I came across Young's Inequality for convolutions, stated as:
Let $f\in L^p, g\in L^q$, where $p,q,r \in [1,\infty]$ and $p^{-1}+q^{-1}=1+r^{-1}$. Then $f*g$ is defined $m$-a.e. on $\mathbb{R}^d$, $f*g\in L^r$, and $||f*g||_r\leq||f||_p||g||_q$.
A proof was given for why $||f*g||_r\leq||f||_p||g||_q$, but the reasons why $f*g$ is defined $m$-a.e. on $\mathbb{R}^d$ and $f*g\in L^r$ were never addressed. Could someone please explain why these two statements are true?