Approximating measure by partitions

Question

Let $(X,\mathcal{B},\mu)$ be a non-atomic Borel probability space. We may assume that $X\subseteq \mathbb{R}^d$ is the open (or closed) unit ball, if it helps.

Let $\mathcal{C}$ be a countable collection of sets such that for $\mu$-a.e. $x\in X$, $\exists \{c_n\}_{n\geq 1}\subset \mathcal{C}$ s.t. $\{x\}= \bigcap_{n\geq1}c_n$, $c_{n+1}\subseteq c_n$ ($n\geq1$), and $\mathrm{diam}(c_n)\xrightarrow[n\rightarrow\infty]{}0$.

My question is the following:

Does there exist a positive constant $\alpha>0$, s.t. $\forall r>0$, $\exists$ a $\textbf{disjoint}$ sub-collection $ \mathcal{C}_{r}\subset \mathcal{C}$ s.t. $\forall c\in \mathcal{C}_{r}$, $\mathrm{diam}(c)\leq r$, and $\mu(\bigcup \mathcal{C}_{r})\geq \alpha$ ?

Any tips, insights, or counter-examples would be greatly appreciated!

Thanks ahead

Answer 1

No, such positive $\alpha$ may not exist.

Fix some fery fast growing function $F$. Let $X$ be a unit square, and choose $\mathcal{C}$ consisting of $2^{-F(n)}$-neigborhoods of $2^{-n}$ by $2^{-n}$ $+$-shapes with dyadic rational centers. They clearly satisfy your first condition, but there cannot be more than $2^{2n}$ disjoint crosses with diameter $2^{-n}$ on a unit square, and you can bound their area above by $2 \times (2^{-F(n)} \times 2^{-n}) := S_n$.

Function $\sum_{n > N} 2^{2n}S_n$ is an obvious upper bound for the area of any disjoint family with diameter of elements less than $2^{-N}$. So if this function has limit 0 ($F = 2^{2^n}$ will suffice) then $\alpha$ does not exist.