I'm not sure if these details matter, but anyway for this particular case, consider a compact abelian group $G$ with operation $\cdot$, a Haar measure $\mu$ on it and $f$ a non-trivial character on $G$. (I'm looking at this question: The integral of a character is $0$)
I've seen stated that for $x,y\in G$,
$$\int_G f(x)\>d\mu(x)=\int_G f(x\cdot y)\>d\mu(x\cdot y).$$
Is this true? If so why? What's the general theory behind it?