Let $(X,\mathcal{B},\mu)$ be a non-atomic probability space. We may assume $X\subseteq \mathbb{R}^d$ is an open set. Let $\mathcal{C}$ be a 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. $\forall r_0>0$, $\exists$ a $\textbf{disjoint}$ sub-collection $ \mathcal{C}_{r_0}\subset \mathcal{C}$ s.t. $\forall c\in \mathcal{C}_{r_0}$ $\mathrm{diam}(c)\leq r_0$, and $\mu(\bigcup \mathcal{C}_{r_0})\geq \alpha$.
A similar, but not exactly the same, kind of statement is the Vitalli-Lebesgue theorem, whose statement applies to the Lebesgue measure, and which imposes additional requirements on $\mathcal{C}$, but implies that $\alpha=1$.
Any tips, or counter-examples will be welcomed. I am aware that the question may seem obvious or trivial, but so far my attempts at solving it have failed, and I have no new ideas at how to approach it.
Thanks ahead