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