Difference between revisions of "Radon measure"

Latest revision as of 19:35, 1 January 2021

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

A concept introduced originally by J. Radon (1913), whose original constructions referred to measures on the Borel $\sigma$-algebra of the Euclidean space $\mathbb R^n$.

Definition

A measure $\mu$ (cf. Measure in a topological vector space) 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 neighborhood 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}\}\, \end{equation} (see [Sc]).

If $X$ is locally compact every finite Radon measure on $X$ is also outer regular, i.e. \begin{equation}\label{e:outer} \mu (N) = \inf \{\mu (U): U\supset N, U \mbox{ open}\}\, , \end{equation} (cp. therefore with Definition 2.2.5 of [Fe] and Definition 1.5 of [Ma]).

The property \ref{e:tight} is called inner regularity or also tightness of the measure $\mu$ (see Tight measure), whereas property \ref{e:outer} is called outer regularity. Some authors require also that the measure $\mu$ be finite. If $X$ has a countable basis, Radon measures as defined above are necessarily $\sigma$-finite.

Radon space

A topological space $X$ is called a Radon space if every finite measure defined on the $\sigma$-algebra $\mathcal{B} (X)$ of Borel sets is a Radon measure. For instance the Euclidean space is a Radon space (cp. with Theorem 1.11 and Corollary 1.12 of [Ma]).

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 (see Descriptive set theory), or more generally Suslin space (see measure) in the sense of Bourbaki, is Radon.

Duality with continuous functions

Following N. Bourbaki (and ideas going back to W.H. Young and Ch. de la Vallée-Poussin), a (non-negative) Radon measure on 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 $f\geq 0$ (cp. with Section 2.2 of Chapter III in [Bo] or Section 9 of Chapter III in [HS]). One can prove with the help of the Riesz representation theorem 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 in the sense of the definition above.

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. MR1857292 Zbl 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
[Fe]	H. Federer, "Geometric measure theory", Springer-Verlag (1979). MR0257325 Zbl 0874.49001
[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
[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

How to Cite This Entry:
Radon measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon_measure&oldid=27265

This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

@@ Line 10: / Line 10: @@
 {{TEX|done}}
-A finite measure $\mu$ (cf. [[Measure in a topological vector space|Measure in a topological vector space]]) defined on the [[Algebra of sets|σ-algebra]] $\mathcal{B} (X)$ of [[Borel set|Borel sets]]
+A concept introduced originally by J. Radon (1913), whose original constructions referred to measures on the Borel $\sigma$-algebra of the Euclidean space $\mathbb R^n$.
-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
+===Definition===
-\[
+A measure $\mu$ (cf. [[Measure in a topological vector space|Measure in a topological vector space]]) defined on the [[Algebra of sets|σ-algebra]] $\mathcal{B} (X)$ of [[Borel set|Borel sets]] of a topological Hausdorff space $X$ which is locally finite (i.e. for any point $x\in X$ there is a neighborhood which has finite measure) and having the following property:
-\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact}\}\, .
+\begin{equation}\label{e:tight}
-\]
+\mu (B)= \sup \{\mu(K): K\subset B, K \mbox{ 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.
+\end{equation}
+(see {{Cite|Sc}}).
-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.
+If $X$ is locally compact every finite Radon measure on $X$ is also outer regular, i.e.
+\begin{equation}\label{e:outer}
+\mu (N) = \inf \{\mu (U): U\supset N, U \mbox{ open}\}\, ,
+\end{equation}
+(cp. therefore with Definition 2.2.5 of {{Cite|Fe}} and Definition 1.5 of {{Cite|Ma}}).
-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
+The property \ref{e:tight} is called ''inner regularity'' or also ''tightness'' of the measure $\mu$ (see [[Tight measure]]), whereas property \ref{e:outer} is called ''outer regularity''. Some authors require also that the measure $\mu$ be finite. If $X$ has a countable [[Basis|basis]], Radon measures as defined above are necessarily $\sigma$-finite.
+===Radon space===
+A topological space $X$ is called a Radon space if every finite measure defined on the $\sigma$-algebra $\mathcal{B} (X)$ of [[Borel set|Borel sets]] is a Radon measure. For instance the Euclidean space is a Radon space (cp. with Theorem 1.11 and Corollary 1.12 of {{Cite|Ma}}).
+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|metrizable space]]). In particular, any [[Polish space]] (see [[Descriptive set theory]]), or more generally [[Suslin space]] (see [[measure]]) in the sense of Bourbaki, is Radon.
+===Duality with continuous functions===
+Following N. Bourbaki (and ideas going back to W.H. Young and Ch. de la Vallée-Poussin), a (non-negative) Radon measure on 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.
+$L(f)\geq 0$ whenever $f\geq 0$ (cp. with Section 2.2 of Chapter III in {{Cite|Bo}} or Section 9 of Chapter III in {{Cite|HS}}).  One can prove with the help of the [[Riesz representation theorem]] 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 in the sense of the definition above.
 ====References====
@@ Line 37: / Line 47: @@
 |-
 |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|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|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}}
 |}

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