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An [[Additive function|additive function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808601.png" /> defined on a family of sets in a topological space whose total variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808602.png" /> (cf. [[Total variation of a function|Total variation of a function]]) satisfies the condition
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{{MSC|28A}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808603.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808604.png" /> denotes the interior of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808606.png" /> the closure of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808607.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086010.png" /> are in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086011.png" />). Every bounded additive regular set function, defined on a semi-ring of sets in a compact topological space, is countably additive (Aleksandrov's theorem).
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{{TEX|done}}
  
The property of regularity can also be related to a measure, as a special case of a set function, and one speaks of a regular measure, defined on a topological space. For example, the [[Lebesgue measure|Lebesgue measure]] is regular.
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In general this terminology is used for [[Set function|set functions]], i.e. maps $\mu$ defined on a class $\mathcal{C}$ of subsets of a set $X$ and taking values in the extended real line (or, more generally, a normed vector space), with respect to which general sets in the domain of definition $\mathcal{C}$ enjoy some suitable ''approximation properties'' with a relevant subclass of sets $\mathcal{A}\subset \mathcal{C}$. Such approximation properties imply usually that for a generic set $C\in \mathcal{A}$ there is a set $A\in \mathcal{A}$ such that $|\mu (C\triangle A)|$ is small. Often the set $X$ is a topological space and the class $\mathcal{A}$ is related to the topology of $X$.
  
====References====
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The precise meaning of the term depends, however, on the nature of the set function and on the author. The following are notable examples:
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Wiley  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.D. Aleksandrov,   "Additive set-functions in abstract spaces"  ''Mat. Sb.'' , '''9'''  (1941)  pp. 563–628  (In Russian)</TD></TR></table>
 
  
 +
* If $X$ is a locally compact topological space and $\mu$ a [[Set function|set function]] $\mu: \mathcal{C} \to [0, \infty]$ defined on the closed sets $\mathcal{C}$ which is finitely additive and finite on compact sets, then $\mu$ is called (by some authors) a ''regular content'' if
 +
\[
 +
\mu (D) = \inf\; \{ \mu (C): D\subset {\rm int}\, (C)\} \qquad \forall D\in \mathcal{C}\, .
 +
\]
 +
(See for instance Section 54 of {{Cite|Ha}}). A regular content is countably additive (cp. with Theorem A of Section 54 in {{Cite|Ha}}).
  
 +
* If $X$ is a topological space and $\mu$ a finitely additive set function $\mu: \mathcal{C} \to [0, \infty]$ defined on a ring of sets, then $\mu$ is called (by some authors) regular, if
 +
\[
 +
\mu (D) = \inf\; \{ \mu (C): D\subset {\rm int}\, (C)\} = \sup\; \{\mu (C): \overline{C} \subset D\}\, .
 +
\]
 +
This definition can be extended to additive set functions taking values in $[-\infty, \infty]$ be requiring the same identities for their [[Total variation of a measure|total variation]]. If $X$ is locally compact, $\mu$ is regular and it is finite on compact sets, then $\mu$ is $\sigma$-additive. This theorem is called ''Aleksandrov's Theorem'' by some authors (its proof can be reduced, for instance, to the aforementioned Theorem A in Section 54 of {{Cite|Ha}}).
  
====Comments====
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* If $X$ is a topological space and $\mu: \mathcal{C}\to [0, \infty]$ a [[Measure|measure]], then $\mu$ is called (by some authors) outer (respectively inner) regular if the Borel sets belong to the [[Algebra of sets|$\sigma$-algebra]] $\mathcal{C}$ and
Although a set function is called regular if it satisfies a property of approximation from below or above involving  "nice"  sets, the precise meaning of  "regular"  usually depends on the context (and on the author). For example, a (Carathéodory) [[Outer measure|outer measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086012.png" /> is called regular if for every part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086014.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086015.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086016.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086017.png" />-measurable set containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086018.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086019.png" /> is a topological space, the outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086020.png" /> is called Borel regular if Borel sets are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086021.png" />-measurable and if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086022.png" /> above can be taken Borel. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086023.png" /> is a metrizable space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086024.png" /> is a finite measure on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086025.png" />-field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086026.png" /> is always regular in the sense of the article above. In this setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086027.png" /> is often called inner regular, or just regular, if for any Borel subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086028.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086029.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086030.png" /> a countable union of compact sets included in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086031.png" />, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086032.png" /> is a [[Radon measure|Radon measure]]. Instead of calling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086033.png" /> Radon, one nowadays most often says that it is tight.
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\[
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\mu (D) = \inf\;\{\mu (C) : D\subset C \mbox{ and $C$ is open}\} \qquad \forall D\in \mathcal{C}
 +
\]
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\[
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\left(\mbox{resp.}\qquad \mu (D) = \sup\;\{\mu (C) : C\subset D \mbox{ and $C$ is closed}\} \qquad \forall D\in \mathcal{C}\, \right).
 +
\]
 +
(See Section 52 in {{Cite|Ha}}). If the measure is both inner and outer regular than it is called regular. Some authors also require, additionally, that $X$ is locally compact and $\mu$ is finite on compact sets. Other authors use the terminology [[Radon measure]] (see for instance Definition 1.5(4) of {{Cite|Ma}}) and some others the terminology ''tight measure''.
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Variants of these definitions apply to [[Signed measure|signed measures]] or [[Vector measure|vector measures]] $\mu$: in such cases the assumptions above are required to hold for the [[Total variation of a measure|total variation]] of $\mu$.
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* If $X$ is a topological space and $\mu:\mathcal{P} (X) \to [0, \infty]$ a [[Outer measure|outer measure]], then $\mu$ is called Borel outer measure if the Borel sets are $\mu$-measurable (see [[Outer measure]] for the relevant definition) and Borel regular if, in addition, for every $C\subset X$ there is a Borel set $B$ with $C\subset B$ and $\mu (B)=\mu (C)$. See for instance Section 1.1 of {{Cite|EG}}.
 +
 
