Here is the problem: Let $\Omega$ be a nonempty set and $\mathcal A$ $=$ $\mathcal P$($\Omega$). Define $\mu$ on $\mathcal A$ by $\mu$($E$) equal to $N$($E$) if $E$ is finite and $\infty$ if $E$ is infinite, where $N$($E$) denotes the number of elements of E. Prove that $\mu$ is measurable on $\mathcal A$.
I have already proved the first two conditions for $\mu$ to be measurable on $\mathcal A$.
However, I am stuck on proving the third condition [i.e $\mu($$\bigcup_{n=1}^{\infty}E_n$) $=$ $\sum_{n=1}^{\infty}\mu$($E_n$)]
This is what I'm thinking: Let $\{E_n\}_{n=1}^{\infty}$ be a sequence of pairwise disjoint sets, and consider the following three cases:
1.) Suppose for some $k$ $\in$ $\Bbb N$, we have $N$($E_k$) = $\infty$.
2.) Suppose that for all $n$ $\in$ $\Bbb N$, we have $N$($E_n$) $<$ $\infty$, and that there are infinitely many $n$ such that $N$($E_n$) $>$ $0$.
3.) Suppose that for all $n$ $\in$ $\Bbb N$, we have $N$($E_n$) $<$ $\infty$ and that there are finitely many $n$ such that $N$($E_n$) $>$ $0$.
