Convolution
Definition 1. Let $f, g$ be piecewise continuous functions on $[0, \infty)$. The convolution of $f$ and $g$, denoted by $f * g$, is a function defined by the integral $$f * g = \int_0^t f(\tau) g(t - \tau) \, d \tau$$
Remark.
(i) $f * g = g * f$
(ii) $f * (g_1 + g_2) = f * g_1 + f * g_2$
(iii) $(f * g) * h = f * (g * h)$
(iv) $f * 0 = 0 * f = 0$
(v) $\int_0^t f(\tau) \, d \tau = f * 1$. In general, $f * 1 \neq f$
Convolution Theorem
Theorem 1. If $f$ and $g$ are piecewise continuous on $[0, \infty)$ and of exponential order, then $$\mathcal{L} \{ f * g \} = \mathcal{L} \{ f(t) \} \mathcal{L} \{ g(t) \}$$
Proof.
Remark. $$\mathcal{L} \left\{ \int_0^t f(\tau) \, d \tau \right\} = \mathcal{L} \{ f * 1 \} = \mathcal{L} \{ f(t) \} \mathcal{L} \{ 1 \} = \frac{F(s)}{s}$$