Tight measure

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2020 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 \[ \beta (A_1) - \beta (A_2) = \sup \{\beta (K): K \subset A_1\setminus A_2, K\in \mathcal{K}\}\, \qquad \forall A_1, A_2 \in \mathcal{A} \mbox{ with } A_1\supset A_2\, . \]


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[LC] L. LeCam, "Convergence in distribution of probability processes" Univ. of Calif. Publ. Stat. , 2 : 11 (1957) pp. 207–236
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[OU] J.C. Oxtoby, S. Ulam, "On the existence of a measure invariant under a transformation" Ann. of Math. , 40 (1939) pp. 560–566
[Sc] L. Schwartz, "Radon measures on arbitrary topological spaces and cylindrical measures". Tata Institute of Fundamental Research Studies in Mathematics, No. 6. Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. MR0426084 Zbl 0298.2800
[To] F. Topsøe, "Topology and measure" , Springer (1970)
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