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 by Constantin 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 (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.
Contents
- Formal definitions
- Outer measure and topology
- Construction of outer measures
- Method I
- Method II
- References
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:
- Any interval of reals [a, b] has measure b − a
- The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
- Translation invariance: For any set A and any real x, the sets A and A+x have the same measure (where
A + x = { a + x : a ∈ A } ) - Countable additivity: for any sequence (Aj) of pairwise disjoint subsets of R
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.
Formal definitions
An outer measure on a set
defined on all subsets of
This allows us to define the concept of measurability as follows: a subset
where
Theorem. The
Outer measure and topology
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.)
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.
Method I
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 {Ai} 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.
Method II
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, φδ ≥ φδ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus
exists (possibly infinite).
Theorem. φ0 is a metric outer measure on X.
This is the construction used in the definition of Hausdorff measures for a metric space.