Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. I have following problem:
Given is the map $W:\Omega\rightarrow C[0,1]$ (it is not given but I think it is implicit a Wiener process). Then the claim is that the map is actually a Wiener measure on the space $(C([0,1]),\mathcal{B}(C[0,1]))$ under $\mathbb{P}$. But this is true only if $W$ is a measurable map. This is the case if for any set in $E\subset\mathcal{B}(C[0,1]):W^{-1}(E)\subset \mathcal{F}$. As an hint I have that $W$ should be identically $0$ on $\mathbb{P}$-nullsets. My idea was to use continuity of the Wiener process to conclude continuity and hence measurability of $W$ but I don't know how... Some advice would be appreciated. Thanks :)