Let $\mu$ a Radon measure on $\mathbb{R}^n$, $C$ a bounded and measurable set and $F_i$ a countable family of closed ball. I also have that $\frac{\mu(C)}{\eta(n)}\le \mu (C \cap \bigcup \{ \bar{B} \colon \bar{B} \in F_i \})$ I'm not able to prove the following:
As $\mu(C)<\infty$, we can select a finite subfamily $F_i′\subset F_i$ with the property that \begin{equation} \frac{\mu(C)}{2\eta(n)}\le \mu (C \cap \bigcup \{ \bar{B} \colon \bar{B} \in F_i' \}) \end{equation}
The point is simply that if $F_i = \{\overline{B}_1, \overline{B}_2, \ldots\}$, by countable additivity $$\mu(C \cap \bigcup_{i=1}^\infty \overline{B}_i) = \lim_{N\to \infty} \mu(C \cap \bigcup_{i=1}^N \overline{B}_i)$$