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[parent] example of function not Lebesgue Measurable with measurable level sets (Example)

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

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

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

\begin{displaymath} f^{-1} \left( \left\{ x \right\} \right) = \begin{cases} \{x... ...x \le 3 \wedge x-2 \in V\ \{\} & \text{otherwise} \end{cases}\end{displaymath}

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



"example of function not Lebesgue Measurable with measurable level sets" is owned by cvalente.
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See Also: measurable function, Vitali's Theorem, measurable function


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Cross-references: measurable set, measurable function, measurable, singleton, empty set, level sets, function, Vitali's Theorem

This is version 4 of example of function not Lebesgue Measurable with measurable level sets, born on 2006-04-18, modified 2006-04-19.
Object id is 7841, canonical name is ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets.
Accessed 2558 times total.

Classification:
AMS MSC: 28B15 (Measure and integration :: Set functions, measures and integrals with values in abstract spaces :: Set functions, measures and integrals with values in ordered spaces)
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