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

[edit] Definition

Let $(Ω, T)$ be a topological space, and let $M$ be a collection of measures defined on $σ(T)$ , the smallest sigma algebra containing the topology $T$ (so that every open set in $Ω$ is measurable). The collection $M$ is called tight if, for any $\varepsilon > 0$ , there is a compact set $K_{\varepsilon} \subseteq \Omega$ such that, for all measures $\mu \in M$ , $\mu (\Omega \setminus K_{\varepsilon}) < \varepsilon$ . Very often, the measures in question are probability measures, so the last part can be written as $\mu (K_{\varepsilon}) \geq 1 - \varepsilon$ .

[edit] Examples

[edit] Compact spaces

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

[edit] A collection of point masses

Consider the real line $\mathbb{R}$ with its usual Borel topology. Let $δ x$ denote the Dirac delta, 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 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 $K_{\varepsilon}$ 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.

[edit] A collection of Gaussian measures

Consider $n$ -dimensional Euclidean space $\mathbb{R}^{n}$ with its usual Borel topology and sigma algebra. Consider a collection of Gaussian measures $\{ \gamma_{i} | i \in I \}$ , where the measure $γ i$ has expected value (mean) $\mu_{i} \in \mathbb{R}^{n}$ and variance $\sigma_{i}^{2} > 0$ . Then the collection $\{ \gamma_{i} | i \in I \}$ 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