Outer measure

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.

[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

One might expect to define a generalized measuring function

The purpose of constructing an outer measure on all subsets of

is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.

such that 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.

[3] Some textbooks, such as Halmos (1950) and Folland (1999), instead define an outer measure on

is an outer measure in the sense of the original definition.

However, the formal logical development of the theory shows that the situation is more complicated.

A formal implication of the axiom of choice is that for any definition of area as an outer measure which includes as a special case the standard formula for the area of a rectangle, there must be subsets of the plane which fail to be measurable.

In particular, the above "expected principle" is false, provided that one accepts the axiom of choice.

A similar proof shows that: The properties given here can be summarized by the following terminology: Given any outer measure

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.

Given a set X, an outer measure μ on X is said to be regular if any subset

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 taking the countable intersection of

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

There are several procedures for constructing outer measures on a set.

Suppose the family C and the function p are as above and define That is, the infimum extends over all sequences {Ai} of elements of C which cover E, with the convention that the infimum is infinite if no such sequence exists.

The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures.

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.

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