Outer measure

In mathematics, in particular in measure theory, an outer 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 developed 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 and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension.

Related Topics:
Mathematics - Measure theory - Carathéodory - Measurable set - Hausdorff - Hausdorff dimension

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Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than mere intervals or open balls in R3. One might expect to define a generalized measuring function φ that fulfils the following three requirements:

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• Any interval of reals has measure b − a
• The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
• Countable additivity, For any sequence {A_j}_j of pairwise disjoint subsets of X

:: arphileft(igcup_{i=1}^infty A_i ight) = sum_{i=1}^infty arphi(A_i)

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It turns out the second and third requirements together for all sets are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of X is to suitably pick out a class of subsets (to be called measurable) in such a way that fulfils the countably additivity property.

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Outer measure: Formal definitions ►