Tightness of measures

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In mathematics, tightness is a concept in measure theory, the intuitive idea being that a given collection of measures does not "escape to infinity."

1 Definition
2 Examples
3 Tightness and convergence
4 References

[edit] Definition

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 measures defined on Σ. The collection M is called tight if, for any η > 0, there is a compact subset K_η of X such that, for all measures μ in M,

$\mu (X \setminus K_{\eta}) < \eta.$

Very often, the measures in question are probability measures, so the last part can be written as

μ(K η) > 1 - η.

[edit] Examples

[edit] Compact spaces

If X is a compact space, then every collection of probability measures on X is tight.

[edit] 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.

[edit] 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.

[edit] 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

[edit] 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.

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Category: Measure theory