Let $\mu$ be a nonatomic probability measure on a Banach space $X$. Is it true that for $\mu$ a.e. $x \in X$, the function $g\_x: (0, \infty) \to \mathbb R$ given by


$$g\_x (r) := \mu(B\_r (x))$$


is continuous in $r$?


*Note: Here $B\_r (x)$ denotes the open ball of radius $r$ around $x$.*



Are nonatomic probability measures on a Banach space nicely shrinking a.e?
No, consider $X=\mathbf{R}^2$ with the $\ell\_\infty$ norm, and let $\mu$ be a non-atomic probability measure giving mass $\frac 1 2$ to both segments $I\_0=[-1,1]\times\{0\}$ and $I\_1=[-1,1]\times\{2\}$. Then for every $x$ in the support of $\mu$ (so in particular for a.e. $x$), $g\_x$ is discontinuous at $2$: $g\_x(2)=\frac 1 2$ but $g\_x(2+\varepsilon) = 1$ for every $\varepsilon>0$.


