# example of function not Lebesgue Measurable with measurable level sets

$$f(x)=\{\begin{array}{cc}x\hfill & \text{if}x\notin V\hfill \\ 2+x\hfill & \text{if}x\in V\hfill \end{array}$$ |

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

$${f}^{-1}\left(\left\{x\right\}\right)=\{\begin{array}{cc}\{x\}\hfill & \text{if}\mathrm{\hspace{0.25em}0}\le x\le 1\wedge x\notin V\hfill \\ \{2-x\}\hfill & \text{if}\mathrm{\hspace{0.25em}2}\le x\le 3\wedge x-2\in V\hfill \\ \{\}\hfill & \text{otherwise}\hfill \end{array}$$ |

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

Title | example of function not Lebesgue Measurable with measurable level sets |
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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 |