# limit superior of sets

Let ${A}_{1},{A}_{2},\mathrm{\dots}$ be a sequence of sets. The limit superior of sets is defined by

$$lim\; sup{A}_{n}=\bigcap _{n=1}^{\mathrm{\infty}}\bigcup _{k=n}^{\mathrm{\infty}}{A}_{k}.$$ |

It is easy to see that $x\in lim\; sup{A}_{n}$ if and only if $x\in {A}_{n}$ for infinitely many values of $n$. Because of this, in probability theory the notation $[{A}_{n}\mathrm{i}.\mathrm{o}.]$ is often used to refer to $lim\; sup{A}_{n}$, where i.o. stands for infinitely often.

The limit inferior of sets is defined by

$$lim\; inf{A}_{n}=\bigcup _{n=1}^{\mathrm{\infty}}\bigcap _{k=n}^{\mathrm{\infty}}{A}_{k},$$ |

and it can be shown that $x\in lim\; inf{A}_{n}$ if and only if $x$ belongs to ${A}_{n}$ for all but finitely many values of $n$.

Title | limit superior of sets |
---|---|

Canonical name | LimitSuperiorOfSets |

Date of creation | 2013-03-22 13:13:22 |

Last modified on | 2013-03-22 13:13:22 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 8 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 28A05 |

Classification | msc 60A99 |

Defines | limit inferior of sets |

Defines | infinitely often |

Defines | i.o. |