Erdős conjecture on arithmetic progressions - Revision history

en>Joel B. Lewis at 03:51, 7 December 2014

2014-12-07T03:51:46Z

← Older revision		Revision as of 05:51, 7 December 2014
Line 1:		Line 1:
	~~In [[mathematics]], more precisely in [[measure theory]], '~~''Lebesgue's decomposition theorem'''<ref>{{harv\|Halmos\|1974\|loc=Section 32, Theorem C}}</ref><ref>{{harv\|Hewitt\|Stromberg\|1965\|loc=Chapter V, § 19, (19.42) Lebesque Decomposition Theorem}}</ref><ref>{{harv\|Rudin\|1974\|loc=Section 6.9, The Theorem of Lebesgue-Radon-Nikodym}}</ref> states that for every two [[sigma-finite measure\|σ-finite]] [[signed measure]]s ~~<math>\mu</math> and <math>\nu</math>~~ on a [~~[measurable space]] <math>(\Omega,\Sigma),<~~/~~math> there exist two σ-finite signed measures <math>\nu_0<~~/~~math> and <math>\nu_1<~~/~~math> such that:~~		Golda is what's written on my birth [http://cartoonkorea.com/ce002/1093612 love psychic readings] certification although it is not the title on my birth certification. Distributing production is exactly where her main earnings arrives from. My wife and I reside in Mississippi but now I'm considering other choices. The preferred hobby for him and his children is to perform lacross and he would never give it up.<br><br>Feel free to surf to my web blog :: accurate psychic [http://modenpeople.co.kr/modn/qna/292291 accurate psychic readings] readings ([http://netwk.hannam.ac.kr/xe/data_2/85669 source website])

	* <math>\nu=\nu_0+\nu_1\, </~~math>~~
	* <math>\nu_0\ll\mu</math> (that is, <math>\nu_0</math> is [[absolutely continuous]~~] with respect to <math>\mu</math>)~~
	* <math>\nu_1\perp\mu</math> (that is~~, <math>\nu_1</math> and <math>\mu</math> are [[singular measure\|singular]]).~~

	~~These two measures are uniquely determined by <math>\mu</math> and <math>\nu</math>.~~

	~~==Refinement==~~
	~~Lebesgue's decomposition theorem can be refined in a number of ways.~~

	~~First,~~ the ~~decomposition of the [[singular measure\|singular]] part of a regular [[Borel measure]]~~ on ~~the [[real line]] can be refined:<ref>{{harv\|Hewitt\|Stromberg\|1965\|loc=Chapter V, § 19, (19~~.~~61) Theorem}}</ref>~~
	~~:<math>\, \nu = \nu_{\mathrm{cont}} + \nu_{\mathrm{sing}} + \nu_{\mathrm{pp}}</math>~~
	where
	* ''ν''<sub>cont</sub> is the '''absolutely continuous''' part
	* ''ν''<sub>sing</sub> is the '''singular continuous''' part
	* ''ν''<sub>pp</sub> is the '''pure point''' part (a [[discrete measure]]).

	~~Second, absolutely continuous measures are classified by the [[Radon–Nikodym theorem]],~~ and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The [[Cantor measure]] (the [[probability measure]] on the [[real line]] whose [[cumulative distribution function]] is the [[Cantor function]]) is an example of a singular continuous measure.

	~~== Related concepts ==~~
	~~=== Lévy–Itō decomposition ===~~
	~~{{main\|Lévy–Itō decomposition}}~~
	The ~~analogous decomposition~~ for ~~a [[stochastic processes]]~~ is ~~the [[Lévy–Itō decomposition]]: given a [[Lévy process]] ''X,''~~ it ~~can be decomposed as a sum of three independent [[Lévy process\|Lévy processes]] <math>X=X^{(1)}+X^{(2)}+X^{(3)}~~<~~/math~~> ~~where:~~
	* <~~math~~>~~X^{(1)}</math> is a [[Brownian motion]] with drift, corresponding~~ to ~~the absolutely continuous part;~~
	* <math>X^{(2)}</math> is a [[compound Poisson process]], corresponding to ~~the pure point part;~~
	* <math>X^{(3)}</math> is a [[square integrable]] pure jump [[Martingale (probability theory)\|martingale]] that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.

