Let $G$ be a locally compact Hausdorff group. Let $C_c(G)$ be the space of complex-valued compactly supported continuous functions on $G$. Let $M(G)$ be the space of Radon measures on $G$. We fix $d\mu$ a Haar measure on $G$. Then there is a natural map \begin{align} C_c(G)&\to M(G)\\ f&\mapsto fd\mu \end{align} Question : is the image dense, for a natural topology of $M(G)$, say the inductive limit weak * one ?
Any hint or reference would be highly appreciated.
The answer is Yes. I appreciate Evangelopoulos Foivos's hint. The following should be a possible reasoning.
I will assume that $G$ is compact, since everything can be deduced from an inductive limit process. By Riesz representation theorem, $M(G)$ is the dual of $C(G):=C_c(G)$.
Recall that a total set of a vector space is a set of linear functionals $T$ with the property that if a vector $x\in X$ satisfies $h(x)=0$ for all $h\in T$ then $x=0$.
Back to the question. By $G$ is dense in $X^*$ in weak* sense if and only if $G$ is total set, we need to show that the image is a total set of $M(G)$. If $f_0\in C(G)$ is a vector such that $\int_G f_0f\,d\mu=0$ for all $f\in C(G)$, then we can just take $f$ to be the complex conjugate of $f_0$ to see $f_0=0$. Hence the image is a total set of $M(G)$.