# 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“.

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## . . . 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.