Let $\displaystyle A\subset\mathbb{R}$ is measurable set, and let $\displaystyle B=\{|x|:x\in A\}$. Show that $\displaystyle B$ is also measurable set.
Thanks.
Let C be $\displaystyle \{-|x| | x\in A\}$. Then it is easy to see that $\displaystyle C= \{-x | x \in B\}$ so if either B or C is measurable, so is the other. But it is also true that $\displaystyle A= B\cup C$ so that if neither B nor C is measurable, then A is not measurable.
But what if $\displaystyle A=(\mathbb{Q}\cap [-1,0])\cup ((0,1]-\mathbb{Q}) $? Then $\displaystyle B=[0,1]$, and $\displaystyle C=[-1,0]$, but $\displaystyle B\cup C$ is not equal to $\displaystyle A$. Sorry if I misunderstood something, od done something wrong.
Thanks.