I’m reading through an introductory operator algebra paper currently, and I’m a little lost on how one proves continuity of the convolution on $C_c(G)$ for any locally compact topological group $G$.
The paper says “For $f,g \in C_c(G)$ and $t \in G$, define: $$ (f \ast g)(t) = \int_{G} f(s)g(s^{-1}t) d\mu(s).$$ Then $f \ast g$ is continuous by the Dominated convergence theorem.”
I follow this reasoning for $G$ first countable, since then for $(t_k)_{k=1}^{\infty}$ converging to $t$, we can define $h_k(s) = f(s)g(s^{-1}t_k)$ and apply the DCT (maybe I’m not doing this correctly?). But for $G$ not first countable, I don’t see how you could apply DCT, since convergent nets of measurable functions need not even be measurable.
I’d be very appreciative if someone could point out what I’m missing (I feel like this may need a uniform convergence argument). Thank you.