Difference between revisions of "Tight measure"
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[[Category:Classical measure theory]] | [[Category:Classical measure theory]] | ||
| − | + | ''inner regular measure, Radon measure'' | |
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| − | A measure $\mu$ defined on the [[Algebra of sets|σ-algebra]] $\mathcal{B} (X)$ of [[Borel set | + | A measure $\mu$ defined on the [[Algebra of sets|σ-algebra]] $\mathcal{B} (X)$ of [[Borel set]]s 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} | \begin{equation}\label{e:tight} | ||
| − | \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ | + | \mu (B)= \sup \{\mu(K): K\subset B, K \mbox{ compact}\} \qquad \forall B\in \mathcal{B}\,. |
\end{equation} | \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. | 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. | ||
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On a locally compact space $X$ any tight finite measure $\mu$ is also outer regular, i.e. | On a locally compact space $X$ any tight finite measure $\mu$ is also outer regular, i.e. | ||
\begin{equation}\label{e:outer} | \begin{equation}\label{e:outer} | ||
| − | \mu (N) = \inf \{\mu (U): U\supset N, \mbox{ | + | \mu (N) = \inf \{\mu (U): U\supset N,\, U \mbox{ open}\}\, \qquad \forall N\in \mathcal{B}, |
\end{equation} | \end{equation} | ||
(cp. therefore with Definition 2.2.5 of {{Cite|Fe}} and Definition 1.5 of {{Cite|Ma}}). | (cp. therefore with Definition 2.2.5 of {{Cite|Fe}} and Definition 1.5 of {{Cite|Ma}}). | ||
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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 {{Cite|To}}. The terminology "tight" was introduced by L. LeCam, see {{Cite|LC}}. | 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 {{Cite|To}}. The terminology "tight" was introduced by L. LeCam, see {{Cite|LC}}. | ||
| − | More generally, let $\mathcal{A}\supset \mathcal{K}$ be two [[ | + | More generally, let $\mathcal{A}\supset \mathcal{K}$ be two [[paving]]s on a set $X$ and |
$\beta$ a [[Set function|set function]] on $\mathcal{A}$ taking values in $[0, \infty]$. Then $\beta$ is tight with respect to $\mathcal{K}$ if | $\beta$ a [[Set function|set function]] on $\mathcal{A}$ taking values in $[0, \infty]$. Then $\beta$ is tight with respect to $\mathcal{K}$ if | ||
\[ | \[ | ||
\beta (A_1) - \beta (A_2) = \sup \{\beta (K): K \subset A_1\setminus A_2, K\in \mathcal{K}\}\, \qquad | \beta (A_1) - \beta (A_2) = \sup \{\beta (K): K \subset A_1\setminus A_2, K\in \mathcal{K}\}\, \qquad | ||
| − | \forall A_1, A_2 \in \mathcal{A} \mbox{ with | + | \forall A_1, A_2 \in \mathcal{A} \mbox{ with } A_1\supset A_2\, . |
\] | \] | ||
Latest revision as of 22:40, 31 December 2017
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 \[ \beta (A_1) - \beta (A_2) = \sup \{\beta (K): K \subset A_1\setminus A_2, K\in \mathcal{K}\}\, \qquad \forall A_1, A_2 \in \mathcal{A} \mbox{ with } A_1\supset A_2\, . \]
References
| [Bo] | N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001 |
| [DS] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 |
| [Bic] | K. Bichteler, "Integration theory (with special attention to vector measures)" , Lect. notes in math. , 315 , Springer (1973) |
| [Bi] | P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201 |
| [Fe] | H. Federer, "Geometric measure theory", Springer-Verlag (1979). MR0257325 Zbl 0874.49001 |
| [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) |
Tight measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tight_measure&oldid=28977