Suppose that $A ⊆ E ⊆ B$, where $A$ and $B$ are Lebesgue measurable subsets on $R^n.$ If m(A) = m(B). Give an example showing that the same conclusion does not hold if A and B have infinite measure, then $E$ is measurable.
I can't find an example like that, I know if A and B have infinite measure, then axioms of equality will fails. Thus, we can't get $E$ is measurable.
Another one of mine question is that the union of two nonmeasurable is measurable or nonmeasurable? Or both be possible?
If we consider Vitali set, we have $R=V∪V^∁$. Therefore, it is measurable. Will it always true for all case?
First question: Take $A=(0,\infty), B=\mathbb R$ and $E=A\cup F$ where $F\subset (-\infty, 0)$ is non-measurable.
Second question: Consider the union of a non-measurable subset of $(0,1)$ and a non-measurable subset of $(1,2)$.