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.