	~~==See also==~~
	* [[Decomposition_of_spectrum_(functional_analysis)\|Decomposition of spectrum]]
	* [[Hahn decomposition theorem]] and the corresponding Jordan decomposition theorem

	~~==Citations==~~
	~~{{Reflist\|2}}~~

	~~==References==~~
	* {{Citation
	~~\| last = Halmos~~
	~~\| first = Paul R.~~
	~~\| author-link = Paul Halmos~~
	~~\| title = Measure Theory~~
	~~\| place = New York, Heidelberg, Berlin~~
	~~\| publisher = Springer-Verlag~~
	~~\| series = Graduate Texts in Mathematics~~
	~~\| volume = 18~~
	~~\| origyear = 1950~~
	~~\| year = 1974~~
	~~\| isbn = 978-0-387-90088-9~~
	~~\| mr = 0033869~~
	~~\| zbl = 0283.28001}}~~
	* {{Citation
	~~\| last = Hewitt~~
	~~\| first = Edwin~~
	~~\| author-link = Edwin Hewitt~~
	~~\| last2 = Stromberg~~
	~~\| first2 = Karl~~
	~~\| title = Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable~~
	~~\| place = Berlin, Heidelberg, New York~~
	~~\| publisher = Springer-Verlag~~
	~~\| series = Graduate Texts in Mathematics~~
	~~\| volume = 25~~
	~~\| year = 1965~~
	~~\| isbn = 978-0-387-90138-1~~
	~~\| mr = 0188387~~
	~~\| zbl = 0137.03202}}~~
	* {{Citation
	~~\| last = Rudin~~
	~~\| first = Walter~~
	~~\| author-link = Walter Rudin~~
	~~\| title = Real and Complex Analysis~~
	~~\| place = New York, Düsseldorf, Johannesburg~~
	~~\| publisher = McGraw-Hill Book Comp.~~
	~~\| series = McGraw-Hill Series in Higher Mathematics~~
	~~\| edition = 2nd~~
	~~\| year = 1974~~
	~~\| isbn = 0-07-054233-3~~
	~~\| mr = 0344043~~
	~~\| zbl = 0278.26001}}~~

	~~==External links==~~
	* [http://~~www~~.~~encyclopediaofmath~~.~~org~~/~~index.php/Lebesgue_decomposition Lebesgue decomposition] at [http:~~//~~www.encyclopediaofmath.org/ Encyclopedia of Mathematics~~]
	* [http://~~www~~.~~encyclopediaofmath~~.~~org/index~~.~~php~~/~~Absolutely_continuous_measures Absolutely continuous measures] at [http:~~//~~www.encyclopediaofmath.org/ Encyclopedia of Mathematics]~~

	~~{{PlanetMath attribution\|id=4003\|title=Lebesgue decomposition theorem}}~~

	~~[[Category:Integral calculus]]~~
	~~[[Category:Theorems in measure theory]~~]

en>Tosha at 22:08, 31 January 2014

2014-01-31T22:08:20Z

← Older revision		Revision as of 00:08, 1 February 2014
Line 1:		Line 1:
	~~Hi there~~, ~~I am Andrew Berryhill~~. The ~~preferred hobby~~ for ~~him~~ and ~~his kids~~ is ~~style~~ and he'll be ~~starting some thing else along with it~~. ~~Invoicing~~ is ~~my profession~~. ~~Alaska~~ is ~~exactly~~ where ~~he's usually been living.~~<br><br>~~my web site~~: [http://~~modenpeople~~.co.kr/~~modn~~/~~qna~~/~~292291 tarot readings~~]		In [[mathematics]], more precisely in [[measure theory]], '''Lebesgue's decomposition theorem'''<ref>{{harv\|Halmos\|1974\|loc=Section 32, Theorem C}}</ref><ref>{{harv\|Hewitt\|Stromberg\|1965\|loc=Chapter V, § 19, (19.42) Lebesque Decomposition Theorem}}</ref><ref>{{harv\|Rudin\|1974\|loc=Section 6.9, The Theorem of Lebesgue-Radon-Nikodym}}</ref> states that for every two [[sigma-finite measure\|σ-finite]] [[signed measure]]s <math>\mu</math> and <math>\nu</math> on a [[measurable space]] <math>(\Omega,\Sigma),</math> there exist two σ-finite signed measures <math>\nu_0</math> and <math>\nu_1</math> such that:

