Let $G$ be a locally compact, Hausdorff group with left Haar measure $\mu$. I would like to prove an equality
$f\ast f'=\int f(g)f'_g d\mu(g)$,
where $f,f' \in L^1(G),\;f_g(x)=f(g^{-1}x) \quad (x,g\in G)$, and this integral is understood in the sense of Pettis (weak integral). I was able to do it under additional assumption of $\sigma-$finitness of $G$. In this case i can use Fubini's theorem, but i believe it is an obstacle i can deal with using density of $C_c(G)\subset L^1(G)$ or other method.
More serious problem is the fact that we cannot identify functionals on $L^1(G)$ with $L^\infty(G)$ functions in a generic case. Is there any nice characterisation of functionals $\Lambda\in(L^1(G))^*$ which might be helpful? For example is $L^\infty(G)\subset L^1(G)^*$ weak$-\ast$ dense?