High Quality Content by WIKIPEDIA articles! 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 Caratheodory to provide a basis for the theory of measurable sets and countably additive measures. Caratheodory'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 Caratheodory'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 or balls in R3.

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