If we have the function $$T_tf(x)=\int f(y)p_t(x,dy),\quad f\in C_0(S), x\in S, t\ge 0 $$ where $\hat{S}$ is a locally compact, separable metric space. Let $S:= \hat{S}\cup \{\infty\}$ be the one-point compactification of $\hat{S}$ and $p_t(x,\cdot)$ are probability measure on S. We know that $f\mapsto T_tf(x)$ is a positive linear functional on $C_0$ with norm $1$ for fixed $x$ and that $(T_t)$ is a Feller-semigroup. That's why we know that the right-hand side of the equation above is continuous, so measurable.
How do we show the measurability of $p_t(x,B)$ for any $t\ge0$ and Borel set $B\subset S$ by using standard approximation and the monotone class theorem? Do we need to approximate the indicator function in some way?
You can do this in two steps:
First, you verify that $x\mapsto p_t(x,F)$ is measurable for $F$ closed. You can do this by approximating the indicator function of $F$ by continuous functions. For example, the sequence $\langle f_n\rangle$ with $f_n$ given by $f_n(x)=\max\{0,1-nd(x,F)\}$ works.
Second, you use the $\pi-\lambda$-theorem to show that whenever $\mathcal{C}$ is a family of Borel sets closed under pairwise intersections and $x\mapsto p_t(x,C)$ is measurable for all $C\in\mathcal{C}$, then $x\mapsto p_t(x,B)$ is measurable for all $B\in\mathcal{C}$.