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* The spherical $\alpha$-dimensional outer measures on a metric space $(X,d)$, see Section 2.1.2 of {{Cite|KP}};
 
* The spherical $\alpha$-dimensional outer measures on a metric space $(X,d)$, see Section 2.1.2 of {{Cite|KP}};
 
* The Gross outer measures, the Caratheodory outer measures, the integral-geometric outer measures and the Gillespie outer measures in $\mathbb R^n$, see Sections  2.1.3-2.1.4-2.1.5 of {{Cite|KP}} (cp. also with 2.10.2-2.10.3-2.10.4 of {{Cite|Fe}}).
 
* The Gross outer measures, the Caratheodory outer measures, the integral-geometric outer measures and the Gillespie outer measures in $\mathbb R^n$, see Sections  2.1.3-2.1.4-2.1.5 of {{Cite|KP}} (cp. also with 2.10.2-2.10.3-2.10.4 of {{Cite|Fe}}).
 +
* The [[Capacity|capacity]] in $\mathbb R^n$ (see Theorem 1 in Section 4.7 of {{Cite|EG}}).
 
In all these examples the adjective ''outer'' is dropped when the outer measures are restricted to their respective measurable sets.
 
In all these examples the adjective ''outer'' is dropped when the outer measures are restricted to their respective measurable sets.
  

Revision as of 06:20, 22 September 2012

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

Definition

An outer measure is a set function $\mu$ such that

  • Its domain of definition is an hereditary $\sigma$-ring (also called $\sigma$-ideal) of subsets of a given space $X$, i.e. a $\sigma$-ring $\mathcal{R}\subset \mathcal{P} (X)$ with the property that for every $E\in \mathcal{R}$ all subsets of $E$ belong to $\mathcal{R}$;
  • Its range is the extended real half-line $[0, \infty$];
  • $\mu (\emptyset) =0$ and $\mu$ is $\sigma$-subadditive (also called countably subadditive), i.e. for every countable family $\{E_i\}\subset \mathcal{R}$ the following inequality holds:

\[ \mu \left(\bigcup_i E_i\right) \leq \sum_i \mu (E_i)\, . \] Cp. with Section 10 of [Ha] and with Section 1.1 of [EG]. The most common outer measures are defined on the full space $\mathcal{P} (X)$ of subsets of $X$. Indeed observe that, if an hereditary $\sigma$-ring is also an algebra, then it must contain $X$ and hence it coincides necessarily with $\mathcal{P} (X)$.

Measurable sets

There is a commonly used procedure to derive a measure from an outer measure $\mu$ on an hereditary $\sigma$-ring $\mathcal{R}$ (cp. with Section 11 of [Ha] and Section 1.1 of [EG]).

Definition 1 If $\mu:\mathcal{R}\to [0, \infty]$ is an outer measure, then a set $M\in \mathcal{R}$ is called $\mu$-measurable if \[ \mu (A\cap M) + \mu (A\setminus M) = \mu (A) \qquad \forall A\in \mathcal{R}\, . \]

Theorem 2 If $\mu:\mathcal{R}\to [0, \infty]$ is an outer measure, then the class $\mathcal{M}$ of $\mu$-measurable sets is a $\sigma$-ring and $\mu$ is countably additive on $\mathcal{M}$, i.e. \[ \mu \left(\bigcup_i E_i\right) = \sum_i \mu (E_i) \] whenever $\{E_i\}\subset \mathcal{M}$ is a countable collection of pairwise disjoint sets.

When $\mathcal{R} = \mathcal{P} (X)$, then it follows trivially from the definition that $X\in \mathcal{M}$: thus $\mathcal{M}$ is in fact a $\sigma$-algebra. Therefore $(X, \mathcal{M}, \mu)$ is a measure space.

Caratheodory criterion

An important class of outer measures on metric spaces $X$ are the ones satisfying the so-called Caratheodory criterion (called metric outer measures or Caratheodory outer measures, see Caratheodory measure): for such $\mu$ the Borel sets are $\mu$-measurable.

Definition 3 An outer measure $\mu: \mathcal{P} (X) \to [0, \infty]$ on a metric space $(X,d)$ is a metric outer measure if \[ \mu (A\cup B) = \mu (A) + \mu (B) \] for every pair of sets $A, B\subset X$ which have positive distance (i.e. such that $\inf \{d(x,y): x\in A, y\in B\} > 0$).

Theorem 4 If $\mu$ is a metric outer measure, then every Borel set is $\mu$-measurable. Moreover, the restriction of $\mu$ to the $\mu$-measurable sets is called, by some author, Caratheodory measure.

Cp. with Theorem 5 of [EG]. The converse is also true: if $\mu$ is an outer measure on the class $\mathcal{P} (X)$ of subsets of a metric space $X$ such that the Borel sets are $\mu$-measurable, then $\mu$ is a metric outer measure (cp. with Remark (8c) in Section 11 of [Ha]).

