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 measuretheoretic 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 dimensionlike 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 or balls in R^{3}. One might expect to define a generalized measuring function φ that fulfils the following three requirements:
It turns out the second and third requirements together for all sets are incompatible conditions; see nonmeasurable 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.
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An outer measure is a function
defined on all subsets of a set X, that satisfies the following conditions:
This allows us to define the concept of measurability as follows: a subset E of X is φmeasurable (or Carathéodorymeasurable by φ) iff for every subset A of X
Theorem. The φmeasurable sets form a σalgebra 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.
Suppose (X, d) is a metric space and φ an outer measure on X. If φ has the property that
whenever
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 φmeasurable. (The Borel sets of X are the elements of the smallest σalgebra generated by the open sets.)
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 nonnegative 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
That is, the infimum extends over all sequences (A_{i})_{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 nonnegative extended real valued function on C which vanishes on the empty set. For each δ > 0, let
and
Obviously, φ_{δ} ≥ φ_{δ'} when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus
exists.
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.
