Suppose $f:[0,1]\to \mathbb R$ is Lebesgue measurable and $f(x)\in\{0,1\} $for almost all $x\in[0,1]$.
Do we always have $f $ being represented by some indicator function $\chi_{A} $, where $A$ is the union of finitely or countably many disjoint measurable subsets of $[0,1]$?
Thanks.