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 ) {\displaystyle (X,T)} be a Hausdorff space, and let Σ {\displaystyle \Sigma } be a σ-algebra on X {\displaystyle X} that contains the topology T {\displaystyle T} . (Thus, every open subset of X {\displaystyle X} is a measurable set and Σ {\displaystyle \Sigma } is at least as fine as the Borel σ-algebra on X {\displaystyle X} .) Let M {\displaystyle M} be a collection of (possibly signed or complex) measures defined on Σ {\displaystyle \Sigma } . The collection M {\displaystyle M} is called tight (or sometimes uniformly tight) if, for any ε > 0 {\displaystyle \varepsilon >0} , there is a compact subset K ε {\displaystyle K_{\varepsilon }} of X {\displaystyle X} such that, for all measures μ ∈ M {\displaystyle \mu \in M} ,
| μ | ( X ∖ K ε ) < ε . {\displaystyle |\mu |(X\setminus K_{\varepsilon })<\varepsilon .}
where | μ | {\displaystyle |\mu |} is the total variation measure of μ {\displaystyle \mu } . Very often, the measures in question are probability measures, so the last part can be written as
μ ( K ε ) > 1 − ε . {\displaystyle \mu (K_{\varepsilon })>1-\varepsilon .\,}
If a tight collection M {\displaystyle M} consists of a single measure μ {\displaystyle \mu } , then (depending upon the author) μ {\displaystyle \mu } may either be said to be a tight measure or to be an inner regular measure. If Y {\displaystyle Y} is an X {\displaystyle X} -valued random variable whose probability distribution on X {\displaystyle X} is a tight measure then Y {\displaystyle Y} is said to be a separable random variable or a Radon random variable. Another equivalent criterion of the tightness of a collection M {\displaystyle M} is sequential weak compactness. We say the family M {\displaystyle M} of probability measures is sequentially weakly compact if for every sequence { μ n } {\displaystyle \left\{\mu _{n}\right\}} from the family, there is a subsequence of measures that converges weakly to some probability measure μ {\displaystyle \mu } . It can be shown that a family of measures is tight if and only if it is sequentially weakly compact.
Examples
Compact spaces If X {\displaystyle X} is a metrizable compact space, then every collection of (possibly complex) measures on X {\displaystyle X} is tight. This is not necessarily so for non-metrisable compact spaces. If we take [ 0 , ω 1 ] {\displaystyle [0,\omega _{1}]} with its order topology, then there exists a measure μ {\displaystyle \mu } on it that is not inner regular. Therefore, the singleton { μ } {\displaystyle \{\mu \}} is not tight.
Polish spaces If X {\displaystyle X} is a Polish space, then every finite measure on X {\displaystyle X} is tight; this is Ulam's theorem. Furthermore, by Prokhorov's theorem, a collection of probability measures on X {\displaystyle 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 {\displaystyle \mathbb {R} } with its usual Borel topology. Let δ x {\displaystyle \delta _{x}} denote the Dirac measure, a unit mass at the point x {\displaystyle x} in R {\displaystyle \mathbb {R} } . The collection
M 1 := { δ n ∣ n ∈ N } {\displaystyle M_{1}:=\{\delta _{n}\mid n\in \mathbb {N} \}}
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