Let $X$ be a compact metric space (with balls $B_{\varepsilon }(x)$), $\mu $ a Borel probability measure, and $A$ a Borel set with positive probability.
Do we have that $\lim_{\varepsilon \rightarrow 0}\frac{\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}=1$ for a.e. $x\in A?$
Note: It seems the result is not true in general for Polish spaces and Borel measures, I don't know if the compactness and the finiteness of the measure changes this. There seems to be some hope as it seems to be true for any Borel measure on the interval http://www.personal.psu.edu/axk29/502/Problems9.pdf .