Outer measure

In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.^[1]^[2] Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.

Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in $\mathbb {R}$ or balls in $\mathbb {R} ^{3}$ . One might expect to define a generalized measuring function $\varphi$ on $\mathbb {R}$ that fulfills the following requirements:

It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of $X$ is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.

null empty set: $\mu (\varnothing )=0$

countably subadditive: for arbitrary subsets $A,B_{1},B_{2},\ldots$ of $X,$
${\text{if }}A\subseteq \bigcup _{j=1}^{\infty }B_{j}{\text{ then }}\mu (A)\leq \sum _{j=1}^{\infty }\mu (B_{j}).$

Given a set $X,$ let $2^{X}$ denote the collection of all subsets of $X,$ including the empty set $\varnothing .$ An outer measure on $X$ is a set function

Note that there is no subtlety about infinite summation in this definition. Since the summands are all assumed to be nonnegative, the sequence of partial sums could only diverge by increasing without bound. So the infinite sum appearing in the definition will always be a well-defined element of $[0,\infty ].$ If, instead, an outer measure were allowed to take negative values, its definition would have to be modified to take into account the possibility of non-convergent infinite sums.

An alternative and equivalent definition.^[3] Some textbooks, such as Halmos (1950), instead define an outer measure on $X$ to be a function $\mu :2^{X}\to [0,\infty ]$ such that

if $A\subseteq X$ is $\mu$ -measurable then its $X\setminus A\subseteq X$ is also $\mu$ -measurable.

complement

Regular outer measures[edit]

Definition of a regular outer measure[edit]

Given a set $X$ , an outer measure $μ$ on $X$ is said to be regular if any subset $A\subseteq X$ can be approximated 'from the outside' by $μ$ -measurable sets. Formally, this is requiring either of the following equivalent conditions:

Inner measure

Aliprantis, C.D.; Border, K.C. (2006). Infinite Dimensional Analysis (3rd ed.). Berlin, Heidelberg, New York: . ISBN 3-540-29586-0.

Springer Verlag

(1968) [1918]. Vorlesungen über reelle Funktionen (in German) (3rd ed.). Chelsea Publishing. ISBN 978-0828400381.

Carathéodory, C.

Evans, Lawrence C.; Gariepy, Ronald F. (2015). Measure theory and fine properties of functions. Revised edition. CRC Press, Boca Raton, FL. pp. xiv+299. 978-1-4822-4238-6. {{cite book}}: |work= ignored (help)

ISBN

(1996) [1969]. Geometric Measure Theory. Classics in Mathematics (1st ed reprint ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 978-3540606567.

Federer, H.

(1978) [1950]. Measure theory. Graduate Texts in Mathematics (2nd ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 978-0387900889.

Halmos, P.

Munroe, M. E. (1953). (1st ed.). Addison Wesley. ISBN 978-1124042978.

Introduction to Measure and Integration

; Fomin, S. V. (1970). Introductory Real Analysis. Richard A. Silverman transl. New York: Dover Publications. ISBN 0-486-61226-0.

Outer measure

complement

Regular outer measures[edit]

Definition of a regular outer measure[edit]

Inner measure

Springer Verlag

Carathéodory, C.

ISBN

Federer, H.

Halmos, P.

Introduction to Measure and Integration

Kolmogorov, A. N.

Outer measure

Caratheodory measure