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

[diddy_kaufen]

Homework Statement

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

Homework Equations

None.

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 $$(|f|)^{-1}(S)=(f^2)^{-1}(S)=X \text{ or } \varnothing$$

 

[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.