Let $M$ be a Riemannian manifold, and let $C = \{ c : [0,1] \to M \mid c \text{ is continuous}\}$. Endow $C$ with the Wiener measure $\mathbb P_x$ concentrated on the curves $c \in C$ with $c(0) = x \in M$.
If $B \subseteq C$ is Borel, why is the function $x \mapsto \mathbb P_x (B)$ measurable?
Things are clear if $B$ is cylindrical, but how do I reason for arbitrary Borel subsets? I need this in order to be able to show that the set function $B \mapsto \int _M \mathbb P_x (B) \ \mathrm d x$ defines a measure on $C$, so I need to show first that the integral exists.