# a Lebesgue measurable but non-Borel set

We give an example of a subset of the real numbers which is Lebesgue measurable, but not Borel measurable.

Let $S$ consist of the set of all irrational real numbers with continued fraction^{} of the form

$$x={a}_{0}+\frac{1}{{a}_{1}+{\displaystyle \frac{1}{{a}_{2}+{\displaystyle \frac{1}{\mathrm{\ddots}}}}}}$$ |

such that there exists an infinite^{} sequence^{} $$ where each ${a}_{{i}_{k}}$ divides ${a}_{{i}_{k+1}}$.
It can be shown that this set is Lebesgue measurable, but not Borel measurable.

In fact, it can be shown that $S$ is an analytic set^{} (http://planetmath.org/AnalyticSet2), meaning that it is the image of a continuous function^{} $f:X\to \mathbb{R}$ for some Polish space^{} $X$ and, consequently, $S$ is a universally measurable set.

This example is due to Lusin (1927).

Title | a Lebesgue measurable but non-Borel set |
---|---|

Canonical name | ALebesgueMeasurableButNonBorelSet |

Date of creation | 2013-03-22 18:37:01 |

Last modified on | 2013-03-22 18:37:01 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 28A05 |

Classification | msc 28A20 |