Here is a question that has been posted on another forum (in french) that I couldn't answer and about which I'm really curious.
So here it is, let's be given as a state space $\Omega=\{f\in C([0,1],R) s.t. f(0)=0\}$ associated with the borelian $\sigma$-field $\mathcal{F}=\mathcal{B}(O)$, where $O$ is the topology over $\Omega$ defined thanks to uniform convergence norm.
Let $X_t$ be the canonical process $X_s(f)=f(s)$ for $f\in \Omega$ and let $\mathcal{F}_t=\sigma(X_s;s\le t)$ be given as the natural filtration associated with the canonical process $X$.
Is it true that $\mathcal{F}_t$ is a right-continuous filtration?
PS: The usual way to construct continuous processes is by constructing first a process that has good distributional properties then to take a filtration that is both complete and right continuous and finally take a modification of the initial process that is continuous.
Regards