TITLE: Measurable functions in a countable co-countable $\sigma$-algebra
QUESTION [7 upvotes]: I found a interesting problem that says: Let $(X,S)$ be a measurable space where $X=\mathbb R$ and $S$ is the countable co-countable $\sigma$-algebra in $\mathbb R$, i.e. $S=\{A\subset\mathbb R: A\ \lor\ \mathbb R-A\ \ is\ finite\ or\ countable\}$. So the point is to describe the S-measurable functions $\ f:X\to \Bbb R^*$, where $\Bbb R^*$ are the extended real numbers.

REPLY [2 votes]: Let $f$ be a measurable function. Divide the line into a countable number of half-closed intervals and look at their preimages under $f$. Exactly one of them is uncountable. Break that interval into two and look at their preimages. Exactly one of them is uncountable. Continue like this. By the nested interval theorem there is exactly one point whose preimage is uncountable and cocountable. So the function is constant except on a countable set.