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



Proof of Young's convolutions inequality for a general measure on $\mathbb R^d$
Note that if your measure is not translation-invariant, the convolution product is not commutative. 


This allows for a simple counterexample with $r=q=\infty$, $p=1$,
$d\mu(x)={\bf 1}\_{[0,1]}(x) dx$.


Define $f\*g(x)= \int f(x-s)g(s)d\mu(s) \ (\neq g\*f(x))$. 


Take $g\equiv 1, \ f={\bf 1}\_{[1,2]}$.
This gives $||f||\_1=0$, but the spreading in the convolution makes the left term non zero.


$\max\_x \ \ \int\_0^1 {\bf 1}\_{[1,2]}(x-s)ds =1$.


Just add $\varepsilon e^{-|x|}dx$ to $d\mu$ to get a counterexample with a measure of full support. I think that the reasonable setting for a Young inequality is the case of a translation invariant measure on a locally compact group. Jonas Mayer gave a reference (Hewitt and Ross) for that case in the comments. 


