Let $C(\mathbb{R}^n;\mathbb{R}^n)$ be the set of continuous functions from the $\mathbb{R}^n$ to itself, let C_0 denote the linear subspace consisting only of those functions that take the value zero at the infimum of the set $\mathbb{R}^n$, and let $\nu$ denote the classical Wiener measure on $C_0$.
Is it true that for any non-empty open subset $U$ of $C(\mathbb{R}^n;\mathbb{R}^n)$ and for any non-empty open subset $V$ of $C_0$ that
- $\nu^n\times\nu(U)>0$ (where $\nu^n$ is the standard Gaussian measure on $\mathbb{R}^n$),
- $\nu(V)>0$?
It seems intuitive to me, since $\nu$ is Gaussian on finite-dimensional projections but I can't seem to prove it.