Radon measure

The radon measure or radon measure is a term from the mathematical branch of measure theory . It is a special measure on Borel's σ-algebra of a Hausdorff space with certain regularity properties. However, the term is not used consistently in the specialist literature. The preferred definition in this article is (according to Jürgen Elstrodt ) "particularly advantageous for the treatment of Riesz's notation ". The radon measures are named after the mathematician Johann Radon .

definition

A definition (by Laurent Schwartz and Jürgen Elstrodt ) reads:

A Radon measure is a measure on Borel's σ-algebra of a Hausdorff space that is locally finite and regular from within .

For a measure locally finite means: For each there is an environment with . ${\displaystyle \mu }$ ${\displaystyle x\in X}$ ${\displaystyle U}$ ${\displaystyle \mu (U)<\infty }$

Regular from the inside means:

${\displaystyle \mu (A)=\sup\{\mu (K)\mid K\subset A,\ K\ {\textrm {kompakt}}\}}$

for all measurable quantities . ${\displaystyle A}$

Other meanings

In addition to the above definition, it is sometimes required that the measure should be finite.

Some authors use the term "Radon measure" for a Borel measure in which every compact set has finite measure. They refer to a measure as a Borel measure if it is defined on the Borel σ-algebra of a topological space. For a locally compact Hausdorff space, this Radon measure is then locally finite and thus corresponds in this special case to a Borel measure (in the sense of a locally finite measure on the Borel σ-algebra of a Hausdorff space).

In English locally finite measures on the Borel σ-algebra of a Hausdorff space that are regular from the inside (i.e. Radon measures in the sense of the definition given here) are called tight measures . However, they do not correspond to the strict measures that are common in the German-speaking area.

Unless explicitly stated otherwise, this article deals with the properties of radon measures in the sense of the definition given above.

Examples

Examples of measures with this regularity property are:

• The Lebesgue-Stieltjes measures on the Borel quantities are exactly the radon measures . ${\displaystyle \mathbb {R} ^{n}}$
• The hair measure on locally compact Hausdorff topological groups .

The concept of the Radon measure comes naturally when one examines positive linear functionals " " (so-called Radon integrals ) on (the continuous, real-valued functions with a compact carrier ) on a locally compact Hausdorff space. In such locally compact spaces the property of the local finiteness of a measure is equivalent to the finiteness of the measure on compact sets (see Borel measure ). ${\displaystyle \textstyle \left(f\geq 0\Rightarrow \int f\geq 0\right)}$ ${\displaystyle \textstyle \int }$ ${\displaystyle C_{c}(X)}$

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .

Individual evidence

1. ↑ Jürgen Elstrodt: Measure and integration theory. 7th edition. Springer, Berlin / Heidelberg 2011, ISBN 978-3-642-17904-4 , p. Vii.
2. ^ Radon, Johann Karl August . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
3. ↑ Laurent Schwartz : Radon measures on arbitrary topological spaces and cylindrical measures (= Studies in Mathematics. Vol. 6). Oxford University Press, London 1973, ISBN 0-19-560516-0 .
4. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 313.
5. ↑ Eric W. Weisstein : Radon Measure . In: MathWorld (English).
6. ↑ Tight measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).