Let $W$ be a separable Banach space and $\mu$ a Gaussian Borel measure on $W$ which is centered and non-degenerate. For $F : W \to \mathbb{R}$ bounded Borel and $t \ge 0$, let $$P_t F(x) = \int_W F(x+\sqrt{t}y)\\,\mu(dy)$$ be the transition semigroup of Brownian motion on $W$.

Is $P_t$ well defined as an operator from $L^\infty(W,\mu)$ to itself? That is, if $F=0$ $\mu$-almost everywhere, must the same be true of $P_t F$?

Of course, if $W$ is finite dimensional the answer is yes, because the measures $\mu(x + \cdot)$ are mutually absolutely continuous as $x$ ranges over $W$. This suggests that the answer may be no in infinite dimensions, but I can't seem to find a counterexample.