# F non-measurable but |f| and f^2 are

Tags:
1. Feb 12, 2017

### diddy_kaufen

1. The problem statement, all variables and given/known data

Find a function in a measurable space that is non-measurable, but |f| and f2 are measurable.

2. Relevant equations

None.

3. The attempt at a solution

I am trying to understand the following answer to the problem:

(source: http://math.stackexchange.com/a/1233792/413398)

I do not understand why for all subsets $S \subseteq \mathbb{R}$, we have $$(|f|)^{-1}(S)=(f^2)^{-1}(S)=X \text{ or } \varnothing$$

2. Feb 12, 2017

### Q.B.

Hi,

$f^2$ or $abs(f)$ both send $a$ and $b$ to $1$. So $1$ has $a$ and $b$ as antecedents, and any other real has no antecedent. So if $S$ contains $1$ it has $X=\{a,b\}$ as reciprocal image, and the empty set if it doesn't contain it.