Tightness of measures
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 Kε of X such that, for all measures μ in M,
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 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.
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
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 δn-measure zero for large enough n. On the other hand, the collection
is tight: the compact interval [0, 1] will work as Kη 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
Tightness and convergence
- Finite-dimensional distribution
- Prokhorov's theorem
- Tightness in classical Wiener space
- Tightness in Skorokhod space
A generalization of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures (μδ)δ>0 on a Hausdorff topological space X is said to be exponentially tight if, for any η > 0, there is a compact subset Kη of X such that
- 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)