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