Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$.
$$\mu(B(x,r)) < \delta r^n.$$
Under what conditions can we find an $\lambda$ (independent of $\delta$) such that for all $r<R$ there exist (at most) countably many pairwise disjoint balls $B_i \subset X$ with radius smaller or equal to $r$, and
$$\sum_{i} \mu(B_i) > \lambda \mu(X).$$
This is the case, for example, when the measure is doubling, or more generally when the space is a Vitali space. However, I am interested in a more general framework. The good news is that I only need this (in some sense) weaker form of a covering theorem. Another question would be whether this statement already implies that the space is a Vitali space.
Thanks in advance!