 +
==References==
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Al}}|| A.D. Aleksandrov,  "Additive set-functions in abstract spaces" ''Mat. Sb.'' , '''9'''  (1941)  pp. 563–628
 +
|-
 +
|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz,  "Linear operators. General theory" , '''1'''  , Interscience (1958)  {{MR|0117523}}
 +
|-
 +
|valign="top"|{{Ref|EG}}||    L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of    functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,    1992. {{MR|1158660}} {{ZBL|0804.2800}}
 +
|-
 +
|valign="top"|{{Ref|Fe}}||    H. Federer, "Geometric measure  theory". Volume 153 of Die  Grundlehren  der mathematischen  Wissenschaften. Springer-Verlag New  York Inc., New  York, 1969.   {{MR|0257325}} {{ZBL|0874.49001}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}||      P. Mattila, "Geometry of sets  and measures in Euclidean spaces.  Fractals and rectifiability".     Cambridge Studies in Advanced  Mathematics, 44. Cambridge University      Press, Cambridge,  1995.   {{MR|1333890}} {{ZBL|0911.28005}}
 +
|-
 +
|}

Revision as of 12:39, 23 September 2012

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

In general this terminology is used for set functions, i.e. maps $\mu$ defined on a class $\mathcal{C}$ of subsets of a set $X$ and taking values in the extended real line (or, more generally, a normed vector space), with respect to which general sets in the domain of definition $\mathcal{C}$ enjoy some suitable approximation properties with a relevant subclass of sets $\mathcal{A}\subset \mathcal{C}$. Such approximation properties imply usually that for a generic set $C\in \mathcal{A}$ there is a set $A\in \mathcal{A}$ such that $|\mu (C\triangle A)|$ is small. Often the set $X$ is a topological space and the class $\mathcal{A}$ is related to the topology of $X$.

The precise meaning of the term depends, however, on the nature of the set function and on the author. The following are notable examples:

  • If $X$ is a locally compact topological space and $\mu$ a set function $\mu: \mathcal{C} \to [0, \infty]$ defined on the closed sets $\mathcal{C}$ which is finitely additive and finite on compact sets, then $\mu$ is called (by some authors) a regular content if

\[ \mu (D) = \inf\; \{ \mu (C): D\subset {\rm int}\, (C)\} \qquad \forall D\in \mathcal{C}\, . \] (See for instance Section 54 of [Ha]). A regular content is countably additive (cp. with Theorem A of Section 54 in [Ha]).

  • If $X$ is a topological space and $\mu$ a finitely additive set function $\mu: \mathcal{C} \to [0, \infty]$ defined on a ring of sets, then $\mu$ is called (by some authors) regular, if

\[ \mu (D) = \inf\; \{ \mu (C): D\subset {\rm int}\, (C)\} = \sup\; \{\mu (C): \overline{C} \subset D\}\, . \] This definition can be extended to additive set functions taking values in $[-\infty, \infty]$ be requiring the same identities for their total variation. If $X$ is locally compact, $\mu$ is regular and it is finite on compact sets, then $\mu$ is $\sigma$-additive. This theorem is called Aleksandrov's Theorem by some authors (its proof can be reduced, for instance, to the aforementioned Theorem A in Section 54 of [Ha]).

  • If $X$ is a topological space and $\mu: \mathcal{C}\to [0, \infty]$ a measure, then $\mu$ is called (by some authors) outer (respectively inner) regular if the Borel sets belong to the $\sigma$-algebra $\mathcal{C}$ and

\[ \mu (D) = \inf\;\{\mu (C) : D\subset C \mbox{ and '"`UNIQ-MathJax31-QINU`"' is open}\} \qquad \forall D\in \mathcal{C} \] \[ \left(\mbox{resp.}\qquad \mu (D) = \sup\;\{\mu (C) : C\subset D \mbox{ and '"`UNIQ-MathJax32-QINU`"' is closed}\} \qquad \forall D\in \mathcal{C}\, \right). \] (See Section 52 in [Ha]). If the measure is both inner and outer regular than it is called regular. Some authors also require, additionally, that $X$ is locally compact and $\mu$ is finite on compact sets. Other authors use the terminology Radon measure (see for instance Definition 1.5(4) of [Ma]) and some others the terminology tight measure. Variants of these definitions apply to signed measures or vector measures $\mu$: in such cases the assumptions above are required to hold for the total variation of $\mu$.

  • If $X$ is a topological space and $\mu:\mathcal{P} (X) \to [0, \infty]$ a outer measure, then $\mu$ is called Borel outer measure if the Borel sets are $\mu$-measurable (see Outer measure for the relevant definition) and Borel regular if, in addition, for every $C\subset X$ there is a Borel set $B$ with $C\subset B$ and $\mu (B)=\mu (C)$. See for instance Section 1.1 of [EG].

References

[Al] A.D. Aleksandrov, "Additive set-functions in abstract spaces" Mat. Sb. , 9 (1941) pp. 563–628
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ma] P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
How to Cite This Entry:
Regular set function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_set_function&oldid=28130
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article