Let $f_i$ be a sequence of smooth functions on $S^2$ such that the measures $\mu_i=f_i \;d\mathrm{vol}_{S^2}$ converge weakly to $d\mathrm{vol}_{S^2}$. Now suppose $\epsilon_i$ is a sequence going to zero. My question now is if for each $x\in S^2$ $\limsup_i \epsilon_i^{-2}\mu_i(B_{\epsilon_i}(x))=1.$
(I've chosen to phrase the question on $S^2$ because I want it on a compact set w/o boundary, so that nothing funny can happen at the boundary or "at infinity".)