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 &sigma;-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 &sigma;-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.

Another equivalent criterion of the tightness of a collection $$M$$ is sequentially weakly compact. We say the family $$M$$ of probability measures is sequentially weakly compact if for every sequence $$\left\{\mu_n\right\}$$ from the family, there is a subsequence of measures that converges weakly to some probability measure $$\mu$$. It can be shown that a family of measure is tight if and only if it is sequentially weakly compact.

Compact spaces
If $$X$$ is a metrizable 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 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


 * 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 $$(\mu_{\delta})_{\delta > 0}$$ 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


 * $$\limsup_{\delta \downarrow 0} \delta \log \mu_{\delta} (X \setminus K_{\varepsilon}) < - \varepsilon.$$