101.170.170.166 at 06:19, 31 December 2014

2014-12-31T06:19:06Z

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	"John Howell brings important experience to Stateside Associates," said President and CEO Constance Campanella. "We could not be more excited about integrating his huge experience in telecom, authorities affairs and regulatory problem management with the comprehensive data providers we offer our clients." At Sprint Nextel Mr. Howell's duty extended to corporate compliance with major FCC mandates, together with enhanced E-911 and wireless native quantity portability.		Also visit my site: [http://www.kalamazoooptometry.com/mediac/p/r/jimmychoo/ ジミーチュウ靴アウトレット]

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	11 and Hurricane Katrina. In addition to serving as a daily speaker on regulatory matters earlier than industry boards and state commissions, Mr. Howell was appointed by Governor Warner to serve on the Virginia E-911 Providers Board on which he additionally served because the Chair of the Legislative Subcommittee which reviewed and rewrote the Board's enabling legislation. He has gained awards for his work, together with for Dash Nextel Operational Excellence and for excellent efficiency in managing the FCC Network Reliability and Interoperability Council.

en>Dicklyon: Reverted 1 edit by 150.250.121.253 (talk): & is the alignment point, not the equal sign. (TW)

2014-03-03T03:46:26Z

Reverted 1 edit by 150.250.121.253 (talk): & is the alignment point, not the equal sign. (TW)

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129.16.106.189 at 14:10, 29 January 2014

2014-01-29T14:10:04Z

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In [[Measure (mathematics)|measure theory]], the '''Lebesgue measure''', named after [[france|French]] mathematician [[Henri Lebesgue]], is the standard way of assigning a [[measure (mathematics)|measure]] to [[subset]]s of ''n''-dimensional [[Euclidean space]]. For ''n'' = 1, 2, or 3, it coincides with the standard measure of [[length]], [[area]], or [[volume]]. In general, it is also called '''''n''-dimensional volume''', '''''n''-volume''', or simply '''volume'''.<ref>The term ''[[volume]]'' is also used, more strictly, as a [[synonym]] of 3-dimensional volume</ref> It is used throughout [[real analysis]], in particular to define [[Lebesgue integration]]. Sets that can be assigned a Lebesgue measure are called '''Lebesgue measurable'''; the measure of the Lebesgue measurable set ''A'' is denoted by λ(''A'').

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.<ref>{{cite journal |author=Henri Lebesgue |title=Intégrale, longueur, aire |year=1902 |publisher=Université de Paris }}</ref>

The Lebesgue measure is often denoted ''dx'', but this should not be confused with the distinct notion of a [[volume form]].

== Definition ==

Given a subset <math>E\subset\mathbb{R}</math>, with the length of an (open, closed, semi-open) interval <math>I = [a,b]</math> given by <math>l(I)=b - a</math>, the Lebesgue outer measure <math>\lambda^*(E)</math> is defined as

:<math>\lambda^*(E) = \text{inf} \left\{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of open intervals with } E\subset \bigcup_{k=1}^\infty I_k\right\}</math>.

The Lebesgue measure of E is given by its Lebesgue outer measure <math>\lambda(E)=\lambda^*(E)</math> if, for every <math> A\subset\mathbb{R}</math>,

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

== Examples ==

* Any [[closed interval]] [''a'', ''b''] of [[real number]]s is Lebesgue measurable, and its Lebesgue measure is the length ''b''−''a''. The [[open interval]] (''a'', ''b'') has the same measure, since the [[set difference|difference]] between the two sets consists only of the end points ''a'' and ''b'' and has [[measure zero]].
* Any [[Cartesian product]] of intervals [''a'', ''b''] and [''c'', ''d''] is Lebesgue measurable, and its Lebesgue measure is (''b''−''a'')(''d''−''c''), the area of the corresponding [[rectangle]].
* The Lebesgue measure of the set of [[rational numbers]] in an interval of the line is 0, although the set is [[Dense set|dense]] in the interval.
* The [[Cantor set]] is an example of an [[uncountable set]] that has Lebesgue measure zero.
* [[Vitali set]]s are examples of sets that are [[non-measurable set|not measurable]] with respect to the Lebesgue measure. Their existence relies on the [[axiom of choice]].

