Is a function measurable on subset if its measurable on set?

Question

Suppose we have a set $E=E_1\cup E_2$, where $E_1$ and $E_2$ are disjoint and measurable. $E$ is also measurable. Also, we have a function $f$ which is measurable on $E$. The question: is it measurable on $E_1$ and on $E_2$? Why?

Answer 1

Finally I've got the answer.

$\forall k \in \mathbb{R}$ write $E_1 = A \cup B, E_2 = C \cup D$ where $A = E_1(f<k), C = E_2(f<k)$, then $E = A \cup B \cup C \cup D$. We know that $E(f < k)$ is measurable, it means that $A \cup C$ is measurable. Let's look at $ E \setminus E_1 \setminus E(f < k) = D$. $D \in \mathcal{A} $ (sigma algebra), but $D = E_2(f >= k)$. Then $f$ is measurable on $E_2$. That's it.