I'm reading a proof about the existence of a left invariant mean on certain types of groups, and I came across this part that I don't quite understand:
Let $G$ be a locally compact Hausdorff group.
We have that $m_{\varphi}(f)=\int_{G}f \hspace{0.1cm}d\mu$ for any continuous function $f:G \to \mathbb{C}$ with compact support. (This is shown in the proof but I don't think the details are necessary here)
Then as the set of continuous functions with compact support is dense in $L^{1}(G)$ we have $m_{\varphi}(f)=\int_{G}f \hspace{0.1cm}d\mu$ for all $f \in L^{1}(G)$
What I don't quite understand is why the density of the compact support functions automatically extends the statement to the whole of $L^{1}(G)$. I would really appreciate it if someone could explain this!
I don't know what $m_\varphi$ is, but assuming that it is a continuous function from $L^1(G)$ into $\mathbb C$ then whoever wrote that is using that fact that if two continuous functions $\alpha$ and $\beta$ from a topological space $X$ into a topological space $Y$ are such that, for some dense subset $D$ of $X$,$$(\forall x\in D):\alpha(x)=\beta(x),$$then $\alpha=\beta$. This is so because the set $\{x\in X\mid\alpha(x)=\beta(x)\}$ is a closed subset of $X$ and, since $D$ is dense in $X$, the only closed subset of $X$ containing $D$ is $X$ itself.