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, there is a compact subset of X such that, for all measures \mu \in M, 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 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 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.

Examples

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

A collection of Gaussian measures

Consider n-dimensional Euclidean space with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures where the measure \gamma_{i} has expected value (mean) and covariance matrix. Then the collection \Gamma 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

Exponential tightness

A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures on a Hausdorff topological space X is said to be exponentially tight if, for any, there is a compact subset of X such that