Let $f: \mathbb{R} \to \mathbb{R}$. We have given the Sigma-algebra $\Sigma$, the collection of subsets $A \subset \mathbb{R}$ where either $A$ or $A^{c}$ is countable or finite. I want to show that f is measurable if and only if there exists a contable set $A$ such that for $x \in A^{c}$ $f(x...