Outer Measure

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 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.

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:

1. Any interval of reals has measure b − a
2. The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
3. Translation invariance: For any set A and any real x, the sets A and A+x have the same measure (where )
4. Countable additivity: for any sequence (A_j) of pairwise disjoint subsets of X

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.