View source for Regular set function

{{MSC|28A}}

[[Category:Classical measure theory]]

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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 $X$.

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|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}}).

* 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
\[
\mu (D) = \inf\;\{\mu (C) : D\subset C \mbox{ and } C \mbox{ is open}\} \qquad \forall D\in \mathcal{C}
\]
\[
\left(\mbox{resp.}\qquad \mu (D) = \sup\;\{\mu (C) : C\subset D \mbox{ and } C \mbox{ is closed}\} \qquad \forall D\in \mathcal{C}\, \right).
\]
(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''.
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 $\mu$.

* 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==
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|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}}
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|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}}
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|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}}
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