# Tight measure

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2010 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]

inner regular measure, Radon measure

A measure $\mu$ defined on the σ-algebra $\mathcal{B} (X)$ of Borel sets of a topological Hausdorff space $X$ which is locally finite (i.e. for any point $x\in X$ there is a neighbourhood which has finite measure) and having the following property: \begin{equation}\label{e:tight} \mu (B)= \sup \{\mu(K): K\subset B, K \mbox{ compact}\} \qquad \forall B\in \mathcal{B}\,. \end{equation} Many authors also use the terminology Radon for such measures (see Radon measure): however some other authors require also that $\mu$ is finite to call it Radon.

On a locally compact space $X$ any tight finite measure $\mu$ is also outer regular, i.e. \begin{equation}\label{e:outer} \mu (N) = \inf \{\mu (U): U\supset N,\, U \mbox{ open}\}\, \qquad \forall N\in \mathcal{B}, \end{equation} (cp. therefore with Definition 2.2.5 of [Fe] and Definition 1.5 of [Ma]).

If $X$ is a separable complete metric space, every probability measure on $X$ for which the Borel sets are measurable is tight (Ulam's tightness theorem), cp. with [To]. The terminology "tight" was introduced by L. LeCam, see [LC].

More generally, let $\mathcal{A}\supset \mathcal{K}$ be two pavings on a set $X$ and $\beta$ a set function on $\mathcal{A}$ taking values in $[0, \infty]$. Then $\beta$ is tight with respect to $\mathcal{K}$ if $\beta (A_1) - \beta (A_2) = \sup \{\beta (K): K \subset A_1\setminus A_2, K\in \mathcal{K}\}\, \qquad \forall A_1, A_2 \in \mathcal{A} \mbox{ with } A_1\supset A_2\, .$

How to Cite This Entry:
Tight measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tight_measure&oldid=42665