Existence of a certain type of function on locally compact groups

Question

I have seen it stated that on a locally compact group $G$ with $\mu$ its (left) Haar measure, there exists a positive, compactly supported function $f\in C_c(G)$ with $\int_G f(s)d\mu(s)=1$ satisfying that $f(s)=f(s^{-1})$ for every $s\in G$.

This feels like a basic result, but I somehow can't work it out.

All hints are appreciated.

Answer 1

Let $g$ be any (real) function in $C_c(G)$. Let $f(g)=M[g^{2}(s)+g^{2}(s^{-1})$]. Then $f \in C_c(G)$ because, if $g$ is supported by a compact set $K$, then $f$ is supported by $K \cup K^{-1}$. We can always choose $M$ so that $\int f(s)d\mu(s)=1$.