I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(Z,d)$ be a compact metric space. Let $m$ be a Borel probability measure on $m$. Let $\Omega$ be a collection of balls with the following property.
There exists A,D,C>0 such that
a)If $B(x_1,r_1)$ and $B(x_2,r_2)$ are in $\Omega$ with $r_2>r_1/A$ then $m(B(x_2,r_2))\geq D m(B(x_1,r_1))$
b)For each $r>0$ the set of balls in $\Omega$ of radius between $Cr$ and $C^{-1}r$ cover $Z$.
c) For each $x\in Z$ there exist balls in $\Omega$ of arbitrarily small radius centered at $x$.
Then for any measurable $A\subset Z$ for almost any $a\in A$
$$\sup_{r\to 0:B(a,r)\in \Omega}m(B(a,r)\cap A)/m(B(a,r))\to 1$$ Does such a result (or a similar one) exist?
