I'm currently working on a research project based on John B. Conway's "A Course in Functional Analysis", specifically Fourier theory on locally compact groups. In his book, Conway claims that $L^{1}(G)$ has the structure of a Banach algebra with the convolution of functions as a product. Most of the properties follow immediately from standard results in measure theory. However, I'm struggling with the proof of the inequality $\Vert f\ast g\Vert\leq\Vert f\Vert\Vert g\Vert$. All the proofs I've seen at some point rely on a change in the order of integration, which is easily dealt with by making use of Fubini's theorem, but this only works when the space is $\sigma$-finite, which apparently isn't true for an arbitrary choice of $G$. It doesn't look like Conway implicitly assumes that this condition is satisfied since in later sections he does calculations involving discrete groups, which fail to satisfy $\sigma$-finiteness. So I'm guessing there must be a way around this. I saw on a similar thread that you can use a special case of Young's inequality, but after looking into it, the proof of this inequality is also dependent on the ability to change the order of integration.
On the other hand, since the product operation in Banach algebras is required to be associative, I have a similar problem when trying to prove this property for the convolution, since the standard proof of associativity of convolution also makes use of Fubini's theorem. So... what gives? Everything works out fine when the space is $\sigma$-finite, but seems to break down when we drop this hypothesis.
I'd be grateful if you could give me some advice on how to work around these.
Edit: I found in the same book that by proving that the set of all finite Borel Measures over $G$ is a Banach algebra with an adequate norm and product then if follows that $L^{1}(G)$ is a Banach algebra with the convolution as the product. So, that seems to solve the problem. However, out of curiosity, can we prove this directly, without resorting to a more general object?