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[1] by Carathéodory[2] to provide a basis for the theory of measurable sets and countably additive measures. 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:

  1. Any interval of reals [a, b] has measure b − a
  2. The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
  3. Translation invariance: For any set A and any real x, the sets A and A+x have the same measure (where A%2Bx = \{ a%2Bx: a\in A\})
  4. Countable additivity: for any sequence (Aj) of pairwise disjoint subsets of X
\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.

Contents

Formal definitions

An outer measure on a set X is a function

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

defined on all subsets of X, that satisfies the following conditions:

\varphi(\varnothing) = 0
A\subseteq B\quad\text{implies}\quad\varphi(A) \leq \varphi(B).
\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) %2B \varphi(A \setminus 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 so 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) %2B \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.

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 (Ai)i of elements of C which cover E (with the convention that if no such sequence exists, then the infimum is infinite). Then φ is an outer measure on X.

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, φδ ≥ φδ' when δ ≤ 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.

Theorem. φ0 is a metric outer measure on X.

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

See also

References

  1. ^ See pp379, C.D. Aliprantis, K.C. Border, Infinite Dimensional Analysis, 3rd ed, Springer 2006. ISBN 3-540-29586-0
  2. ^ C. Carathéodory, Vorlesungen über reelle Funktionen, 1st ed, Berlin: Leipzig 1918, 2nd ed, New York: Chelsea 1948.