Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is finite on the bounded subsets of $X$. Let $\mathcal{F}$ a family of closed subsets of $X$.
Theorem 2.8.2 from "Geometric Measure Theory" of Federer. Let $\mathcal{K}$ be a countable family of subsets of $X$ and let $\sigma\colon\mathcal{K}\to(0,1)$ a function such that for each $K\in\mathcal{K}$ it holds the following property: for each open subset $U\subseteq X$ there exists a countable and disjointed subfamily $\mathcal{G}\subseteq\mathcal{F}$ such that $\bigcup_{G\in\mathcal{G}}G\subseteq U$ and $\mu\left((U\cap K)\setminus\bigcup_{G\in\mathcal{G}}G\right)\le\sigma(K)\mu(U\cap K)$. Then $\mathcal{F}$ is $\mu$-adequate for $\bigcup_{K\in\mathcal{K}}K$.
The Proof. Since $X$ is the union of countably many open bounded subsets, we may assume that each member of $K$ is bounded. Since $\mathcal{K}$ is countable, there exists a sequence $\{ C_n\mid n\in\mathbb{N}\} $ such that $\{C_n\}_{n\in\mathbb{N}}=\mathcal{K}$ and such that for all $K\in\mathcal{K}$ the set $\{n\in\mathbb{N}\mid C_n=K\}$ is infinite. Let $U$ be an open subset of $X$. We define open sets $U_n$ and finite disjointed subfamilies $\mathcal{G}_n$ of $\mathcal{F}$ by induction, starting with $U_0=U$ and $\mathcal{G}_0=\emptyset$, in such a way that
$U_n=U_{n-1}\setminus\bigcup\mathcal{G}_{n-1}$ and $\mu\left((U_n\cap C_n)\setminus\bigcup\mathcal{G}_n\right)\le[\sigma(C_n)]^{\frac{1}{2}}\mu(U_n\cap C_n)$ whenever $n\ge 1$: this is possible because $\bigcup\mathcal{G}_{n-1}$ is closed, then $U_n$ is open, hence $ \mathcal{F}$ has a countable disjointed subfamily $\mathcal{H}$ such that $\bigcup\mathcal{H}\subseteq U_n$ and
$\mu\left((U_n\cap C_n)\setminus\bigcup\mathcal{G}_n\right)\le\sigma(C_n)\mu(U_n\cap C_n)$
and because $\mu(C_n)<+\infty$ and the members of $\mathcal{H}$ are $\mu$-measurable, hence $\mathcal{G}_n$ can be chosen as a sufficiently large finite subfamily of $\mathcal{H}$.
Can anyone explain how can I use the facts that "$\mu(C_n)<+\infty$ and the members of $\mathcal{H}$ are $\mu$-measurable" in order to deduce the the existence of $\mathcal{G}_n$, please?