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A non-negative [[Set function|set function]], denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o0706301.png" />, defined on a countably-additive class of sets that contains in addition to a set itself also any one of its subsets, and having the following properties:
+
{{MSC|28A}}
  
monotony, i.e.
+
[[Category:Classical measure theory]]
  
<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/o/o070/o070630/o0706302.png" /></td> </tr></table>
+
{{TEX|done}}
  
countable semi-additivity, i.e.
+
==Definition==
 +
An outer measure is a [[Set function|set function]] $\mu$ such that
 +
*  Its domain of definition is an hereditary [[Ring of  sets|$\sigma$-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 {{Cite|Ha}} and with Section 1.1 of {{Cite|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 of sets|algebra]], then it must  contain $X$ and hence it coincides necessarily with $\mathcal{P} (X)$.
  
<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/o/o070/o070630/o0706303.png" /></td> </tr></table>
+
===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 {{Cite|Ha}} and Section 1.1 of {{Cite|EG}}).  
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o0706304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o0706305.png" /> is the empty set.
+
'''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
 +
\[
 +
\mu (A\cap M) + \mu (A\setminus M) = \mu (A) \qquad \forall A\in \mathcal{R}\, .
 +
\]
  
An outer measure defined on all subsets of a metric space is said to be an outer measure in the sense of Carathéodory, or a metric outer measure, if
+
'''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.
  
<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/o/o070/o070630/o0706306.png" /></td> </tr></table>
+
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  [[Algebra of sets|$\sigma$-algebra]]. Therefore $(X, \mathcal{M}, \mu)$  is a [[Measure space|measure space]].
  
provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o0706307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o0706308.png" /> is the distance between the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o0706309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o07063010.png" />. If an outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o07063011.png" /> is given, it is possible to specify the class of measurable sets on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o07063012.png" /> becomes a [[Measure|measure]] (cf. also [[Carathéodory measure|Carathéodory measure]]).
+
==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 set|Borel sets]] are $\mu$-measurable.
  
Outer measures result, in particular, from the construction of the extension of a measure from a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o07063013.png" /> onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070630/o07063014.png" />-ring generated by it.
+
'''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 $\inf \{d(x,y): x\in A, y\in B\} > 0$).
  
In the classical theory of the [[Lebesgue measure|Lebesgue measure]] [[#References|[1]]] the outer measure of a set is defined as the greatest lower bound of the measures of the open sets containing the given set; the inner measure of a set is defined as the least upper bound of the measures of the closed sets contained in the given set.
+
'''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]].
  
====References====
+
Cp. with Theorem 5 of {{Cite|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 {{Cite|Ha}}).
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.P. Natanson,  "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
 
  
 +
===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 {{Cite|EG}}.
  
 +
==Constructions of outer measures==
  
====Comments====
+
===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 } A\right\}\, .
 +
\end{equation}
  
====References====
+
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 {{Cite|Ha}}). Here we use instead the convention that $\mu (A) = \infty$ when $A\not \in \mathcal{H}$ (cp. for instance with {{Cite|Mu}}).
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"H.L. Royden,   "Real analysis" , Macmillan (1968)</TD></TR></table>
+
 
 +
'''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 {{Cite|Ha}}  (NB: the proof given in {{Cite|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
 +
\[
 +
\mu^\delta  (A) := \inf \left\{ \sum_i \nu (E_i) : \{E_i\}_{i\in \mathbb N} \subset  \mathcal{C} \mbox{ covers } A
 +
\mbox{ and } {\rm diam}\, (E_i) \leq  \delta\, \right\}\,
 +
\]
 +
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 {{Cite|KP}} (cp. also with {{Cite|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  {{Cite|EG}}: although the reference handles the cases of Hausdorff outer measures, the proof extends verbatim to the setting above).
 +
 
 +
'''Remark 9'''
 +
The  [[Hausdorff measure|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). More generally one can consider functions of type $\nu (A) = h ({\rm diam}\, (A))$, where $h: \mathbb R^+\to \mathbb R^+$ is a monotone function and ${\rm diam}\, (A)$ denotes the diameter of $A$.
 +
 
 +
==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 {{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 Sobolev $p$-[[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.
 +
 
 +
==References==
 +
{|
 +
|-
 +
|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|KP}}|| S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
 +
|-
 +
|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}}
 +
|-
 +
|valign="top"|{{Ref|Mu}}|| M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
 +
|-
 +
|}

Latest revision as of 10:08, 16 August 2013

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 } A\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 \[ \mu^\delta (A) := \inf \left\{ \sum_i \nu (E_i) : \{E_i\}_{i\in \mathbb N} \subset \mathcal{C} \mbox{ covers } A \mbox{ and } {\rm diam}\, (E_i) \leq \delta\, \right\}\, \] 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). More generally one can consider functions of type $\nu (A) = h ({\rm diam}\, (A))$, where $h: \mathbb R^+\to \mathbb R^+$ is a monotone function and ${\rm diam}\, (A)$ denotes the diameter of $A$.

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 Sobolev $p$-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=14673
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article