Regular and Borel regular outer measures

Several authors call regular those outer measures $\mu$ on $\mathcal{P} (X)$ such that for every $E\subset X$ there is a $\mu$-measurable set $F$ with $E\subset F$ and $\mu (E) = \mu (F)$. They moreover call $\mu$ Borel regular if the Borel sets are $\mu$-measurable and for every $E\subset X$ there is a Borel set $G$ with $E\subset G$ and $\mu (E) = \mu (G)$. Cp. with Section 1.1 of [EG].

Constructions of outer measures

Outer measures induced by set functions

A common procedure to construct outer measures $\mu$ from a set function $\nu$ is the following.

Definition 5 If $\mathcal{C}$ is class of subsets of $X$ containing the empty set and $\nu : \mathcal{C}\to [0, \infty]$ a set function with $\nu (\emptyset) =0$, for every set $A\subset X$ we define \begin{equation}\label{e:extension} \mu (A) = \inf \left\{ \sum_i \mu (E_i) : \{E_i\}_{i\in \mathbb N} \subset \mathcal{C} \mbox{ covers '"`UNIQ-MathJax74-QINU`"'}\right\}\, . \end{equation}

Observe that the class $\mathcal{H}$ of subsets of $X$ for which there is a countable covering in $\mathcal{C}$ is an hereditary $\sigma$-ring and some authors restrict the definition of $\mu$ to $\mathcal{H}$ (cp. with Section 10 of [Ha]). Here we use instead the convention that $\mu (A) = \infty$ when $A\not \in \mathcal{H}$ (cp. for instance with [Mu]).

Theorem 6 If $\nu$ and $\mu$ are as in Definition 5, then $\mu$ is an outer measure on $\mathcal{P} (X)$. If in addition

  • $\mathcal{C}$ is a ring and $\nu$ is a finitely additive set function, then $\mu (E) = \nu (E)$ for every $E\in \mathcal{C}$;
  • $\mathcal{C}$ is a $\sigma$-ring and $\nu$ is countably additive, then the elements of $\mathcal{C}$ are $\mu$-measurable.

Cp. with Theorem A of Section 10 and Theorem A in Section 12 of [Ha] (NB: the proof given in [Ha] of $\sigma$-subadditivity of $\mu$ does not use the assumption that $\nu$ is finitely additive).

Caratheodory constructions of metric outer measures

A second common procedure yields metric outer measures in metric spaces $(X, d)$ and goes as follows.

Definition 7 If $\mathcal{C}$ is a class of subsets of $X$ containing the empty set, $\nu : \mathcal{C}\to [0, \infty]$ a set function with $\nu (\emptyset) =0$ and $\delta\in ]0, \infty]$, then for every $A\subset X$ we define \[ \mu^\delta (A) := \inf \left\{ \sum_i \mu (E_i) : \{E_i\}_{i\in \mathbb N} \subset \mathcal{C} \mbox{ covers '"`UNIQ-MathJax106-QINU`"' and '"`UNIQ-MathJax107-QINU`"'}\right\}\, \] and \[ \mu (A) := \lim_{\delta\downarrow 0}\; \mu^\delta (A)\, . \]

Observe that the latter limit exists because $\mu^\delta (A)$ is a nonincreasing function of $\delta$. This construction is often called Caratheodory construction. See Section 2.1 of [KP] (cp. also with [Fe]).

Theorem 8 Let $\nu$ and $\mu$ be as in Definition 7. Then $\mu$ is a metric outer measure.

(Cp. with Claims 1,2 and 3 in the proof of Theorem 1 in Section 2.1 of [EG]: although the reference handles the cases of Hausdorff outer measures, the proof extends verbatim to the setting above).

Remark 9 The Hausdorff outer measure $\mathcal{H}^\alpha$ is given by such $\mu$ as in Definition 7 when we choose $\mathcal{C} = \mathcal{P} (X)$ and $\nu (A) = c_\alpha ({\rm diam}\, (A))^\alpha$ (where $c_\alpha$ is an appropriate normalization constant).

Examples

Very common examples of outer measures are

  • The Lebesgue outer measure on $\mathbb R^n$, see Lebesgue measure;
  • The Haudorff $\alpha$-dimensional outer measures on a metric space $(X,d)$, see Hausdorff measure;
  • The spherical $\alpha$-dimensional outer measures on a metric space $(X,d)$, see Section 2.1.2 of [KP];
  • The Gross outer measures, the Caratheodory outer measures, the integral-geometric outer measures and the Gillespie outer measures in $\mathbb R^n$, see Sections 2.1.3-2.1.4-2.1.5 of [KP] (cp. also with 2.10.2-2.10.3-2.10.4 of [Fe]).
  • The capacity in $\mathbb R^n$ (see Theorem 1 in Section 4.7 of [EG]).

In all these examples the adjective outer is dropped when the outer measures are restricted to their respective measurable sets.

References

[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
[KP] S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
[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
[Mu] M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
How to Cite This Entry:
Outer measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Outer_measure&oldid=28090
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article