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 ∈ M ,
| μ | ( X ∖ K ε ) < ε . where | μ | is the total variation measure of μ . Very often, the measures in question are probability measures, so the last part can be written as
μ ( K ε ) > 1 − ε . 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.
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 , ω 1 ] with its order topology, then there exists a measure μ on it that is not inner regular. Therefore, the singleton { μ } is not tight.
If X is a compact 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.
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 := { δ n | n ∈ 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 := { δ 1 / n | n ∈ N } is tight: the compact interval [ 0 , 1 ] will work as K ε for any ε > 0 . In general, a collection of Dirac delta measures on R n is tight if, and only if, the collection of their supports is bounded.
Consider n -dimensional Euclidean space R n with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures
Γ = { γ i | i ∈ I } , where the measure γ i has expected value (mean) m i ∈ R n and covariance matrix C i ∈ R n × n . Then the collection Γ is tight if, and only if, the collections { m i | i ∈ I } ⊆ R n and { C i | i ∈ I } ⊆ R n × n 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
Finite-dimensional distributionProkhorov's theoremLévy–Prokhorov metricweak convergence of measuresTightness in classical Wiener spaceTightness in Skorokhod spaceA strengthening 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 ε > 0 , there is a compact subset K ε of X such that
lim sup δ ↓ 0 δ log μ δ ( X ∖ K ε ) < − ε . 