== Properties ==
[[File:Translation of a set.svg|thumb|300px|Translation invariance: The Lebesgue measure of <math>A</math> and <math>A+t</math> are the same.]]
The Lebesgue measure on '''R'''''n'' has the following properties:

# If ''A'' is a [[cartesian product]] of [[interval (mathematics)|intervals]] ''I''1 × ''I''2 × ... × ''I''''n'', then ''A'' is Lebesgue measurable and <math>\lambda (A)=|I_1|\cdot |I_2|\cdots |I_n|.</math> Here, |''I''| denotes the length of the interval ''I''.
# If ''A'' is a [[disjoint union]] of [[countable|countably many]] disjoint Lebesgue measurable sets, then ''A'' is itself Lebesgue measurable and λ(''A'') is equal to the sum (or [[infinite series]]) of the measures of the involved measurable sets.
# If ''A'' is Lebesgue measurable, then so is its [[Complement (set theory)|complement]].
# λ(''A'') ≥ 0 for every Lebesgue measurable set ''A''.
# If ''A'' and ''B'' are Lebesgue measurable and ''A'' is a subset of ''B'', then λ(''A'') ≤ λ(''B''). (A consequence of 2, 3 and 4.)
# Countable [[Union (set theory)|unions]] and [[Intersection (set theory)|intersections]] of Lebesgue measurable sets are Lebesgue measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions need not be closed under countable unions: <math>\{\emptyset, \{1,2,3,4\}, \{1,2\}, \{3,4\}, \{1,3\}, \{2,4\}\}</math>.)
# If ''A'' is an [[open set|open]] or [[closed set|closed]] subset of '''R'''''n'' (or even [[Borel set]], see [[metric space]]), then ''A'' is Lebesgue measurable.
# If ''A'' is a Lebesgue measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure (see the [[regularity theorem for Lebesgue measure]]).
# Lebesgue measure is both [[Locally finite measure|locally finite]] and [[Inner regular measure|inner regular]], and so it is a [[Radon measure]].
# Lebesgue measure is [[Strictly positive measure|strictly positive]] on non-empty open sets, and so its [[Support (measure theory)|support]] is the whole of '''R'''''n''.
# If ''A'' is a Lebesgue measurable set with λ(''A'') = 0 (a [[null set]]), then every subset of ''A'' is also a null set. [[A fortiori]], every subset of ''A'' is measurable.
# If ''A'' is Lebesgue measurable and ''x'' is an element of '''R'''''n'', then the ''translation of ''A'' by x'', defined by ''A'' + ''x'' = {''a'' + ''x'' : ''a'' ∈ ''A''}, is also Lebesgue measurable and has the same measure as ''A''.
# If ''A'' is Lebesgue measurable and <math>\delta>0</math>, then the ''dilation of <math>A</math> by <math>\delta</math>'' defined by <math>\delta A=\{\delta x:x\in A\}</math> is also Lebesgue measurable and has measure <math>\delta^{n}\lambda\,(A).</math>
# More generally, if ''T'' is a [[linear transformation]] and ''A'' is a measurable subset of '''R'''''n'', then ''T''(''A'') is also Lebesgue measurable and has the measure <math>|\det(T)|\, \lambda\,(A)</math>.

All the above may be succinctly summarized as follows:

: The Lebesgue measurable sets form a [[sigma-algebra|σ-algebra]] containing all products of intervals, and λ is the unique [[Complete measure|complete]] [[translational invariance|translation-invariant]] [[measure (mathematics)|measure]] on that σ-algebra with <math>\lambda([0,1]\times [0, 1]\times \cdots \times [0, 1])=1.</math>

The Lebesgue measure also has the property of being [[Sigma-finite measure|σ-finite]].

== Null sets ==
{{main|Null set}}
A subset of '''R'''''n'' is a ''null set'' if, for every ε > 0, it can be covered with countably many products of ''n'' intervals whose total volume is at most ε. All [[countable]] sets are null sets.

If a subset of '''R'''''n'' has [[Hausdorff dimension]] less than ''n'' then it is a null set with respect to ''n''-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the [[Euclidean metric]] on '''R'''''n'' (or any metric [[Lipschitz]]{{dn|date=May 2012}} equivalent to it). On the other hand a set may have [[topological dimension]] less than ''n'' and have positive ''n''-dimensional Lebesgue measure. An example of this is the [[Smith–Volterra–Cantor set]] which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.

