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inner regular measure 2010 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 $\mathbb R^n$.

## Contents

### 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.

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.
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.