Which of the following is true?
1) $\{ x\in \mathbb{R}: x^6-6x^4 \text{ is irrational}\}$ is a Lebesgue-measurable subset of $\mathbb{R}$
2) If $A$ is a lebesgue measurable subset of $\mathbb{R}$ and $B$ is a lebesgue non-measurable subset of $\mathbb{R}$ such that $B \subset A$, then it is necessary that $m^*(A\setminus B)>0$
3) If $A$ and $B$ are disjoint subset of $\mathbb{R}$ such that $A$ is lebesgue measurable and $B$ is a lebesgue non-measurable, then it is possible that $m^*(A \cup B)< m^*(A)+m^*(B)$
Remarks:
1) Can we think given set as inverse image of continuous function where $f(x)=x^6-6x^4$, Please help
3) my intuition says is not true for non-measurability of $B$ , Please help
I have no idea about 2, please help
I'm not an expert, but I think that I found a proof of 2 today.
Assume that $A$ is measurable, $B \subseteq A$ is not measurable, and $m^*(A \setminus B) = 0.$ If I get it correctly then all $m^*$-null sets are $m$-measurable and (obviously) are $m$-null. Thus $A \setminus B$ is measurable. This however implies that $B = A \setminus (A \setminus B)$ is measurable. Contradiction! Thus, $m^*(A \setminus B) > 0.$