View source for Tight measure

{{TEX|done}}{{MSC|28A33}}
[[Category:Classical measure theory]]

''inner regular measure, Radon measure''

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}
\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 {{Cite|Fe}} and Definition 1.5 of {{Cite|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 {{Cite|To}}. The terminology  "tight"  was introduced by L. LeCam, see {{Cite|LC}}.

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

====References====
{|
|-
|valign="top"|{{Ref|Bo}}||   N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley   (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}}   {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}}   {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}}   {{ZBL|1095.28001}} {{ZBL|0156.06001}}
|-
|valign="top"|{{Ref|DS}}||   N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,  '''1'''  , Interscience (1958) {{MR|0117523}}
|-
|valign="top"|{{Ref|Bic}}||  K. Bichteler,   "Integration theory (with special attention to vector  measures)" , ''Lect. notes in math.'' , '''315''' , Springer  (1973)
|-
|valign="top"|{{Ref|Bi}}||   P. Billingsley, "Convergence of probability measures" , Wiley (1968)   {{MR|0233396}} {{ZBL|0172.21201}}
|-
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory", Springer-Verlag (1979). {{MR|0257325}} {{ZBL|0874.49001}}
|-
|valign="top"|{{Ref|LC}}|| L. LeCam,   "Convergence in distribution of probability processes"   ''Univ. of Calif. Publ. Stat.'' , '''2''' :  11  (1957)  pp. 207–236
|-
|valign="top"|{{Ref|Ma}}||   P. Mattila, "Geometry of sets and measures in euclidean spaces.   Cambridge Studies in Advanced Mathematics, 44. Cambridge University   Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
|-
|valign="top"|{{Ref|OU}}|| J.C. Oxtoby,   S. Ulam,   "On the existence of a measure invariant under  a transformation"  ''Ann. of Math.'' , '''40'''  (1939)  pp. 560–566
|-
|valign="top"|{{Ref|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.  {{MR|0426084}} {{ZBL|0298.2800}}  
|-
|valign="top"|{{Ref|To}}||  F. Topsøe,   "Topology and measure" , Springer  (1970)
|-
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