example of function not Lebesgue Measurable with measurable level sets

Consider $V$ as in Vitali’s theorem. Define the function $f:[0,1]\to[0,+\infty[$ by:

$f(x)=\begin{cases}x&\text{if}\>x\notin V\\ 2+x&\text{if}\>x\in V\end{cases}$

The level sets of $f$ will either be the empty set, or a singleton and thus measurable.

$f^{-1}\left(\left\{x\right\}\right)=\begin{cases}\{x\}&\text{if}\>0\leq x\leq 1% \wedge x\notin V\\ \{2-x\}&\text{if}\>2\leq x\leq 3\wedge x-2\in V\\ \{\}&\text{otherwise}\end{cases}$

$f$ is not a measurable function since $f^{-1}([2,+\infty[)=V$ and $V$ is not a measurable set.

Title	example of function not Lebesgue Measurable with measurable level sets
Canonical name	ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets
Date of creation	2013-03-22 15:51:22
Last modified on	2013-03-22 15:51:22
Owner	cvalente (11260)
Last modified by	cvalente (11260)
Numerical id	7
Author	cvalente (11260)
Entry type	Example
Classification	msc 28B15
Related topic	measurableFunctions
Related topic	VitalisTheorem
Related topic	MeasurableFunctions