Is Young's inequality true for an arbitrary measure on $\mathbb R^d$? If so, where can I find a proof of it? In particular, where can I find the proof of the discrete version (i.e the version for $\ell^p$ spaces) of this inequality?

Here is the statement of the inequality (from Wikipedia):

Suppose $f$ is in $L^p(\mathbb R^d)$ and $g$ is in $L^q(\mathbb R^d)$ and $$ \frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1$$

with $1\leq p, q, r\leq \infty$

then $$ || f * g || _r \leq ||f||_p ||g||_q.$$