Namespaces
Variants
Actions

Difference between revisions of "Tight measure"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928201.png" /> be a topological space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928202.png" /> the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928204.png" />-field generated by the open sets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928205.png" /> the [[Paving|paving]] (i.e. family of subsets) of all compact sets. A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928206.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928207.png" /> is tight if
+
''inner regular measure, Radon measure''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928208.png" /></td> </tr></table>
+
[[Category:Classical measure theory]]
  
A finite tight measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t0928209.png" /> is a [[Radon measure|Radon measure]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282010.png" /> is a separable complete metric space, every probability measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282011.png" /> is tight (Ulam's tightness theorem), [[#References|[a2]]]. The terminology  "tight"  was introduced by L. LeCam, [[#References|[a5]]].
+
{{TEX|done}}
 +
{{MSC|28A33}}
  
More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282012.png" /> be two pavings on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282014.png" /> a set function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282015.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282016.png" /> is tight with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282017.png" /> if
+
[[Category:classical measure theory]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092820/t09282018.png" /></td> </tr></table>
+
{{TEX|done}}
 +
 
 +
A measure $\mu$ defined on the [[Algebra of  sets|σ-algebra]] $\mathcal{B} (X)$ of [[Borel set|Borel sets]] of a  topological space $X$ which is locally finite (i.e. for any point $x\in  X$ there is a neighborhood which has finite measure) and having the  following property:
 +
\begin{equation}\label{e:tight}
 +
\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact}\}\,.
 +
\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 $X$ such measures are also outer regular, i.e.
 +
\begin{equation}\label{e:outer}
 +
\mu (B) = \inf \{\mu (U): U\supset N, \mbox{ $U$ open}\}\, ,
 +
\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|pavings]] 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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Billingsley,   "Convergence of probability measures" , Wiley  (1968) pp. 9ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Topsøe,   "Topology and measure" , Springer (1970) pp. xii</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Bichteler,  "Integration theory (with special attention to vector measures)" , ''Lect. notes in math.'' , '''315''' , Springer  (1973)  pp. §24</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"J.C. Oxtoby,  S. Ulam,  "On the existence of a measure invariant under a transformation"  ''Ann. of Math.'' , '''40'''  (1939)  pp. 560–566</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. LeCam,   "Convergence in distribution of probability processes" ''Univ. of Calif. Publ. Stat.'' , '''2''' :  11 (1957) pp. 207–236</TD></TR></table>
+
{|
 +
|-
 +
|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)
 +
|-
 +
|}

Revision as of 17:10, 29 November 2012

inner regular measure, Radon measure 2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]

A measure $\mu$ defined on the σ-algebra $\mathcal{B} (X)$ of Borel sets of a topological space $X$ which is locally finite (i.e. for any point $x\in X$ there is a neighborhood which has finite measure) and having the following property: \begin{equation}\label{e:tight} \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ '"`UNIQ-MathJax5-QINU`"' compact}\}\,. \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 $X$ such measures are also outer regular, i.e. \begin{equation}\label{e:outer} \mu (B) = \inf \{\mu (U): U\supset N, \mbox{ '"`UNIQ-MathJax8-QINU`"' open}\}\, , \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 \[ \beta (A_1) - \beta (A_2) = \sup \{\beta (K): K \subset A_1\setminus A_2, K\in \mathcal{K}\}\, \qquad \forall A_1, A_2 \in \mathcal{A} \mbox{ with '"`UNIQ-MathJax18-QINU`"'}. \]

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)
How to Cite This Entry:
Tight measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tight_measure&oldid=28960