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ountermeasur
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There are 12 letters in OUNTERMEASUR ( A1E1M3N1O1R1S1T1U1 )
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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 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. * 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 * } * * * {\displaystyle A+x=\{a+x:a\in A\}} * ) * Countable additivity: for any sequence (Aj) of pairwise disjoint subsets of R * * * * φ * * ( * * * ⋃ * * i * = * 1 * * * ∞ * * * * A * * i * * * * ) * * = * * ∑ * * i * = * 1 * * * ∞ * * * φ * ( * * A * * i * * * ) * . * * * {\displaystyle \varphi \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\varphi (A_{i}).} * 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. |