function defined as integral of borel function

Question

I know that $f \in B_b(E)$, where $B_b(E)$ is the set of Borel bounded function on an euclidean space E. I have to show that: \begin{equation} x \to \int_{0}^{+\infty} e^{-at} P_tf(x) dt \end{equation} is a Borel and bounded function, where $a \in (0, +\infty)$, $P_tf \colon B_b(E) \to B_b(E)$ is an operator. How can I argue?