Let $X$ be a Banach space and let $S(t)$, $t geq 0$, be a $C_0$-semigroup on $X$.

Assume that $f : (0,+infty) rightarrow X$ is Bochner integrable.

Is $S(t-s)f(s)$ Bochner integrable on $(0,t)$ and does $t mapsto int_0^t S(t-s)f(s)ds in C^0((0,+infty),X)$ ?

The function $t mapsto int_0^t S(t-s)f(s)ds$ arises when we define the notion of weak solution to an inhomogeneous evolution PDE $$partial_t u(t) = Au(t) + f(t), quad u(0) = u_0$$

where $A$ is the infinitesimal generator of $S(t)$.

If $f$ is continuous, I know that the result is true, but I’m interested in the non-continuous case. I would expect this to be true.

Any proof or reference is welcomed.