Is it possible to write the preimage of a Lebesgue integrable function as the union of nonmeasurable sets?
I guess this is sort of posed below in the opposite way:
Let's say $f$ is a lebesgue measurable function such that $f \neq 0$ for any $x$ in some arbitrary set $E$, and $m(E) > 0$. Can I gurantee that there exists sets $P \cup N = E$ such that $f > 0$ if $x \in P$ and $f < 0$ if $x \in N$ such that both sets are measurable? And if so, how?
EDIT: Suppose $f$ was originally defined on $\mathbb{R}^1$ of which $E$ is a measurable subset, with measure greater than zero.