Radon measure

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 $\sigma$-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 $\mathcal{B} (X)$ is a Radon measure.

Any Radon measure is tight (also called inner regular): for any Borel $B\subset X$ one has \[ \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ '"`UNIQ-MathJax15-QINU`"' compact}\}\, . \] 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, or more generally Suslin space 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

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