Let X be an uncountable set. And let $$\mathcal{A} =\{A\subset X: A \text{ is countable or } X-A \text{ is countable}\}.$$ Then it can be shown that $\mathcal{A}$ is a $\sigma$-algebra. Define $\mu:\mathcal{A}\rightarrow [0,\infty]$ by $\mu(A)=0$ if $A$ is countable or 1 otherwise. The task is to show that $\mu$ so defined is a measure on $\mathcal{A}$.
It is clear that $\mu(\emptyset)=0$. Now we want to show that $\mu$ is countably additive on disjoint sets from $\mathcal{A}$. I'm first trying to show this for two disjoint sets $A$ and $B$ in $\mathcal{A}$. I am running into a problem if I assume that both $X-A$ and $X-B$ is countable.
For if this happens then, $B\subset X-A$ implies $B$ is countable. But both $B$ and $X-B$ cannot be countable, since then $X=B\cup (X-B)$ will be countable, running contrary to the assumption that $X$ is uncountable. How should I address this case?
Best Answer
All you need to realize is that if $A$ and $B$ are both co-countable, then $A\cap B\ne\varnothing$. Thus, if you have pairwise disjoint measurable sets, at most one of them can be co-countable, and the rest must all be countable.