Let $G$ be a locally compact group with Haar measure $\mu$. Is right-translation continuous on $L^1(G,\mu)$?

Let $G$ be a locally compact group with right-invariant Haar measure $\mu$. I know that the space of compactly supported continuous functions $C_c(G)$ is dense in $L^1(G,\mu)$. If $G$ is 1st-countable, I can show that whenever $y_n\to y$ in $G$, we have that for any $\varphi\in C_c(G)$, if we let $\varphi_{y_n}(x)=\varphi(x y_n)$, then $$\varphi_{y_n}\xrightarrow[L^1(G,\mu)]{n\to\infty}\varphi_y$$ by dominated convergence, so that the map $$G\to L^1(G,\mu),\quad \phi\to\phi_y$$ is continuous. But is this still true if $G$ is not 1st countable?

• How does your proof depend on countability? – Mariano Suárez-Álvarez Mar 10 '18 at 2:46
• It requires countability because I need to use dominated convergence. – Monstrous Moonshine Mar 10 '18 at 2:50
• I don't understand. What version of the dominated convergence theorem requires first countability? The theorem does not even involve a topology, no? – Mariano Suárez-Álvarez Mar 10 '18 at 3:08
• Possible duplicate of A net version of dominated convergence? – Eric Wofsey Mar 10 '18 at 3:10
• @MarianoSuárez-Álvarez: We are trying to prove a certain map $G\to L^1(G,\mu)$ is continuous. We can use dominated convergence to prove the map preserves convergence of sequences. But if $G$ is not first countable, that is not enough to conclude our map is continuous. – Eric Wofsey Mar 10 '18 at 3:12