In order to show that a given set ''A'' is Lebesgue measurable, one usually tries to find a "nicer" set ''B'' which differs from ''A'' only by a null set (in the sense that the [[symmetric difference]] (''A'' − ''B'') <math>\cup</math>(''B'' − ''A'') is a null set) and then show that ''B'' can be generated using countable unions and intersections from open or closed sets.

== Construction of the Lebesgue measure ==
The modern construction of the Lebesgue measure is an application of [[Carathéodory's extension theorem]]. It proceeds as follows.

Fix {{nowrap|''n'' ∈ '''N'''}}. A '''box''' in '''R'''''n'' is a set of the form
:<math>B=\prod_{i=1}^n [a_i,b_i] \, ,</math>
where {{nowrap|''bi'' ≥ ''ai''}}, and the product symbol here represents a Cartesian product. The volume of this box is defined to be
:<math>\operatorname{vol}(B)=\prod_{i=1}^n (b_i-a_i) \, .</math>

For ''any'' subset ''A'' of '''R'''''n'', we can define its [[outer measure]] ''λ''*(''A'') by:

:<math>\lambda^*(A) = \inf \Bigl\{\sum_{B\in \mathcal{C}}\operatorname{vol}(B) : \mathcal{C}\text{ is a countable collection of boxes whose union covers }A\Bigr\} .</math>

We then define the set ''A'' to be Lebesgue measurable if for every subset ''S'' of '''R'''''n'',
:<math>\lambda^*(S) = \lambda^*(S \cap A) + \lambda^*(S - A) \, .</math>

These Lebesgue measurable sets form a [[σ-algebra]], and the Lebesgue measure is defined by {{nowrap|''λ''(''A'') {{=}} ''λ''*(''A'')}} for any Lebesgue measurable set ''A''.

The existence of sets that are not Lebesgue measurable is a consequence of a certain set-theoretical [[axiom]], the [[axiom of choice]], which is independent from many of the conventional systems of axioms for [[set theory]]. The [[Vitali set|Vitali theorem]], which follows from the axiom, states that there exist subsets of '''R''' that are not Lebesgue measurable. Assuming the axiom of choice, [[non-measurable set]]s with many surprising properties have been demonstrated, such as those of the [[Banach–Tarski paradox]].

In 1970, [[Robert M. Solovay]] showed that the existence of sets that are not Lebesgue measurable is not provable within the framework of [[Zermelo–Fraenkel set theory]] in the absence of the axiom of choice (see [[Solovay's model]]).<ref>{{Cite journal |last=Solovay |first=Robert M. |title=A model of set-theory in which every set of reals is Lebesgue measurable |journal=[[Annals of Mathematics]] |jstor=1970696 |series=Second Series |volume=92 |year=1970 |issue=1 |pages=1–56 |doi=10.2307/1970696 }}</ref>

== Relation to other measures ==
The [[Borel measure]] agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not [[Complete measure|complete]].

The [[Haar measure]] can be defined on any [[locally compact]] [[topological group|group]] and is a generalization of the Lebesgue measure ('''R'''''n'' with addition is a locally compact group).

The [[Hausdorff measure]] is a generalization of the Lebesgue measure that is useful for measuring the subsets of '''R'''''n'' of lower dimensions than ''n'', like [[submanifold]]s, for example, surfaces or curves in '''R'''³ and [[fractal]] sets. The Hausdorff measure is not to be confused with the notion of [[Hausdorff dimension]].

It can be shown that [[There is no infinite-dimensional Lebesgue measure|there is no infinite-dimensional analogue of Lebesgue measure]].

== See also ==
* [[Lebesgue's density theorem]]
* [[Duffin–Schaeffer conjecture]]

== References ==
{{reflist}}

[[Category:Measures (measure theory)]]

LaTeX - Revision history

101.170.170.166 at 06:19, 31 December 2014

en>Dicklyon: Reverted 1 edit by 150.250.121.253 (talk): & is the alignment point, not the equal sign. (TW)

129.16.106.189 at 14:10, 29 January 2014