Outer measure

In mathematics, in particular in 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. A general theory of outer measures was first introduced by Constantin Carathéodory to provide a 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.

Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in R or balls in R³. One might expect to define a generalized measuring function φ on R that fulfils the following requirements:

Any interval of reals [a, b] has measure b − a
The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
Translation invariance: For any set A and any real x, the sets A and A+x have the same measure (where $A+x = \{ a+x: a\in A\}$ )
Countable additivity: for any sequence (A_j) of pairwise disjoint subsets of R

\varphi\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \varphi(A_i).

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.

Formal definitions

An outer measure on a set $X$ is a function

\varphi: 2^X \rightarrow [0, \infty],

defined on all subsets of $X$ ( $2^{X}$ is another notation for the power set), that satisfies the following conditions:

Null empty set: The empty set has zero outer measure (see also: measure zero).

\varphi(\varnothing) = 0

Monotonicity: For any two subsets $A$ and $B$ of $X$ ,

A\subseteq B\quad\text{implies}\quad\varphi(A) \leq \varphi(B).

Countable subadditivity: For any sequence $\{A_{j}\}$ of subsets of $X$ (pairwise disjoint or not),

\varphi\left(\bigcup_{j=1}^\infty A_j\right) \leq \sum_{j=1}^\infty \varphi(A_j).

This allows us to define the concept of measurability as follows: a subset $E$ of $X$ is φ-measurable (or Carathéodory-measurable by φ) iff for every subset $A$ of $X$

\varphi (A)=\varphi (A\cap E)+\varphi (A\cap E^{c}).

where $E^{c}$ denotes the complement of $E$ .

Theorem. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive complete measure. For a proof of this theorem see the Halmos reference, section 11. This method is known as the Carathéodory construction and is one way of arriving at the concept of Lebesgue measure that is important for measure theory and the theory of integrals.

Outer measure and topology

Suppose $(X, d)$ is a metric space and $φ$ an outer measure on $X$ . If $φ$ has the property that

\varphi(E \cup F) = \varphi(E) + \varphi(F)

whenever

d(E,F) = \inf\{d(x,y): x \in E, y \in F\} > 0,

then $φ$ is called a metric outer measure.

Theorem. If $φ$ is a metric outer measure on $X$ , then every Borel subset of $X$ is $φ$ -measurable. (The Borel sets of $X$ are the elements of the smallest $σ$ -algebra generated by the open sets.)

Construction of outer measures

There are several procedures for constructing outer measures on a set. The classic Munroe reference below describes two particularly useful ones which are referred to as Method I and Method II.

Method I

Let $X$ be a set, $C$ a family of subsets of $X$ which contains the empty set and $p$ a non-negative extended real valued function on $C$ which vanishes on the empty set.

Theorem. Suppose the family $C$ and the function $p$ are as above and define

\varphi(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C\biggr\}.

That is, the infimum extends over all sequences ${A i}$ of elements of $C$ which cover $E$ , with the convention that the infimum is infinite if no such sequence exists. Then $φ$ is an outer measure on $X$ .

Method II

The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures. Suppose $(X, d)$ is a metric space. As above $C$ is a family of subsets of $X$ which contains the empty set and $p$ a non-negative extended real valued function on $C$ which vanishes on the empty set. For each $δ > 0$ , let

C_\delta= \{A \in C: \operatorname{diam}(A) \leq \delta\}

and

\varphi_\delta(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C_\delta\biggr\}.

Obviously, $φ δ \geq φ δ'$ when $δ \leq δ'$ since the infimum is taken over a smaller class as $δ$ decreases. Thus

\lim_{\delta \rightarrow 0} \varphi_\delta(E) = \varphi_0(E) \in [0, \infty]

exists (possibly infinite).

Theorem. $φ 0$ is a metric outer measure on $X$ .

This is the construction used in the definition of Hausdorff measures for a metric space.

Notes

↑ Carathéodory 1968
↑ Aliprantis & Border 2006, pp. S379

References

Aliprantis, C.D.; Border, K.C. (2006). Infinite Dimensional Analysis (3rd ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 3-540-29586-0.
Carathéodory, C. (1968) [1918]. Vorlesungen über reelle Funktionen (in German) (3rd ed.). Chelsea Publishing. ISBN 978-0828400381.
Federer, H. (1996) [1969]. Geometric Measure Theory. Classics in Mathematics (1st ed reprint ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 978-3540606567.
Halmos, P. (1978) [1950]. Measure theory. Graduate Texts in Mathematics (2nd ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 978-0387900889.
Munroe, M. E. (1953). Introduction to Measure and Integration (1st ed.). Addison Wesley. ASIN B004VIH64U. ISBN 978-1124042978.
Kolmogorov, A. N.; Fomin, S. V. (1970). Introductory Real Analysis. Richard A. Silverman transl. New York: Dover Publications. ISBN 0-486-61226-0.

External links

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