In the proof below, how is the inequality
$$\lambda(G) \le \sum_n\sum_k\mu_0(F_{n,k})$$
justified?
screenshot, transcribed below:
A1.8. Proof of Carathéodory's Theorem. Recall that we need to prove the following. Let $S$ be a set, let $\Sigma_0$ be an algebra on $S$, and let $$ \Sigma:=\sigma\left(\Sigma_0\right) . $$ If $\mu_0$ is a countably additive map $\mu_0: \Sigma_0 \rightarrow[0, \infty]$, then there exists a measure $\mu$ on $(S, \Sigma)$ such that $$ \mu=\mu_0 \text { on } \Sigma_0 . $$ Proof. Step 1: Let $\mathcal{G}$ be the $\sigma$-algebra of all subsets of $S$. For $G \in \mathcal{G}$, define $$ \lambda(G):=\inf \sum_n \mu_0\left(F_n\right), $$ where the infimum is taken over all sequences $\left(F_n\right)$ in $\Sigma_0$ with $G \subseteq \bigcup_n F_n$. We now prove that (a) $\lambda$ is an outer measure on $(S, \mathcal{G})$. The facts that $\lambda(\emptyset)=0$ and $\lambda$ is increasing are obvious. Suppose that $\left(G_n\right)$ is a sequence in $\mathcal{G}$, such that each $\lambda\left(G_n\right)$ is finite. Let $\varepsilon>0$ be given. For each $n$, choose a sequence $\left(F_{n, k}: k \in \mathbf{N}\right)$ of elements of $\Sigma_0$ such that $$ G_n \subseteq \bigcup_k F_{n, k}, \quad \sum_k \mu_0\left(F_{n, k}\right)<\lambda\left(G_n\right)+\varepsilon 2^{-n} . $$ Then $G:=\bigcup G_n \subseteq \bigcup_n \bigcup_k F_{n, k}$, so that $$ \lambda(G) \leq \sum_n \sum_k \mu_0\left(F_{n, k}\right)<\sum_n \lambda\left(G_n\right)+\varepsilon . $$ Since $\varepsilon$ is arbitrary, we have proved result (a).
What is trivial from the definition of $\lambda$ is that, arranging the (countably many) $F_{n,k}$ into a sequence $(F_m)$, we have $$\lambda(G) \le \sum_{m=1}^{\infty}F_m.$$ What was not clear to me is why the above series is to equal $$\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}F_{n,k}.$$
I realize now that the equality can be proven as follows: \begin{equation} \tag{$*$} \sum_{m=1}^{\infty}F_m = \sum_{m\in \mathbb{N}}F_m = \sum_{(n,k)\in \mathbb{N}^2}F_{n,k} = \sum_{n=1}^{\infty}\sum_{k=1}^{\infty}F_{n,k} \end{equation} where the third equality in $(*)$ follows from Tonelli's Theorem for Series, and the two expressions in the middle are defined by $$\sum_{\alpha \in A}x_{\alpha} := \sup\left\{\sum_{\alpha \in F}x_{\alpha} : \text{ for a finite subset $F$ of $A$}\right\}$$ so that the second equality in $(*)$ is trivial and the first is a standard result. All of this is treated at the beginning of Tao's An Introduction to Measure Theory.