Convolution theorem for the Laplace transform

Statement

Let $f, g : [0, \infty) \to \mathbb{C}$ be piecewise continuous, and let $M, a$ be real constants with $|f(t)| \le M e^{a t}$ and $|g(t)| \le M e^{a t}$ for all $t \ge 0$. Let the convolution $f * g : [0, \infty) \to \mathbb{C}$ be $$(f * g)(t) = \int_0^t f(\tau)\, g(t - \tau)\, d\tau.$$ Let $F$ and $G$ be the Laplace transforms of $f$ and $g$. Then for each $s \in \mathbb{C}$ with $\operatorname{Re} s > a$, the Laplace transform of $f * g$ converges at $s$, and $\mathcal{L}\{f * g\}(s) = F(s)\, G(s)$.