# 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 ${\displaystyle (X,T)}$  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 ${\displaystyle \mu \in M}$ ,

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

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.

## Examples

### Compact spaces

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

### Polish spaces

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

### A collection of point masses

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

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

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

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

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

### A collection of Gaussian measures

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

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

where the measure ${\displaystyle \gamma _{i}}$  has expected value (mean) ${\displaystyle m_{i}\in \mathbb {R} ^{n}}$  and covariance matrix ${\displaystyle C_{i}\in \mathbb {R} ^{n\times n}}$ . Then the collection ${\displaystyle \Gamma }$  is tight if, and only if, the collections ${\displaystyle \{m_{i}|i\in I\}\subseteq \mathbb {R} ^{n}}$  and ${\displaystyle \{C_{i}|i\in I\}\subseteq \mathbb {R} ^{n\times n}}$  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 ${\displaystyle (\mu _{\delta })_{\delta >0}}$  on a Hausdorff topological space ${\displaystyle X}$  is said to be exponentially tight if, for any ${\displaystyle \varepsilon >0}$ , there is a compact subset ${\displaystyle K_{\varepsilon }}$  of ${\displaystyle X}$  such that

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

## References

• Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
• Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015 (See chapter 2)