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 or balls in . One might expect to define a generalized measuring function on 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.
Given a set let denote the collection of all subsets of including the empty set An outer measure on 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 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 to be a function such that
Proof of equivalence. |
Suppose that is an outer measure in sense originally given above. If and are subsets of with then by appealing to the definition with and for all one finds that The third condition in the alternative definition is immediate from the trivial observation that
Suppose instead that is an outer measure in the alternative definition. Let be arbitrary subsets of and suppose that |
Let be a set with an outer measure One says that a subset of is -measurable (sometimes called Carathéodory-measurable relative to after the mathematician Carathéodory) if and only if
Informally, this says that a -measurable subset is one which may be used as a building block, breaking any other subset apart into pieces (namely, the piece which is inside of the measurable set together with the piece which is outside of the measurable set). In terms of the motivation for measure theory, one would expect that area, for example, should be an outer measure on the plane. One might then expect that every subset of the plane would be deemed "measurable," following the expected principle that
It is straightforward to use the above definition of -measurability to see that
The following condition is known as the "countable additivity of on measurable subsets."
Proof of countable additivity. |
One automatically has the conclusion in the form " " from the definition of outer measure. So it is only necessary to prove the " " inequality. One has Applying the above definition of -measurability with and with one has |
A similar proof shows that:
The properties given here can be summarized by the following terminology:
Given any outer measure on a set the collection of all -measurable subsets of is a σ-algebra. The restriction of to this -algebra is a measure.
One thus has a measure space structure on arising naturally from the specification of an outer measure on This measure space has the additional property of completeness, which is contained in the following statement:
This is easy to prove by using the second property in the "alternative definition" of outer measure.
Let μ be an outer measure on the set X.
Given another set Y and a map f : X→Y, define f_{#} μ : 2^{Y}→[0,∞] by
One can verify directly from the definitions that f_{#} μ is an outer measure on Y.
Let B be a subset of X. Define μ_{B} : 2^{X}→[0,∞] by
One can check directly from the definitions that μ_{B} is another outer measure on X.
If a subset A of X is μ-measurable, then it is also μ_{B}-measurable for any subset B of X.
Given a map f : X→Y and a subset A of Y, if f^{ −1}(A) is μ-measurable then A is f_{#} μ-measurable. More generally, f^{ −1}(A) is μ-measurable if and only if A is f_{#} (μ_{B})-measurable for every subset B of X.
Given a set X, an outer measure μ on X is said to be regular if any subset can be approximated 'from the outside' by μ-measurable sets. Formally, this is requiring either of the following equivalent conditions:
It is automatic that the second condition implies the first; the first implies the second by considering the intersection of a minimizing sequence of subsets.
Given an outer measure μ on a set X, define ν : 2^{X}→[0,∞] by
Then ν is a regular outer measure on X which assigns the same measure as μ to all μ-measurable subsets of X. Every μ-measurable subset is also ν-measurable, and every ν-measurable subset of finite ν-measure is also μ-measurable.
So the measure space associated to ν may have a larger σ-algebra than the measure space associated to μ. The restrictions of ν and μ to the smaller σ-algebra are identical. The elements of the larger σ-algebra which are not contained in the smaller σ-algebra have infinite ν-measure and finite μ-measure.
From this perspective, ν may be regarded as an extension of μ.
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 φ-measurable. (The Borel sets of X are the elements of the smallest σ-algebra 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 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
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.
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
and
Obviously, φ_{δ} ≥ φ_{δ'} when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus
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.