Tight Measure

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 $K_$ of $X$ such that, for all measures $\mu \in M$,

:$, \mu, (X \setminus K_) < \varepsilon.$

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

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

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 ,\omega_1/math> with its order topology, then there exists a measure $\mu$ 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 $\mathbb$ with its usual Borel topology. Let $\delta_$ denote the Dirac measure, a unit mass at the point $x$ in $\mathbb$. The collection

:$M_ := \$

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

:$M_ := \$

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

A collection of Gaussian measures

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

:$\Gamma = \,$

where the measure $\gamma_$ has expected value (mean) $m_ \in \mathbb^$ and covariance matrix $C_ \in \mathbb^$. Then the collection $\Gamma$ is tight if, and only if, the collections $\ \subseteq \mathbb^$ and $\ \subseteq \mathbb^$ 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
* Lévy–Prokhorov metric
* Weak convergence of measures
* Tightness in classical Wiener space
* Tightness in Skorokhod space

Exponential tightness

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

:$\limsup_ \delta \log \mu_ (X \setminus K_) < - \varepsilon.$

References

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* (See chapter 2)

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