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 Hausdorff 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 ${\displaystyle \mu \in M}$,

where ${\displaystyle |\mu |}$ is the total variation measure of ${\displaystyle \mu }$. Very often, the measures in question are probability measures, so the last part can be written as

If a tight collection ${\displaystyle M}$ consists of a single measure ${\displaystyle \mu }$, then (depending upon the author) ${\displaystyle \mu }$ may either be said to be a tight measure or to be an inner regular measure.

If ${\displaystyle Y}$ is an ${\displaystyle X}$-valued random variable whose probability distribution on ${\displaystyle X}$ is a tight measure then ${\displaystyle Y}$ is said to be a separable random variable or a Radon random variable.

Another equivalent criterion of the tightness of a collection ${\displaystyle M}$ is sequential weak compactness. We say the family ${\displaystyle M}$ of probability measures is sequentially weakly compact if for every sequence ${\displaystyle \left\{\mu _{n}\right\}}$ from the family, there is a subsequence of measures that converges weakly to some probability measure ${\displaystyle \mu }$. It can be shown that a family of measures is tight if and only if it is sequentially weakly compact.

If ${\displaystyle X}$ is a metrizable compact space, then every collection of (possibly complex) measures on ${\displaystyle X}$ is tight. This is not necessarily so for non-metrisable compact spaces. If we take ${\displaystyle [0,\omega _{1}]}$ with its order topology, then there exists a measure ${\displaystyle \mu }$ on it that is not inner regular. Therefore, the singleton ${\displaystyle \{\mu \}}$ is not tight.

If ${\displaystyle X}$ is a Polish space, then every finite measure on ${\displaystyle X}$ is tight; this is Ulam's theorem. Furthermore, by Prokhorov's theorem, a collection of probability measures on ${\displaystyle X}$ is tight if and only if it is precompact in the topology of weak convergence.