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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〔See pp379, C.D. Aliprantis, K.C. Border, ''Infinite Dimensional Analysis'', 3rd ed, Springer 2006. ISBN 3-540-29586-0〕 by Constantin Carathéodory〔C. Carathéodory, ''Vorlesungen über reelle Funktionen'', 1st ed, Berlin: Leipzig 1918, 2nd ed, New York: Chelsea 1948.〕 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: # Any interval of reals (''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 ) # Countable additivity: for any sequence (''A''''j'') 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 is a function : defined on all subsets of ( is another notation for the power set), that satisfies the following conditions: *Null empty set: The empty set has zero outer measure (''see also: measure zero''). :: * Monotonicity: For any two subsets and of , :: * Countable subadditivity: For any sequence of subsets of (pairwise disjoint or not), :: This allows us to define the concept of measurability as follows: a subset of is φ-measurable (or Carathéodory-measurable by φ) iff for every subset of : Theorem. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive complete measure. For a proof of this theorem see the Halmos reference, section 11. This method is known as the Carathéodory construction and is one way of arriving at the concept of Lebesgue measure that is important for measure theory and the theory of integrals. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「outer measure」の詳細全文を読む スポンサード リンク
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