# 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 *K*_{ε} of *X* such that, for all measures *μ* in *M*,

where 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 *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 with its order topology, then there exists a measure on it that is not inner regular. Therefore the singleton 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 *δ*_{x} denote the Dirac measure, a unit mass at the point *x* in **R**. The collection

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 *δ*_{n}-measure zero for large enough *n*. On the other hand, the collection

is tight: the compact interval [0, 1] will work as *K*_{η} for any *η* > 0. In general, a collection of Dirac delta measures on **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 **R**^{n} with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

where the measure *γ*_{i} has expected value (mean) *μ*_{i} in **R**^{n} and variance *σ*_{i}^{2} > 0. Then the collection Γ is tight if, and only if, the collections and 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

- Finite-dimensional distribution
- Prokhorov's theorem
- Tightness in classical Wiener space
- Tightness in Skorokhod space

## Exponential tightness

A generalization of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures (*μ*_{δ})_{δ>0} on a Hausdorff topological space *X* is said to be **exponentially tight** if, for any *η* > 0, there is a compact subset *K*_{η} of *X* such that

## References

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