# measurable space

A *measurable space ^{}* is a set $E$ together with a collection

^{}$\mathcal{B}$ of subsets of $E$ which is a sigma algebra.

The elements of $\mathcal{B}$ are called *measurable sets*.

A measurable space is the correct object on which to define a measure^{}; $\mathcal{B}$ will be the collection of sets which actually have a measure. We normally want to ensure that $\mathcal{B}$ contains all the sets we will ever want to use. We usually cannot take $\mathcal{B}$ to be the collection of all subsets of $E$ because the axiom of choice^{} often allows one to construct sets that would lead to a contradiction^{} if we gave them a measure (even zero). For the real numbers, Vitali’s theorem states that $\mathcal{B}$ cannot be the collection of all subsets if we hope to have a measure that returns the length of an open interval.

Title | measurable space |
---|---|

Canonical name | MeasurableSpace |

Date of creation | 2013-03-22 11:57:30 |

Last modified on | 2013-03-22 11:57:30 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 11 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 28A33 |

Defines | measurable set |