My question is: $S$ is a d-dimensional Ito diffusion, suppose $A$ is borel set in $\mathbb{R}^d$, $T_x:=\{t>0, S^x_t\in A \}$ is the first hitting time for $S$ from any point $x$, $h:\mathbb{R}^d\rightarrow \mathbb{R}$ is continuous. Then is $f(x):=E^x[e^{-T_x}h(X_{T_x})]$ a borel measurable function? Moreover, if $\delta$ is a continuous function is $f(x):=E^x[e^{-\delta(T_x)}h(X_{T_x})]$ borel measurable?
What about change $A$ to nearly borel, does the function $f$ become nearly borel?
From the book "Markov Process, Brownian Motion and Time Symmetry" (page 115), I see that, under some condition: $f(x):= E^x[h(X_{T_x})]$ is borel measurable if $h$ is borel measurable, what about adding a discount function $e^{-\delta(T)}$?