Outer measure

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In mathematics, in particular in measure theory, an outer 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 developed by Carathéodory 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 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 mere intervals or open balls in R3. One might expect to define a generalized measuring function φ that fulfils the following three requirements:

  1. Any interval of reals [a, b] has measure ba
  2. The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
  3. Countable additivity, For any sequence {Aj}j of pairwise disjoint subsets of X
<math> \varphi\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \varphi(A_i)<math>

It turns out the second and third requirements together for all sets are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of X is to suitably pick out a class of subsets (to be called measurable) in such a way that fulfils the countably additivity property.

Contents

Formal definitions

An outer measure is defined as a function defined on all subsets of a set X

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

such that

<math> \varphi(\varnothing) = 0<math>
  • Monotonicity
<math> A \subseteq B \Rightarrow \varphi(A) \leq \varphi(B)<math>
  • Countable sub-additivity: for any sequence {Aj}j of subsets of X (pairwise disjoint or not)
<math> \varphi\left(\bigcup_{j=1}^\infty A_j\right) \leq \sum_{j=1}^\infty \varphi(A_j)<math>

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

<math> \varphi(A) = \varphi(A \cap E) + \varphi(A \setminus E). <math>

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

<math> \varphi(E \cup F) = \varphi(E) + \varphi(F)<math>

whenever

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

then φ is called a metric outer measure. The Borel sets of X are the elements of the smallest σ-algebra generated by the open sets.

Theorem. If φ is a metric outer measure on X, then every Borel subset of X is φ-measurable.

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 subset of 2X which contains the empty set and p an extended real valued function on C which vanishes on the empty set.

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

<math> \varphi(E) = \inf \left\{ \sum_{i=1}^\infty p(A_i)\right\}<math>

where 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 subset of 2X which contains the empty set and p an extended real valued function on C which vanishes on the empty set. For each δ > 0, let

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

and

<math> \varphi_\delta(E) = \inf \left\{ \sum_{i=1}^\infty p(A_i)\right\}<math>

where the infimum extends over all sequences {Ai}i of elements of Cδ which cover E. Obviously, φδ ≥ φδ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus

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

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.

References

  • P. Halmos, Measure theory, D. van Nostrand and Co., 1950
  • M. E. Munroe, Introduction to Measure and Integration, Addison Wesley, 1953
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