equations, he defines Brownian motion by specifying a family of probability measures $\nu_{t_1, \ldots, t_k}(F_1, \ldots, F_k)$ that satisfies conditions of Kolmogorov's extension theorem.

He defines $P = P^{x_0}(B_1 \in F_1, \ldots, B_k \in F_k)$ as integral over $F^\times_k = F_1 \times \cdots \times F_k$

$$\int_{F^\times_k} d x_1 \cdots d x_k \prod_{n=1}^k p(t_k-t_{k-1}, x_k-x_{k-1})$$

He then claims

The Brownian motion thus defined is not unique, i.e. there exists several quadruples $(B_t,\Omega, \mathcal{F}, P^x)$ such the above equation holds.

He then says paths of Brownian motion can be chosen continuous. The needed page is viewable through Google Books.

So I've been curious to see an example of $(B_t, \Omega, \mathcal{F}, P^x)$ where the paths need not be continuous.

Thank you.

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@Byron Thanks, it makes perfect sense now. Should I vote to delete this question as a duplicate then ? – Sasha Aug 12 '11 at 20:55
I'm not sure what the correct protocol is. Your question has a duplicate, but on a different site. I'd leave it up to the moderators, i.e., do nothing. – Byron Schmuland Aug 12 '11 at 21:10