Let $C = \{c : [0,1] \to \mathbb R ^n \mid c \text{ is continuous and } c(0)=0 \}$ be endowed with the Wiener measure $P$. Consider an exhaustion $\mathbb R^n = \bigcup _{k \ge 0} U_k$ where each $U_k \ni 0$ is a connected open subset with smooth boundary. Let $C_k = \{ c \in C \mid c([0,1]) \subseteq \overline {U_k} \}$. Clearly $C = \bigcup _{k \ge 0} C_k$, an exhaustion. Endow each $C_k$ with the intrinsic Wiener measure $P_k$ (notice that $P_k$ is not the restriction of $P$ to $C_k$!). It is known that $P_k (A \cap C_k) \to P(A)$ for every Borel $A \subseteq C$.
If $h : (0, \infty) \times \mathbb R^n \to (0, \infty)$ is the heat kernel on $\mathbb R^n$, there exists a disintegration $(\mu_x) _{x \in \mathbb R^n}$ of $P$ such that $$P(A) = \int _{\mathbb R^n} \mathrm d x \, h(1, x) \, \mu_x (A)$$ for every Borel $A \subseteq C$. ($\mu_x$ is concentrated on the curves that end at $x$ for almost every $x \in \mathbb R^n$.)
Similarly, if $h_k : (0, \infty) \times \overline{U_k} \to (0, \infty)$ is the heat kernel on $\overline {U_k}$, there exists a disintegration $(\mu_x ^k) _{x \in \overline{U_k}}$ of $P_k$ such that $$P_k(A) = \int _{\overline {U_k}} \mathrm d x \, h_k(1, x) \, \mu_x ^k (A)$$ for every Borel $A \subseteq C_k$.
I believe that if $A \subseteq C$ is Borel, then $\mu_x ^k (A \cap C_k) \to \mu_x (A)$ for almost every $x \in U_0$ ($U_0$ is the smallest domain in the exhaustion). Is this true? Where can I find a proof?
