View source for Radon measure

''inner regular measure''

[[Category:Classical measure theory]]

{{TEX|done}}
{{MSC|28A33}}

[[Category:classical measure theory]]

{{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]]
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 set|Borel sets]] is a Radon measure.

Any Radon measure is tight (also called inner regular): for any Borel $B\subset X$ one has
\[
\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact}\}\, .
\]
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|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|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====
{|
|-
|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}}
|-
|}