Difference between revisions of "Radon measure"
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| − | A concept introduced originally by J. Radon (1913), whose original constructions | + | 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=== | ===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 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: | + | 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: |
\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}\}\, |
\end{equation} | \end{equation} | ||
(see {{Cite|Sc}}). | (see {{Cite|Sc}}). | ||
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If $X$ is locally compact every finite Radon measure on $X$ is also outer regular, i.e. | If $X$ is locally compact every finite Radon measure on $X$ 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}\}\, , |
\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}}). | ||
| − | The property \ref{e:tight} is called ''inner regularity'' or also ''tightness'' of the measure $\mu$, 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. | + | 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=== | ===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}}). | 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 | + | 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=== | ===Duality with continuous functions=== | ||
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. 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 |
| [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 |
Radon measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon_measure&oldid=28976