Let $\mu$ be a finite Borel measure on $\mathbb R^N$ and let $f\in L^1(\mu)$ be a non-negative function. Let $M_\mu f$ denote the maximal function of $f$ relative to $\mu$, i.e. $(M_\mu f)(x)=0$ if $\mu(B(x,r))=0$ for some $r>0$ and $(M_\mu f)(x) = \sup_{0<r<\infty} \frac{1}{\mu(B(x,r))} \int_{B(x,r)}f \, d\mu$ otherwise. (Here $B(x,r)$ denotes the open ball of radius $r$ centered at $x$.)

Suppose that $a>0$ and $K\subset \mathbb R^N$ is a compact such that $M_\mu f > a$ on $K$. Then for each $x\in K$ there exists $r_x>0$ such that $$\frac{1}{\mu(B(x,r_x))} \int_{B(x,r_x)}f \, d\mu > a. \tag{1}$$

**Question.** Is it possible to choose for each $x\in K$ the radius $r_x>0$ in such a way that (1) holds and the mapping $x\mapsto r_x$ is continuous, or upper semicontinuous, or at least Borel?

This question is inspired by another recent question about Besicovich type covering theorem.