			* <math>\nu=\nu_0+\nu_1\, </math>
			* <math>\nu_0\ll\mu</math> (that is, <math>\nu_0</math> is [[absolutely continuous]] with respect to <math>\mu</math>)
			* <math>\nu_1\perp\mu</math> (that is, <math>\nu_1</math> and <math>\mu</math> are [[singular measure\|singular]]).

			These two measures are uniquely determined by <math>\mu</math> and <math>\nu</math>.

			==Refinement==
			Lebesgue's decomposition theorem can be refined in a number of ways.

			First, the decomposition of the [[singular measure\|singular]] part of a regular [[Borel measure]] on the [[real line]] can be refined:<ref>{{harv\|Hewitt\|Stromberg\|1965\|loc=Chapter V, § 19, (19.61) Theorem}}</ref>
			:<math>\, \nu = \nu_{\mathrm{cont}} + \nu_{\mathrm{sing}} + \nu_{\mathrm{pp}}</math>
			where
			* ''ν''<sub>cont</sub> is the '''absolutely continuous''' part
			* ''ν''<sub>sing</sub> is the '''singular continuous''' part
			* ''ν''<sub>pp</sub> is the '''pure point''' part (a [[discrete measure]]).

			Second, absolutely continuous measures are classified by the [[Radon–Nikodym theorem]], and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The [[Cantor measure]] (the [[probability measure]] on the [[real line]] whose [[cumulative distribution function]] is the [[Cantor function]]) is an example of a singular continuous measure.

			== Related concepts ==
			=== Lévy–Itō decomposition ===
			{{main\|Lévy–Itō decomposition}}
			The analogous decomposition for a [[stochastic processes]] is the [[Lévy–Itō decomposition]]: given a [[Lévy process]] ''X,'' it can be decomposed as a sum of three independent [[Lévy process\|Lévy processes]] <math>X=X^{(1)}+X^{(2)}+X^{(3)}</math> where:
			* <math>X^{(1)}</math> is a [[Brownian motion]] with drift, corresponding to the absolutely continuous part;
			* <math>X^{(2)}</math> is a [[compound Poisson process]], corresponding to the pure point part;
			* <math>X^{(3)}</math> is a [[square integrable]] pure jump [[Martingale (probability theory)\|martingale]] that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.

			==See also==
			* [[Decomposition_of_spectrum_(functional_analysis)\|Decomposition of spectrum]]
			* [[Hahn decomposition theorem]] and the corresponding Jordan decomposition theorem

			==Citations==
			{{Reflist\|2}}

			==References==
			* {{Citation
			\| last = Halmos
			\| first = Paul R.
			\| author-link = Paul Halmos
			\| title = Measure Theory
			\| place = New York, Heidelberg, Berlin
			\| publisher = Springer-Verlag
			\| series = Graduate Texts in Mathematics
			\| volume = 18
			\| origyear = 1950
			\| year = 1974
			\| isbn = 978-0-387-90088-9
			\| mr = 0033869
			\| zbl = 0283.28001}}
			* {{Citation
			\| last = Hewitt
			\| first = Edwin
			\| author-link = Edwin Hewitt
			\| last2 = Stromberg
			\| first2 = Karl
			\| title = Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable
			\| place = Berlin, Heidelberg, New York
			\| publisher = Springer-Verlag
			\| series = Graduate Texts in Mathematics
			\| volume = 25
			\| year = 1965
			\| isbn = 978-0-387-90138-1
			\| mr = 0188387
			\| zbl = 0137.03202}}
			* {{Citation
			\| last = Rudin
			\| first = Walter
			\| author-link = Walter Rudin
			\| title = Real and Complex Analysis
			\| place = New York, Düsseldorf, Johannesburg
			\| publisher = McGraw-Hill Book Comp.
			\| series = McGraw-Hill Series in Higher Mathematics
			\| edition = 2nd
			\| year = 1974
			\| isbn = 0-07-054233-3
			\| mr = 0344043
			\| zbl = 0278.26001}}

