Tightness of measures

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

Let ${\displaystyle (X,T)}$ be a topological space, and let ${\displaystyle \Sigma }$ be a σ-algebra on ${\displaystyle X}$ that contains the topology ${\displaystyle T}$. (Thus, every open subset of ${\displaystyle X}$ is a measurable set and ${\displaystyle \Sigma }$ is at least as fine as the Borel σ-algebra on ${\displaystyle X}$.) Let ${\displaystyle M}$ be a collection of (possibly signed or complex) measures defined on ${\displaystyle \Sigma }$. The collection ${\displaystyle M}$ is called tight (or sometimes uniformly tight) if, for any ${\displaystyle \varepsilon >0}$, there is a compact subset ${\displaystyle K_{\varepsilon }}$ of ${\displaystyle X}$ such that, for all measures ${\displaystyle \mu \in M}$,

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