Let $(S,\mathcal{S})$ be a measurable space, and let $p: S \times \mathcal{S} \to \mathbb{R}$ satisfies the properties: $p(s,\cdot)$ is a measure for each $s \in S$ and $p(\cdot, A)$ is a measurable function on $S$ for each $A \in \mathcal{S}$.
Let $f$ be a measurable function on $S$, assumed to be non-negative or bounded. Let $A \in \mathcal{S}$. In this case, is the function $x \mapsto \int_A f(y) p(x,dy)$ measurable?