			==External links==
			* [http://www.encyclopediaofmath.org/index.php/Lebesgue_decomposition Lebesgue decomposition] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
			* [http://www.encyclopediaofmath.org/index.php/Absolutely_continuous_measures Absolutely continuous measures] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

			{{PlanetMath attribution\|id=4003\|title=Lebesgue decomposition theorem}}

			[[Category:Integral calculus]]
			[[Category:Theorems in measure theory]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-09T14:40:51Z

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Hi there, I am Andrew Berryhill. The preferred hobby for him and his kids is style and he'll be starting some thing else along with it. Invoicing is my profession. Alaska is exactly where he's usually been living.<br><br>my web site: [http://modenpeople.co.kr/modn/qna/292291 tarot readings]

← Older revision		Revision as of 05:51, 7 December 2014
Line 1:		Line 1:
	~~In [[mathematics]], more precisely in [[measure theory]], '~~''Lebesgue's decomposition theorem'''<ref>{{harv\|Halmos\|1974\|loc=Section 32, Theorem C}}</ref><ref>{{harv\|Hewitt\|Stromberg\|1965\|loc=Chapter V, § 19, (19.42) Lebesque Decomposition Theorem}}</ref><ref>{{harv\|Rudin\|1974\|loc=Section 6.9, The Theorem of Lebesgue-Radon-Nikodym}}</ref> states that for every two [[sigma-finite measure\|σ-finite]] [[signed measure]]s ~~<math>\mu</math> and <math>\nu</math>~~ on a [~~[measurable space]] <math>(\Omega,\Sigma),<~~/~~math> there exist two σ-finite signed measures <math>\nu_0<~~/~~math> and <math>\nu_1<~~/~~math> such that:~~		Golda is what's written on my birth [http://cartoonkorea.com/ce002/1093612 love psychic readings] certification although it is not the title on my birth certification. Distributing production is exactly where her main earnings arrives from. My wife and I reside in Mississippi but now I'm considering other choices. The preferred hobby for him and his children is to perform lacross and he would never give it up.<br><br>Feel free to surf to my web blog :: accurate psychic [http://modenpeople.co.kr/modn/qna/292291 accurate psychic readings] readings ([http://netwk.hannam.ac.kr/xe/data_2/85669 source website])

	* <math>\nu=\nu_0+\nu_1\, </~~math>~~
	* <math>\nu_0\ll\mu</math> (that is, <math>\nu_0</math> is [[absolutely continuous]~~] with respect to <math>\mu</math>)~~
	* <math>\nu_1\perp\mu</math> (that is~~, <math>\nu_1</math> and <math>\mu</math> are [[singular measure\|singular]]).~~

	~~These two measures are uniquely determined by <math>\mu</math> and <math>\nu</math>.~~

	~~==Refinement==~~
	~~Lebesgue's decomposition theorem can be refined in a number of ways.~~

	~~First,~~ the ~~decomposition of the [[singular measure\|singular]] part of a regular [[Borel measure]]~~ on ~~the [[real line]] can be refined:<ref>{{harv\|Hewitt\|Stromberg\|1965\|loc=Chapter V, § 19, (19~~.~~61) Theorem}}</ref>~~
	~~:<math>\, \nu = \nu_{\mathrm{cont}} + \nu_{\mathrm{sing}} + \nu_{\mathrm{pp}}</math>~~
	where
	* ''ν''<sub>cont</sub> is the '''absolutely continuous''' part
	* ''ν''<sub>sing</sub> is the '''singular continuous''' part
	* ''ν''<sub>pp</sub> is the '''pure point''' part (a [[discrete measure]]).

