# existence of the essential supremum

We state the existence of the essential supremum^{} for a set $\mathcal{S}$ of extended real valued functions on a $\sigma $-finite (http://planetmath.org/SigmaFinite) measure space^{} $(\mathrm{\Omega},\mathcal{F},\mu )$.

###### Theorem.

Suppose that the measure space $\mathrm{(}\mathrm{\Omega}\mathrm{,}\mathrm{F}\mathrm{,}\mu \mathrm{)}$ is $\sigma $-finite. Then, the essential supremum of $\mathrm{S}$ exists. Furthermore, if $\mathrm{S}$ is nonempty then there exists a sequence ${\mathrm{(}{f}_{n}\mathrm{)}}_{n\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots}}$ in $\mathrm{S}$ such that

$$\mathrm{esssup}\mathcal{S}=\underset{n}{sup}{f}_{n}.$$ | (1) |

Note that, by reversing the inequalities^{}, this result also applies to the essential infimum, except that equation (1) is replaced by

$$\mathrm{essinf}\mathcal{S}=\underset{n}{inf}{f}_{n}.$$ |

Title | existence of the essential supremum |
---|---|

Canonical name | ExistenceOfTheEssentialSupremum |

Date of creation | 2013-03-22 18:39:22 |

Last modified on | 2013-03-22 18:39:22 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 28A20 |

Related topic | EssentialSupremum |