Suppose $G$ is a group acting on a nonatomic standard measure space $(X,\mu)$ (say $[0,1]$ with Lebesgue measure). Assume that the action is nonsingular, i.e. $\mu(E)=0$ implies $\mu(gE)=0$ for all $g\in G$ and measurable subsets $E\subset X$.
Given $g_0\in G$ and $\delta>0$, is it true that there exists $\varepsilon>0$ such that $\mu(E)\leq\varepsilon$ implies $\mu(g_0E)\leq\delta$ for all measurable $E\subset X$?
This seems to make sense, but I don't immediately see how to start (dis)proving it.