Let $G$ be a compact group. I am reading a paper where the author claims the existence of a function $\varphi:G\to\mathbb{C}$ such that $\varphi$ is non negative, $$\int_G \varphi(g)dg=1$$ and such that $\varphi(g)=\varphi(g^{-1})$ (we are using Haar measure). My questions are the following:
1. First of all, does "non negative" make sense here? Maybe they are implicitly assuming $\varphi:G\to\mathbb{R}$?
2. I have no idea how to construct the function. Is this a more general result? Like for any $X$ compact, we can find such a function? And how would the statement be?
Any hint on how to construct the function would be appreciated!
1. Yes, of course.
2. Take any continuous non-null function $f\colon G\longrightarrow\mathbb R$. Let $F=|f|/\int_G|f|$. Then $\int_GF=1$. Finally, let $\varphi(g)=\frac12\bigl(F(g)+F(g^{-1})\bigr)$. Then $\int_G\varphi=1$ and $\varphi(g^{-1})=\varphi(g)$.
By the same argument, for any compact neighborhood $N$ of any point of $G$, you can find such a function $\varphi$ with the property described above such that, furthermore, the support of $\varphi$ is contained in $N$. I am assuming that the Haar measure is such that the measure of any open non-empty subset is greater than $0$.