Regular set function

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

\[ \mu (D) = \inf\;\{\mu (C) : D\subset C \mbox{ and } C \mbox{ is open}\} \qquad \forall D\in \mathcal{C} \] \[ \left(\mbox{resp.}\qquad \mu (D) = \sup\;\{\mu (C) : C\subset D \mbox{ and } C \mbox{ is closed}\} \qquad \forall D\in \mathcal{C}\, \right). \] (See Section 52 in [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].

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