# 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".

## Definitions

Let $(X,T)$  be a Hausdorff space, and let $\Sigma$  be a σ-algebra on $X$  that contains the topology $T$ . (Thus, every open subset of $X$  is a measurable set and $\Sigma$  is at least as fine as the Borel σ-algebra on $X$ .) Let $M$  be a collection of (possibly signed or complex) measures defined on $\Sigma$ . The collection $M$  is called tight (or sometimes uniformly tight) if, for any $\varepsilon >0$ , there is a compact subset $K_{\varepsilon }$  of $X$  such that, for all measures $\mu \in M$ ,

$|\mu |(X\setminus K_{\varepsilon })<\varepsilon .$

where $|\mu |$  is the total variation measure of $\mu$ . Very often, the measures in question are probability measures, so the last part can be written as

$\mu (K_{\varepsilon })>1-\varepsilon .\,$

If a tight collection $M$  consists of a single measure $\mu$ , then (depending upon the author) $\mu$  may either be said to be a tight measure or to be an inner regular measure.

If $Y$  is an $X$ -valued random variable whose probability distribution on $X$  is a tight measure then $Y$  is said to be a separable random variable or a Radon random variable.

## Examples

### Compact spaces

If $X$  is a metrisable compact space, then every collection of (possibly complex) measures on $X$  is tight. This is not necessarily so for non-metrisable compact spaces. If we take $[0,\omega _{1}]$  with its order topology, then there exists a measure $\mu$  on it that is not inner regular. Therefore, the singleton $\{\mu \}$  is not tight.

### Polish spaces

If $X$  is a compact Polish space, then every probability measure on $X$  is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on $X$  is tight if and only if it is precompact in the topology of weak convergence.

### A collection of point masses

Consider the real line $\mathbb {R}$  with its usual Borel topology. Let $\delta _{x}$  denote the Dirac measure, a unit mass at the point $x$  in $\mathbb {R}$ . The collection

$M_{1}:=\{\delta _{n}|n\in \mathbb {N} \}$

is not tight, since the compact subsets of $\mathbb {R}$  are precisely the closed and bounded subsets, and any such set, since it is bounded, has $\delta _{n}$ -measure zero for large enough $n$ . On the other hand, the collection

$M_{2}:=\{\delta _{1/n}|n\in \mathbb {N} \}$

is tight: the compact interval $[0,1]$  will work as $K_{\varepsilon }$  for any $\varepsilon >0$ . In general, a collection of Dirac delta measures on $\mathbb {R} ^{n}$  is tight if, and only if, the collection of their supports is bounded.

### A collection of Gaussian measures

Consider $n$ -dimensional Euclidean space $\mathbb {R} ^{n}$  with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

$\Gamma =\{\gamma _{i}|i\in I\},$

where the measure $\gamma _{i}$  has expected value (mean) $m_{i}\in \mathbb {R} ^{n}$  and covariance matrix $C_{i}\in \mathbb {R} ^{n\times n}$ . Then the collection $\Gamma$  is tight if, and only if, the collections $\{m_{i}|i\in I\}\subseteq \mathbb {R} ^{n}$  and $\{C_{i}|i\in I\}\subseteq \mathbb {R} ^{n\times n}$  are both bounded.

## Tightness and convergence

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

## Exponential tightness

A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures $(\mu _{\delta })_{\delta >0}$  on a Hausdorff topological space $X$  is said to be exponentially tight if, for any $\varepsilon >0$ , there is a compact subset $K_{\varepsilon }$  of $X$  such that

$\limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(X\setminus K_{\varepsilon })<-\varepsilon .$