# Tightness of measures

In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not “escape to infinity“. This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (March 2016)

## . . . Tightness of measures . . .

Let

${displaystyle (X,T)}$ be a Hausdorff space, and let

${displaystyle Sigma }$ be a σ-algebra on

${displaystyle X}$ that contains the topology

${displaystyle T}$ . (Thus, every open subset of

${displaystyle X}$ is a measurable set and

${displaystyle Sigma }$ is at least as fine as the Borel σ-algebra on

${displaystyle X}$ .) Let

${displaystyle M}$ be a collection of (possibly signed or complex) measures defined on

${displaystyle Sigma }$ . The collection

${displaystyle M}$ is called tight (or sometimes uniformly tight) if, for any

${displaystyle varepsilon >0}$ , there is a compact subset

${displaystyle K_{varepsilon }}$ of

${displaystyle X}$ such that, for all measures

${displaystyle mu in M}$ ,

${displaystyle |mu |(Xsetminus K_{varepsilon })$ where

${displaystyle |mu |}$ is the total variation measure of

${displaystyle mu }$ . Very often, the measures in question are probability measures, so the last part can be written as

${displaystyle mu (K_{varepsilon })>1-varepsilon .,}$ If a tight collection

${displaystyle M}$ consists of a single measure

${displaystyle mu }$ , then (depending upon the author)

${displaystyle mu }$ may either be said to be a tight measure or to be an inner regular measure.

If

${displaystyle Y}$ is an

${displaystyle X}$ -valued random variable whose probability distribution on

${displaystyle X}$ is a tight measure then

${displaystyle Y}$

is said to be a separable random variable or a Radon random variable.