en>Yobot: WP:CHECKWIKI error fixes using AWB (10390)

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WP:CHECKWIKI error fixes using AWB (10390)

en>Bbanerje: /* Derivatives of vector valued functions of vectors */ Dot product of two vectors is a scalar, not another vector. Dot of second order tensor with a vector is a vector.

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Derivatives of vector valued functions of vectors: Dot product of two vectors is a scalar, not another vector. Dot of second order tensor with a vector is a vector.

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			In [[mathematics]], in particular in [[measure theory]], an '''outer measure''' or ''exterior measure'' is a [[function (mathematics)\|function]] defined on all subsets of a given [[Set (mathematics)\|set]] with values in the [[extended real line\|extended real numbers]] satisfying some additional technical conditions. A general theory of outer measures was first introduced<ref>See pp379, C.D. Aliprantis, K.C. Border, ''Infinite Dimensional Analysis'', 3rd ed, Springer 2006. ISBN 3-540-29586-0</ref> by [[Constantin Carathéodory]]<ref>[[Constantin Carathéodory\|C. Carathéodory]], ''Vorlesungen über reelle Funktionen'', 1st ed, Berlin: Leipzig 1918, 2nd ed, New York: Chelsea 1948.</ref> to provide a basis for the theory of [[measurable set]]s and [[sigma additive\|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 [[Felix Hausdorff\|Hausdorff]] to define a dimension-like metric [[invariant (mathematics)\|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 '''R'''<sup>3</sup>. 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 <math>A+x = \{ a+x: a\in A\}</math>)
			# [[Countable additivity]]: for any [[sequence]] (''A''<sub>''j''</sub>) of pairwise [[disjoint set\|disjoint subsets]] of '''R'''
			::<math> \varphi\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \varphi(A_i).</math>

			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 {{math\|X}} is a function
			:<math>\varphi: 2^X \rightarrow [0, \infty], </math>
			defined on all subsets of {{math\|X}} ({{math\|2<sup>X</sup>}} 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]]'').
			:: <math> \varphi(\varnothing) = 0</math>
			* [[monotonic function\|Monotonicity]]: For any two subsets {{math\|A}} and {{math\|B}} of {{math\|X}},
			:: <math>A\subseteq B\quad\text{implies}\quad\varphi(A) \leq \varphi(B).</math>
			* Countable [[subadditivity]]: For any sequence {{math\|{A<sub>j</sub>} }} of subsets of {{math\|X}} (pairwise disjoint or not),
			:: <math> \varphi\left(\bigcup_{j=1}^\infty A_j\right) \leq \sum_{j=1}^\infty \varphi(A_j).</math>

			This allows us to define the concept of measurability as follows: a subset {{math\|E}} of {{math\|X}} is φ-measurable (or [[Carathéodory]]-measurable by φ) [[iff]] for every subset {{math\|A}} of {{math\|X}}

			:<math> \varphi(A) = \varphi(A \cap E) + \varphi(A \cap E^c). </math>

			'''Theorem'''. The φ-measurable sets form a [[sigma algebra\|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 so important for measure theory and the theory of [[integral]]s.

			== Outer measure and topology ==

			Suppose {{math\|(X, d)}} is a [[metric space]] and {{math\|φ}} an outer measure on {{math\|X}}. If {{math\|φ}} has the property that

			:<math> \varphi(E \cup F) = \varphi(E) + \varphi(F)</math>

			whenever

			:<math> d(E,F) = \inf\{d(x,y): x \in E, y \in F\} > 0, </math>

			then {{math\|φ}} is called a [[metric outer measure]].

			'''Theorem'''. If {{math\|φ}} is a metric outer measure on {{math\|X}}, then every Borel subset of {{math\|X}} is {{math\|φ}}-measurable. (The [[Borel algebra\|Borel set]]s of {{math\|X}} are the elements of the smallest {{math\|σ}}-algebra generated by the open sets.)

			== Construction of outer measures ==

			There are several procedures for constructing outer measures on a set. The classic Munroe reference below describes two particularly useful ones which are referred to as '''Method I''' and '''Method II'''.

			=== Method I===
			Let {{math\|X}} be a set, {{math\|C}} a family of subsets of {{math\|X}} which contains the empty set and {{math\|p}} a non-negative extended real valued function on {{math\|C}} which vanishes on the empty set.

			'''Theorem'''. Suppose the family {{math\|C}} and the function {{math\|p}} are as above and define

			:<math> \varphi(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg\|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C\biggr\}.</math>

			That is, the [[infimum]] extends over all sequences {{math\|{A<sub>i</i>} }} of elements of {{math\|C}} which cover {{math\|E}}, with the convention that the infimum is infinite if no such sequence exists. Then {{math\|φ}} is an outer measure on {{math\|X}}.

			===Method II ===
			The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures. Suppose {{math\|(X, d)}} is a metric space. As above {{math\|C}} is a family of subsets of {{math\|X}} which contains the empty set and {{math\|p}} a non-negative extended real valued function on {{math\|C}} which vanishes on the empty set. For each {{math\|δ > 0}}, let

			:<math>C_\delta= \{A \in C: \operatorname{diam}(A) \leq \delta\} </math>

			and

			:<math> \varphi_\delta(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg\|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C_\delta\biggr\}.</math>

			Obviously, {{math\|φ<sub>δ</sub> ≥ φ<sub>δ'</sub>}} when {{math\|δ ≤ δ'}} since the infimum is taken over a smaller class as {{math\|δ}} decreases. Thus

			:<math> \lim_{\delta \rightarrow 0} \varphi_\delta(E) = \varphi_0(E) \in [0, \infty]</math>

			exists (possibly infinite).

			'''Theorem'''. {{math\|φ<sub>0</sub>}} is a metric outer measure on {{math\|X}}.

			This is the construction used in the definition of [[Hausdorff measure]]s for a metric space.

			==See also==
			*[[Inner measure]]

			== References ==
			<references/>
			* [[Paul Halmos\|P. Halmos]], ''Measure theory'', D. van Nostrand and Co., 1950
			* M. E. Munroe, ''Introduction to Measure and Integration'', Addison Wesley, 1953
			* A. N. Kolmogorov & S. V. Fomin, translated by Richard A. Silverman, ''Introductory Real Analysis'', Dover Publications, New York, 1970 ISBN 0-486-61226-0

			==External links==
			* [http://www.encyclopediaofmath.org/index.php/Outer_measure Outer measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
			* [http://www.encyclopediaofmath.org/index.php/Caratheodory_measure Caratheodory measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

			[[Category:Measures (measure theory)]]

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en>Yobot: WP:CHECKWIKI error fixes using AWB (10390)

en>Bbanerje: /* Derivatives of vector valued functions of vectors */ Dot product of two vectors is a scalar, not another vector. Dot of second order tensor with a vector is a vector.

en>Helpful Pixie Bot: ISBNs (Build KE)