Outer measure

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).