View source for Outer measure

{{MSC|28A}}

[[Category:Classical measure theory]]

{{TEX|done}}

==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 $\mathcal{P} (X)$.

===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}}). 

'''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  [[Algebra of sets|$\sigma$-algebra]]. Therefore $(X, \mathcal{M}, \mu)$  is a [[Measure space|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 set|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 {{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}}).

===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==

===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 {{Cite|Ha}}). Here we use instead the convention that $\mu (A) = \infty$ when $A\not \in \mathcal{H}$ (cp. for instance with {{Cite|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 {{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
\[
\mu^\delta  (A) := \inf \left\{ \sum_i \mu (E_i) : \{E_i\}_{i\in \mathbb N} \subset  \mathcal{C} \mbox{ covers $A$ and ${\rm diam}\, (E_i) \leq  \delta$}\right\}\, 
\]
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).

==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).
|-
|}