Proof of Young's convolutions inequality for a general measure on $\mathbb R^d$ 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 
$$ || f * g || _r \leq ||f||_p ||g||_q.$$
 A: 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. 
A: I am not able to add comments - so this is to be regarded as such:
In SHARPNESS IN YOUNG'S INEQUALITY
FOR CONVOLUTION by JOHN J. F. FOURNIER
the author proves that, under certain conditions on the underlying locally compact unimodular group, the Young inequality can be improved with a constant $C<1$ in front of the RHS of the standard inequality
