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".

is at least as fine as the Borel σ-algebra on

be a collection of (possibly signed or complex) measures defined on

is the total variation measure of

Very often, the measures in question are probability measures, so the last part can be written as If a tight collection

-valued random variable whose probability distribution on

Another equivalent criterion of the tightness of a collection

is sequentially weakly compact.

of probability measures is sequentially weakly compact if for every sequence

It can be shown that a family of measure is tight if and only if it is sequentially weakly compact.

is a metrizable compact space, then every collection of (possibly complex) measures on

This is not necessarily so for non-metrisable compact spaces.

with its order topology, then there exists a measure

is a Polish space, then every probability measure on

Furthermore, by Prokhorov's theorem, a collection of probability measures on

is tight if and only if it is precompact in the topology of weak convergence.

with its usual Borel topology.

denote the Dirac measure, a unit mass at the point

is not tight, since the compact subsets of

is tight: the compact interval

In general, a collection of Dirac delta measures on

is tight if, and only if, the collection of their supports is bounded.

with its usual Borel topology and σ-algebra.

Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimension.

See A family of real-valued random variables

is tight if and only if there exists an almost surely finite random variable

denotes the stochastic order defined by

[1] A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory.

on a Hausdorff topological space