View source for Radon measure

''inner regular measure''

[[Category:Classical measure theory]]

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{{MSC|28A33}}

[[Category:classical measure theory]]

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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|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}
\mu (B)= \sup \{\mu(K): K\subset B, K \mbox{ compact}\}\, 
\end{equation}
(see {{Cite|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 {{Cite|Fe}} and Definition 1.5 of {{Cite|Ma}}). 

The property \ref{e:tight} is called ''inner regularity'' or also ''tightness'' of the measure $\mu$ (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 $f\geq 0$ (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====
{|
|-
|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|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}}  
|}