Difference between revisions of "Radon measure"
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of a topological space $X$ and having the following property: | of a topological space $X$ and having the following property: | ||
for every $\varepsilon > 0$ there is a compact | 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. | + | 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 $\mathcal{B} (X)$ of [[Borel set|Borel sets]] is a Radon measure. |
Any Radon measure is tight (also called inner regular): for any Borel $B\subset X$ one has | Any Radon measure is tight (also called inner regular): for any Borel $B\subset X$ one has | ||
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\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact}\}\, . | \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 $\mathcal{B} (X)$ 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. |
| − | 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 $X$ has a countable [[Basis|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 | 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 | ||
Revision as of 16:46, 8 August 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 σ-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 $\mathcal{B} (X)$ of Borel sets is a Radon measure.
Any Radon measure is tight (also called inner regular): for any Borel $B\subset X$ one has \[ \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ '"`UNIQ-MathJax13-QINU`"' compact}\}\, . \] 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 (see Descriptive set theory), or more generally Suslin space (see measure) 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 |
Radon measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon_measure&oldid=27265