# 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 topological space, and let Σ be a σ-algebra on X that contains the topology T. (Thus, every open subset of X is a measurable set and Σ 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 Σ. The collection M is called tight (or sometimes uniformly tight) if, for any ε > 0, there is a compact subset Kε of X such that, for all measures μ 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 M consists of a single measure μ, then (depending upon the author) μ 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 ${\displaystyle [0,\omega _{1}]}$ 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 X is a 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 R with its usual Borel topology. Let δx denote the Dirac measure, a unit mass at the point x in R. The collection

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

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

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

is tight: the compact interval [0, 1] will work as Kη for any η > 0. In general, a collection of Dirac delta measures on Rn is tight if, and only if, the collection of their supports is bounded.

### A collection of Gaussian measures

Consider n-dimensional Euclidean space Rn with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

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

where the measure γi has expected value (mean) μi in Rn and variance σi2 > 0. Then the collection Γ is tight if, and only if, the collections ${\displaystyle \{\mu _{i}|i\in I\}\subseteq \mathbb {R} ^{n}}$ and ${\displaystyle \{\sigma _{i}^{2}|i\in I\}\subseteq \mathbb {R} }$ 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 generalization of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures (μδ)δ>0 on a Hausdorff topological space X is said to be exponentially tight if, for any η > 0, there is a compact subset Kη of X such that

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

## References

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