Regular measure

In measure theory, a regular measure is a special measure in a topological space for which certain approximation properties apply. A distinction is made between the regularity from the inside and the regularity from the outside of a measure. If a measure is regular inside and out, it is called regular.

The regularity of measures is not used consistently in the literature, especially in the context of Borel measures . A precise comparison with the definition in the respective context is therefore essential.

definition

Be a Hausdorff space and a σ-algebra on which the Borel σ-algebra contains. ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle X}$

Then all open and all closed subsets of in lie . ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$

Since Hausdorff is, all compact subsets of in are also . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$

A measure is called ${\displaystyle \mu \colon {\mathcal {A}}\rightarrow [0,\infty ]}$

• from the inside regularly , if the following applies to each : ${\displaystyle A\in {\mathcal {A}}}$

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

• Regularly from the outside , if the following applies to each : ${\displaystyle A\in {\mathcal {A}}}$

${\displaystyle \mu (A)=\inf\{\mu (U)\mid A\subset U,\ U\ {\textrm {offen}}\}\,,}$

• regular if it is regular inside and out.

An amount that fulfills one of the three specified properties is correspondingly referred to as an internally regular , externally regular or regular amount . Sometimes one demands the inner regularity only for open sets (in this sense the hair measure is then regular) or demands that the measure is a Borel measure . ${\displaystyle A\in {\mathcal {A}}}$

Other meanings

Measures on a metric space with Borel's σ-algebra are sometimes called closed regular , if for every set and each an open set and a closed set exist with and . Other authors only call these dimensions regular. ${\displaystyle {\mathcal {B}}}$ ${\displaystyle B\in {\mathcal {B}}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle O}$ ${\displaystyle A}$ ${\displaystyle A\subset B\subset O}$ ${\displaystyle \mu (O\setminus A)<\epsilon }$

In English there is also the term "tightness" for regularity from within. However, the "tight measures" do not correspond to the internally regular dimensions or the tight dimensions , but to the Radon dimensions (in the sense of an internally regular, locally finite dimension on Borel's σ-algebra of a Hausdorff space).

Properties and examples

Regular measures allow approximation arguments in many proofs. It is often sufficient to show certain statements for compact or open sets and then to expand these to measurable sets using the two formulas. Many dimensions are regular.

• The Lebesgue measure on the is regular. ${\displaystyle {\mathbb {R} }^{n}}$

• More generally: If a locally compact Hausdorff space with a countable basis and a Borel measure is on , then it is regular. ${\displaystyle X}$ ${\displaystyle \mu }$ ${\displaystyle X}$ ${\displaystyle \mu }$

• A Borel measure in a Polish area is regular.

Regular Borel dimensions

Depending on how you define a Borel measure , there are different concepts of the regularity of Borel measures.

• If a Borel measure is understood to be a locally finite measure on the Borel σ-algebra of a Hausdorff space, then this Borel measure is called a regular Borel measure if it is regular from inside and outside, i.e. regular in the above sense .
• If a Borel measure is understood to be a measure on Borel's σ-algebra of a topological space , then this measure is called a regular Borel measure if ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \mu }$

${\displaystyle \mu (B)=\sup\{\mu (C)\;|\;C\subset B,\;C{\text{ abgeschlossen }}\}}$

applies to each . ${\displaystyle B\in {\mathcal {B}}}$

• If one understands by a Borel measure an external measure with respect to which all Borel sets are Carathéodory-measurable , then the Borel measure is called a regular Borel measure, if for any subset of the superset there exists a Borel set such that is. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \mu (A)=\mu (B)}$

Generalizations

Regularity can also be defined for signed dimensions and complex dimensions ; one then speaks of regular signed dimensions or regular complex dimensions . The regularity is then equivalent to the regularity of the variation or the real / imaginary components.

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 313.
2. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 379.
3. ^ Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , p. 193 , doi : 10.1007 / 978-3-642-22261-0 .
4. ^ RA Minlos: Radon Mesure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
5. ↑ Tight measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
6. ↑ Elstrodt: Measure and Integration Theory. 2011, Chapter VIII. Corollary 1.12
7. ^ VV Sazonov: Borel measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
8. ↑ Eric W. Weisstein : Regular Borel Measure . In: MathWorld (English).

literature

• Heinz Bauer : Measure and integration theory. 2nd, revised edition. de Gruyter, Berlin a. a. 1992, ISBN 3-11-013625-2 .
• 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 .