# Math Help - Measurable set

1. ## Measurable set

Let $A\subset\mathbb{R}$ is measurable set, and let $B=\{|x|:x\in A\}$. Show that $B$ is also measurable set.

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2. Let C be $\{-|x| | x\in A\}$. Then it is easy to see that $C= \{-x | x \in B\}$ so if either B or C is measurable, so is the other. But it is also true that $A= B\cup C$ so that if neither B nor C is measurable, then A is not measurable.

3. But what if $A=(\mathbb{Q}\cap [-1,0])\cup ((0,1]-\mathbb{Q})$? Then $B=[0,1]$, and $C=[-1,0]$, but $B\cup C$ is not equal to $A$. Sorry if I misunderstood something, od done something wrong.

Thanks.