# Averaging measurable functions over actions of amenable groups

Let $G$ be a countable abelian group acting on a space $X$.

It is known that such groups are amenable, i.e., there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$. (For finite groups this is just the average of the values.)

Given a bounded function $f\colon X\to {\mathbf R}$ one can use this mean to define a $G$-invariant function $\overline{f}\colon X\to {\mathbf R}$; one just uses the mean to average over the $G$-orbits.

Question: If $f$ was Borel-measurable, then so is $\overline{f}$ Borel-measurable? (At least if $G$ was countable?)

(This is true for finite groups because the sum and hence the average of measurable functions is measurable.)