Namespaces
Variants
Actions

Difference between revisions of "Radon measure"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 1: Line 1:
 
''inner regular measure''
 
''inner regular measure''
  
A finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771701.png" /> (cf. [[Measure in a topological vector space|Measure in a topological vector space]]) defined on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771702.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771703.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771704.png" />, and having the following property: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771705.png" /> there is a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771706.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771707.png" />. It was introduced by J. Radon (1913), whose original constructions referred to measures on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771708.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r0771709.png" />, the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717010.png" />-algebra of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717012.png" />. A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717013.png" /> is called a Radon space if every finite measure defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717014.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717015.png" /> is a Radon measure.
+
[[Category:Classical measure theory]]
  
====References====
+
{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Eléments de mathématiques. Intégration" , Hermann  (1963)  pp. Chapts. 6 - 8</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>
+
{{MSC|28A33}}
  
 +
[[Category:classical measure theory]]
  
 +
{{TEX|done}}
  
====Comments====
+
A finite measure $\mu$ (cf. [[Measure in a topological vector space|Measure in a topological vector space]]) defined on the $\sigma$-algebra $\mathcal(B) (X)$ of [[Borel set|Borel sets]]
Any Radon measure is tight (also called inner regular): For any Borel subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717017.png" /> one has
+
of a topological space $X$ and having the following property:
 +
for every $\varepsilon > 0$ there is a compact
 +
set $K$ such that $\mu (X\setminus K)<\varepsilon$. It was introduced by J. Radon (1913), whose original constructions referred to measures on the Borel $\sigma$-algebra of the Euclidean space $\mathbb R^n$. A topological space $X$ is called a Radon space if every finite measure defined on the $\sigma$-algebra $\mathcal{B} (X)$ is a 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/r/r077/r077170/r07717018.png" /></td> </tr></table>
+
Any Radon measure is tight (also called inner regular): for any Borel $B\subset X$ one has
 +
\[
 +
\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact.}.
 +
\]
 +
If $\mathcal{B} (X)$ is countably generated, $X$ is a Radon space if and only if it is Borel isomorphic to a universally measurable subset of $[0,1]^{\mathbb N}$ (or any other uncountable compact metrizable space). In particular, any polish space, or more generally Suslin space in the sense of Bourbaki, is Radon.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717019.png" /> is countably generated, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717020.png" /> is a Radon space if and only if it is Borel isomorphic to a universally measurable subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717021.png" /> (or any other uncountable compact metrizable space). In particular, any polish space, or more generally Suslin space in the sense of Bourbaki, is Radon.
+
One can also define non-finite (non-negative) Radon measures; they are tight and take finite values on compact subsets. If $X$ has a countable basis, they are $\sigma$-finite.
  
One can also define non-finite (non-negative) Radon measures; they are tight and take finite values on compact subsets. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717022.png" /> has a countable basis, they are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717023.png" />-finite.
+
Following N. Bourbaki (and ideas going back to W.H. Young and Ch. de la Vallée-Poussin), a (non-negative) Radon measure on, say, a locally compact space $X$ is a continuous linear functional $L$ on the space $C_c (X)$ of continuous functions with compact support
 +
(endowed with its natural inductive topology) which is nonnegative, i.e. such that
 +
$L(f)\geq 0$ whenever $f\geq 0$. One can prove with the help of the [[Riesz representation theorem]] (which deals with the case $X$ compact) that any non-negative and bounded Radon measure in this sense is the restriction to $C_c (X)$ of the integral with respect to a unique (non-finite) Radon measure.
  
Following N. Bourbaki (and ideas going back to W.H. Young and Ch. de la Vallée-Poussin), a (non-negative) Radon measure on, say, a locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717024.png" /> is a (non-negative) continuous linear functional on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717025.png" /> of continuous functions with compact support endowed with its natural inductive topology. One can prove with the help of the Riesz–Markov theorem (which deals with the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717026.png" /> compact) that any non-negative and bounded Radon measure in this sense is the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077170/r07717027.png" /> of the integral with respect to a unique (finite) Radon measure as defined in the article above; the converse is true and trivial.
+
====References====
 +
{|
 +
|-
 +
|valign="top"|{{Ref|AmFuPa}}||  L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The  Clarendon Press, Oxford University Press, New York, 2000.  {{MR|1857292}}{{ZBL|0957.49001}}
 +
|-
 +
|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|Bi}}||  P. Billingsley, "Convergence of probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
 +
|-
 +
|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}}
 +
|-
 +
|}

Revision as of 13:13, 21 July 2012

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

A finite measure $\mu$ (cf. Measure in a topological vector space) defined on the $\sigma$-algebra $\mathcal(B) (X)$ of Borel sets of a topological space $X$ and having the following property: for every $\varepsilon > 0$ there is a compact set $K$ such that $\mu (X\setminus K)<\varepsilon$. It was introduced by J. Radon (1913), whose original constructions referred to measures on the Borel $\sigma$-algebra of the Euclidean space $\mathbb R^n$. A topological space $X$ is called a Radon space if every finite measure defined on the $\sigma$-algebra $\mathcal{B} (X)$ is a Radon measure.

Any Radon measure is tight (also called inner regular): for any Borel $B\subset X$ one has \[ \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ '"`UNIQ-MathJax14-QINU`"' compact.}. \] If $\mathcal{B} (X)$ is countably generated, $X$ is a Radon space if and only if it is Borel isomorphic to a universally measurable subset of $[0,1]^{\mathbb N}$ (or any other uncountable compact metrizable space). In particular, any polish space, or more generally Suslin space in the sense of Bourbaki, is Radon.

One can also define non-finite (non-negative) Radon measures; they are tight and take finite values on compact subsets. If $X$ has a countable basis, they are $\sigma$-finite.

Following N. Bourbaki (and ideas going back to W.H. Young and Ch. de la Vallée-Poussin), a (non-negative) Radon measure on, say, a locally compact space $X$ is a continuous linear functional $L$ on the space $C_c (X)$ of continuous functions with compact support (endowed with its natural inductive topology) which is nonnegative, i.e. such that $L(f)\geq 0$ whenever $f\geq 0$. One can prove with the help of the Riesz representation theorem (which deals with the case $X$ compact) that any non-negative and bounded Radon measure in this sense is the restriction to $C_c (X)$ of the integral with respect to a unique (non-finite) Radon measure.

References

[AmFuPa] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[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
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[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
How to Cite This Entry:
Radon measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon_measure&oldid=27137
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article