I am trying to prove that given a locally compact abelian (Hausdorf) topological group, the characters on it parameterize the multiplicative linear functionals on the banach * algebra $L^1(G, d\mu)$ where $\mu$ is any Haar measure. That is that all multiplicative linear functionals on the convolution banach * algebra are given by
$f \mapsto \int f\chi d\mu$ for some character $\chi$.
So far my progress is this. Let $L$ be an MLF. Then by duality, $L$ is given by integration against some $\phi:G\rightarrow \mathbb{C}$. One can use the given facts to then see that $\phi(x+y)=\phi(x)\phi(y)$ for $\mu\times\mu$ a.e. $x, y \in G$. (So I guess this is already only true when $\mu$ is sigma finite. What about when it's not? Then according to work I have done one can still say that it holds that for a.e. $\mu (x)$ we have a.e. $\mu (y)$ that...) Trouble is to see that it holds everywhere and $\phi$ is continuous, or more precisely that $\phi$ is the class of a cts function.
I am trying to use this to either see the big name theorems, or if I decide not to pursue them, then just for interest because I compute many Fourier and Laplace transforms, and want them to be unified. Thus, I'm not interested in any proofs using Pontryagin duality, Peter Weyl, or such things, unless they are established on the way without circularity.
Another issue is to see that $f \mapsto \int f\chi d\mu$ is an MLF. This I can do except the part where I need to see it's nonzero. I don't know how to do this without the sigma-finiteness assumption as I am afraid of Haar measures taking values in $\{0, \infty\}$.