	~~Second, absolutely continuous measures are classified by the [[Radon–Nikodym theorem]],~~ and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The [[Cantor measure]] (the [[probability measure]] on the [[real line]] whose [[cumulative distribution function]] is the [[Cantor function]]) is an example of a singular continuous measure.

	~~== Related concepts ==~~
	~~=== Lévy–Itō decomposition ===~~
	~~{{main\|Lévy–Itō decomposition}}~~
	The ~~analogous decomposition~~ for ~~a [[stochastic processes]]~~ is ~~the [[Lévy–Itō decomposition]]: given a [[Lévy process]] ''X,''~~ it ~~can be decomposed as a sum of three independent [[Lévy process\|Lévy processes]] <math>X=X^{(1)}+X^{(2)}+X^{(3)}~~<~~/math~~> ~~where:~~
	* <~~math~~>~~X^{(1)}</math> is a [[Brownian motion]] with drift, corresponding~~ to ~~the absolutely continuous part;~~
	* <math>X^{(2)}</math> is a [[compound Poisson process]], corresponding to ~~the pure point part;~~
	* <math>X^{(3)}</math> is a [[square integrable]] pure jump [[Martingale (probability theory)\|martingale]] that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.

	~~==See also==~~
	* [[Decomposition_of_spectrum_(functional_analysis)\|Decomposition of spectrum]]
	* [[Hahn decomposition theorem]] and the corresponding Jordan decomposition theorem

	~~==Citations==~~
	~~{{Reflist\|2}}~~

	~~==References==~~
	* {{Citation
	~~\| last = Halmos~~
	~~\| first = Paul R.~~
	~~\| author-link = Paul Halmos~~
	~~\| title = Measure Theory~~
	~~\| place = New York, Heidelberg, Berlin~~
	~~\| publisher = Springer-Verlag~~
	~~\| series = Graduate Texts in Mathematics~~
	~~\| volume = 18~~
	~~\| origyear = 1950~~
	~~\| year = 1974~~
	~~\| isbn = 978-0-387-90088-9~~
	~~\| mr = 0033869~~
	~~\| zbl = 0283.28001}}~~
	* {{Citation
	~~\| last = Hewitt~~
	~~\| first = Edwin~~
	~~\| author-link = Edwin Hewitt~~
	~~\| last2 = Stromberg~~
	~~\| first2 = Karl~~
	~~\| title = Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable~~
	~~\| place = Berlin, Heidelberg, New York~~
	~~\| publisher = Springer-Verlag~~
	~~\| series = Graduate Texts in Mathematics~~
	~~\| volume = 25~~
	~~\| year = 1965~~
	~~\| isbn = 978-0-387-90138-1~~
	~~\| mr = 0188387~~
	~~\| zbl = 0137.03202}}~~
	* {{Citation
	~~\| last = Rudin~~
	~~\| first = Walter~~
	~~\| author-link = Walter Rudin~~
	~~\| title = Real and Complex Analysis~~
	~~\| place = New York, Düsseldorf, Johannesburg~~
	~~\| publisher = McGraw-Hill Book Comp.~~
	~~\| series = McGraw-Hill Series in Higher Mathematics~~
	~~\| edition = 2nd~~
	~~\| year = 1974~~
	~~\| isbn = 0-07-054233-3~~
	~~\| mr = 0344043~~
	~~\| zbl = 0278.26001}}~~

	~~==External links==~~
	* [http://~~www~~.~~encyclopediaofmath~~.~~org~~/~~index.php/Lebesgue_decomposition Lebesgue decomposition] at [http:~~//~~www.encyclopediaofmath.org/ Encyclopedia of Mathematics~~]
	* [http://~~www~~.~~encyclopediaofmath~~.~~org/index~~.~~php~~/~~Absolutely_continuous_measures Absolutely continuous measures] at [http:~~//~~www.encyclopediaofmath.org/ Encyclopedia of Mathematics]~~

	~~{{PlanetMath attribution\|id=4003\|title=Lebesgue decomposition theorem}}~~

	~~[[Category:Integral calculus]]~~
	~~[[Category:Theorems in measure theory]~~]