Let $m$ be a probability measure on $\mathbb{R}^n$.
Consider a function $\ f: \mathbb{R}^n \rightarrow \mathbb{R}$.
Say under what conditions the following inequality holds. $$ m\left(\left\{ x \in \mathbb{R}^m : \sup_{ y \in \{x\} + \epsilon \mathbb{B} } f(y) >0 \right\} \right) \ \leq \ m\left(\left\{ x \in \mathbb{R}^m : \ f(x) >0 \right\} \right) + g(\epsilon) $$ for some strictly increasing $g: \mathbb{R}_{\geq 0} \rightarrow [0,1]$ such that $g(0) = 0$.
Comments. I think that at least $m$ has to be atomless.
This question is similar to that one.