Continuity of volume of balls as function of radius (metric measure spaces) Let $(X,d,m)$ be a proper metric measure space, i.e.

*

*closed ($d$-)balls are compact

*$m$ is a Borel atomless measure, finite and non-zero on every ball of positive radius

(If necessary, we can assume that $(X,d)$ is complete.)
Question: What are (mild) sufficient conditions such that

for every $x_0\in X$ there exists $r_0=r_0(x_0)$ such that $F_{x_0}\colon r\mapsto m B_r(x_0)$ is continuous on $[0,r_0)$

Comments:

*

*It is clear that the assertion does not hold on arbitrary metric measure spaces (see here).


*Furthermore, it is always true that for every $x_0\in X$ the function $F_{x_0}$ has up to countably many discontinuity points (see here).


*Also, $F_{x_0}$ is continuous at $0$, since $m$ is assumed atomless


*The property is stable under multiplication by $L^p_{\rm loc}(m)$ densities ($p>1$, but maybe not $p=1$), in the sense that if the assertion is true for $m$, then it is true for $fm$ for any density $f$ as above


*It seems reasonable to expect the property to hold for $\mathsf{CD}(K,N)$ spaces, $\mathsf{MCP}(K,N)$ spaces, and maybe spaces satisfying a Bishop–Gromov-type or a Brunn–Minkowski-type inequality, but I would very much like some simpler assumptions: e.g. doubling (and, if needed, weak (1,1)-Poincaré, however I do not see why it could be needed).
5'. Bishop–Gromov seems a reasonable assumption because in that case $x\mapsto m B_r(x)$ is continuous for each fixed $r$ (see here, Lemma A.1)


*A proof/reference for $\mathsf{MCP}(K,N)$ spaces would also be much appreciated.


*While this seems reasonable to expect, I am not able to show the sought property for the Hausdorff measure on Ahlfors regular spaces (definition here)
 A: The statement is true on geodesic Monge spaces.
This includes spaces satisfying the measure contraction property $\mathsf{MCP}$ (hence the curvature-dimension $\mathsf{CD}$ condition for finite-dimension) and also spaces satisfying E. Milman's quasi-curvature dimension condition $\mathsf{QCD}$.
In the following, a metric measure space is a complete and separable metric space $(X,d)$ endowed with a non-zero measure $m$ finite on bounded sets.
Definition (Monge space)
A metric measure space $(X,d,m)$ is a Monge space
if for every pair of probability measures $\mu_0,\mu_1$ on $X$ with $\mu_0\ll m$
(a) there exists a unique optimal dynamical plan $\pi\in \mathrm{OptGeo(\mu_0,\mu_1)}$;
(b) $(X,d,m)$ has good transport behaviour, i.e. $\pi$ is induced by a map: $\pi=T_*\mu_0$ for some map $T\colon X\to \mathrm{Geo}(X,d)$;
(c) $(X,d,m)$ has the strong interpolation property, i.e. $\pi$ satisfies $(e_t)_*\pi\ll m$ for all $t\in [0,1)$.
Corollary 2.5 here shows

Let $(X,d,m)$ be a geodesic Monge space. Then, every ball $B\subset X$ is a continuity set for $m$. In particular, for every $r\geq 0$, the sphere
$$S_r(x):=\{y\in X: d(x,y)=r\}$$
is $m$-negligible.

