# tight and relatively compact measures

Tight and relatively compact measuresFernando Sanz

###### Definition 1.

Let $\mathcal{M}=\{{\mu}_{i},i\in I\}$ be a family of finite measures on the Borel subsets of a metric space $\mathrm{\Omega}$. We say that $\mathcal{M}$ is tight iff for each $\u03f5>0$ there is a compact set $K$ such that $$ for all $i$. We say that $\mathcal{M}$ is relatively compact iff each sequence in $\mathcal{M}$ has a subsequence converging weakly to a finite measure on $\mathcal{B}(\mathrm{\Omega})$.

If $\{{F}_{i},i\in I\}$ is a family of distribution functions^{},
relative compactness or tightness of $\{{F}_{i}\}$ refers to relative
compactness or tightness of the corresponding measures.

###### Theorem.

Let $\mathrm{\{}{F}_{i}\mathrm{,}i\mathrm{\in}I\mathrm{\}}$ be a family of distribution functions with $$ for all $i$. The family is tight iff it is relatively compact.

###### Proof.

Coming soon…(needs other theorems before) ∎

Title | tight and relatively compact measures |
---|---|

Canonical name | TightAndRelativelyCompactMeasures |

Date of creation | 2013-03-22 17:19:37 |

Last modified on | 2013-03-22 17:19:37 |

Owner | fernsanz (8869) |

Last modified by | fernsanz (8869) |

Numerical id | 6 |

Author | fernsanz (8869) |

Entry type | Definition |

Classification | msc 60F05 |

Related topic | LindebergsCentralLimitTheorem |