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,

|\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 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 $[0,\omega_1]$ 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 R with its usual Borel topology. Let δ_x denote the Dirac measure, a unit mass at the point x in R. The collection

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 δ_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_η 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

\Gamma = \{ \gamma_{i} | i \in I \},

where the measure γ_i has expected value (mean) μ_i in Rⁿ and variance σ_i² > 0. Then the collection Γ is tight if, and only if, the collections $\{ \mu_{i} | i \in I \} \subseteq \mathbb{R}^{n}$ and $\{ \sigma_{i}^{2} | i \in I \} \subseteq \mathbb{R}$ 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 (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

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

References

Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.
Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR 1102015 (See chapter 2)