tight and relatively compact measures


Tight and relatively compact measuresFernando Sanz

Definition 1.

Let ={μi,iI} be a family of finite measures on the Borel subsets of a metric space Ω. We say that is tight iff for each ϵ>0 there is a compact set K such that μi(Ω-K)<ϵ for all i. We say that is relatively compact iff each sequence in has a subsequence converging weakly to a finite measure on (Ω).

If {Fi,iI} is a family of distribution functionsMathworldPlanetmath, relative compactness or tightness of {Fi} refers to relative compactness or tightness of the corresponding measures.

Theorem.

Let {Fi,iI} be a family of distribution functions with Fi()-Fi(-)<M< 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