Question:
Let $N\in\mathbb{N}$ and
$$
\Omega = \Big\{(i, j)\ \big|\ 1\le i,j \le N\Big\}\subseteq \mathbb{N}\times\mathbb{N}
$$
Define $\mu^*: \mathcal{P}(\Omega) \to [0, \infty]$ by letting $\mu^*(A)$ be the the number of integers $i$ so that there exists some $j$ with $(i, j) \in A$.
(a) Show that $\mu^*$ is an outer measure.
(b) Determine the $\sigma$-algebra of all $\mu^*$-measurable sets.
Attempt:
From my interpretation of $\mu^*$, it is a function that counts the number of unique $1$st-entry in the set. So, for instance,
$$
\mu^*\bigg(\Big\{(3,7),\ (5,1),\ (3,4)\Big\}\bigg) = 2
$$
since there is only $2$ unique $1$st-entry for this set (namely $3$ and $5$).
I'm able to show that $\mu^*(\emptyset) = 0$ and monotonicity. However, I am running into a bit of trouble showing countable sub-additivity for this outer measure. $$ \mu^*\left(\bigcup_{k=1}^\infty A_k\right) \le \sum_{k=1}^\infty \mu^*(A_k) \text{ for all } A_k\in\Omega $$
Also, for (b), I have no idea how to use Carathéodory's criterion to determine the $\sigma$-algebra.
Carathéodory's criterion: Let $\mu^*$ be an outer measure on $\Omega$ and $$ \Sigma = \bigg\{E \subseteq \Omega\ \Big|\ \mu^*(A) = \mu^*(A\cap E) + \mu^*(A\cap E^c)\text{ for all }A\subseteq\Omega\bigg\} $$ Then $\Sigma$ is a $\sigma$-algebra and $\mu^*$ is a measure on $\Sigma$.
Note that $|\Omega|=N^2$ and, thus, $\mathcal{P}(\Omega)$ is finite and you need to consider only finite unions. For two sets $A_1$ and $A_2$ the number of distinct first coordinates in $A_1\cup A_2$ is clearly less or equal than the sum of that in $A_1$ and $A_2$ individually. Use induction to conclude that the subadditivity holds for any number of sets $A_1,A_2,\ldots, A_k$.
As for (b), consider the collection $$ \mathcal{F}=\{E\subset \Omega:\mu^*(E)= |E|/N\}. $$ Basically, if $E\in \mathcal{F}$ and $(i,j)\in E$ for some $j$, then $(i,k)\in E$ for all $k\ne j$. To show that $\mathcal{F}=\Sigma$, notice that if $i$ appears in at least one element of $E$, but $(i,j)\notin E$ for some $j$, $$ N< \mu^*(\Omega\cap E)+\mu^*(\Omega\cap E^c). $$