Let $I_1:=(0,1)$, $I_2=(a,a+1)$, where $a\geq 1$, be open intervals of ${\Bbb R}$. Let $M:=I_1\cup I_2$. Choose $$f:M\rightarrow{\Bbb R},\qquad x\mapsto\cases{0 &if $x\in I_1$\cr 1&if $x\in I_2$\cr}$$

$f$ is continuous. Choose the metric to be the standard Euclidean one. Choose $\mu$ to be the one-dimensional Lebesgue-measure (i.e. ${\rm d}\mu={\rm d}x$).

In this case, the r.h.s. of your equation will be equal to $1$. Your l.h.s. refers to a single sample trajectory of the Wiener-process. If this trajectory, considered as a trajectory in ${\Bbb R}$, starts in $I_1$, it cannot enter $I_2$. So the l.h.s. will be equal to $0$.