We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. The convolution product of $f$ and $g$ is defined by
$$\forall x \in \mathbb{R}, \quad (f * g)(x) = \int_{-\infty}^{+\infty} f(t) g(x-t) \,\mathrm{d}t$$
Show that $f * g$ is defined on $\mathbb{R}$ and that
$$\forall x \in \mathbb{R}, \quad (f * g)(x) = \int_{-\infty}^{+\infty} f(x-t) g(t) \,\mathrm{d}t = (g * f)(x)$$