Here is an excerpt from the book Principles of Harmonic Analysis by Deitmar and Echterhoff:
Write $C_c^+(G)$ for the set of all positive functions $f \in C_c(G)$ [Note that $G$ is a locally compact group]. For any two functions $f,g \in C_c^+(G)$ with $g \neq 0$ there are finitely many elements $s_j \in G$ and positive numbers $c_j$ such that for every $x \in G$ one has $f(x) \le \sum_{i=1}^n c_j g(s_j^{-1}x)$.
My first question is, how is it possible to have $g \neq 0$ on $G$, unless $G$ is compact, since $g$ is a function with compact support? Secondly, how do I prove the claim? I could use a/some hint(s).
The second part is a standard sort of compactness argument. Let $U=\{x:g(x)>0\}$. This is an open subset of $G$. Then for $s\in U$, $sU=\{x:g(s^{-1}x)>0\}$ is also open.
Let $K$ be the support of $f$. It is compact, by hypothesis. So there are finitely many $s_i$ such that $K\subseteq s_1U\cup\cdots\cup s_n U$. Define $h(x)=\sum_{i=1}^n g(s_i^{-1}(x))$. Then $h(x)>0$ on $K$, and another compactness argument gives some $c>0$ such that $f(x)\le cg(x)$ on $K$, and so on all of $G$.