Complex measure - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:23:57Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 14:23, 5 May 2014
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	~~In [[mathematics]], more precisely in [[measure theory]], an '''atom'''~~ is ~~a measurable set which has positive measure~~ and ~~contains no set of smaller but positive measure. A measure which has no atoms is called '''non-atomic''' or '''atomless'''.~~		Oscar is what my spouse enjoys to call me and I totally dig that title. My day job is a meter reader. South Dakota is where I've usually been living. Body developing is 1 of the things I love most.<br><br>My website: [http://www.ddaybeauty.com/node/14766 std home test]

	~~==Definition==~~
	~~Given a [[measurable space]] <math>(X, \Sigma)</math> and a [[measure (mathematics)\|measure]] <math>\mu</math> on~~ that ~~space, a set <math>A</math> in <math>\Sigma</math> is called an '''atom''' if~~

	~~: <math> \mu (A) >0\, </math>~~

	~~and for any measurable subset <math>B</math> of <math>A</math> with~~

	~~: <math> \mu(A) > \mu (B) \, </math>~~

	~~one has <math> \mu(B)=0.</math>~~

	~~== Examples==~~
	* Consider the set ''X''={1, 2, ..., 9, 10} and let the sigma-algebra <math>\Sigma</math> be the [[power set]] of ''X''. ~~Define the measure <math>\mu</math> of a set to be its [[cardinality]], that~~ is~~, the number of elements in the set. Then, each of the [[singleton (mathematics)\|singleton]]s {''i''}, for ''i''=1,2, ..., 9, 10 is an atom.~~
	* Consider the [[Lebesgue measure]] on the [[real line]]. This measure has no atoms.

	~~== Non-atomic measures==~~
	~~A measure which has no atoms is called '''non-atomic'''. In other words,~~ a ~~measure is non-atomic if for any measurable set <math>A</math> with <math> \mu (A) >0</math> there exists a measurable subset ''B'' of ''A'' such that~~

	~~: <math> \mu(A) > \mu (B) > 0. \, </math>~~

	A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set ''A'' with <math> \mu (A) >0</math> one can construct a decreasing sequence of measurable sets

	~~:<math>A=A_1\supset A_2 \supset A_3 \supset \cdots</math>~~

	~~such that~~

	~~:<math>\mu(A)=\mu(A_1) > \mu(A_2) > \mu(A_3) > \cdots > 0~~. ~~</math>~~

	~~This may not be true for measures having atoms; see the first example above.~~

	~~It turns out that non-atomic measures actually have a [[Continuum (theory)\|continuum]] of values. It can be proved that if μ is a non-atomic measure and ''A''~~ is ~~a measurable set with <math>\mu (A) >0,</math> then for any real number~~ '~~'b'' satisfying~~

	~~: <math>\mu (A) \geq b \geq0\, </math>~~

	~~there exists a measurable subset ''B'' of ''A'' such that~~

	~~: <math>\mu(B)=b~~.~~\,</math>~~

	~~This theorem~~ is due to [[Wacław Sierpiński]].<ref>{{cite journal \|first=W. \|last=Sierpinski \|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3125.pdf \|title=Sur les fonctions d'ensemble additives et continues \|journal=Fundamenta Mathematicae \|volume=3 \|issue= \|pages=240–246 \|year=1922\|language=French }}</ref><ref>{{Cite book \|last=Fryszkowski \|first=Andrzej \|title=Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications) \|year=2005 \|publisher=Springer \|location=New York \|isbn=1~~-4020-2498-3 \|page=39}}</ref>~~
	~~It is reminiscent~~ of the ~~[[intermediate value theorem]] for continuous functions.~~

	~~'''Sketch of proof''' of Sierpiński's theorem on non-atomic measures~~. ~~A slightly stronger statement, which however makes the proof easier, is that if~~ <~~math~~>~~(X,\Sigma, \mu)~~<~~/math~~> ~~is a non-atomic measure space and <math>\mu(X)=c</math>, there exists a function <math>S~~:[~~0, c]\to\Sigma</math> that is monotone with respect to inclusion, and a right-inverse to <math>\mu~~:~~\Sigma\to[0,\,c]<~~/~~math>. That is, there exists a one-parameter family of measurable sets S(t) such that for all <math>0\leq t \leq t'\leq c</math>~~
	~~:<math>S(t)\subset S(t'),<~~/~~math>~~
	~~:<math>\mu\left (S(t)\right)=t~~.~~</math>~~
	~~The proof easily follows from [[Zorn's lemma]] applied to the set of all monotone partial sections to <math>\mu</math> :~~
	~~:<math>\Gamma:=\{S:D\to\Sigma\; :\; D\subset[0,\,c],\, S\; \mathrm{ monotone }, \forall t\in D\; (\mu\left (S(t)\right)=t)\},</math>~~
	~~ordered by inclusion of graphs, <math>\mathrm{graph}(S)\subset \mathrm{graph}(S')~~.</math> It's then standard to show that every chain in <math>\Gamma</math> has an upper bound in <math>\Gamma</math>, and that any maximal element of <math>\Gamma</math> has domain <math>[0,c],</math> proving the claim.

	~~== See also ==~~
	* [[Atom (order theory)]] — an analogous concept in order theory
	* [[Dirac delta function]]
	* [[Elementary event]], also known as an '''atomic event'''

	~~== Notes==~~
	~~<references~~ />

	~~==References==~~
	* {{Cite book \| author=Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. \| authorlink= \| coauthors= \| title=Real analysis \| year=1997 \| publisher=Prentice-Hall \| location=Upper Saddle River, N.J. \| isbn=0-13-458886-X \| page=108}}
	* {{Cite book \| author=Butnariu, Dan; Klement, E. P. \| authorlink= \| coauthors= \| title=Triangular norm-based measures and games with fuzzy coalitions \| year=1993 \| publisher=Kluwer Academic \| location=Dordrecht \| isbn=0-7923-2369-6 \| page=87}}

	~~[[Category:Measure theory]~~]

en>Leonard Huang at 08:35, 29 November 2013

2013-11-29T08:35:02Z

← Older revision		Revision as of 10:35, 29 November 2013
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	~~Hello~~, ~~my title~~ is ~~Andrew~~ and ~~my wife doesn't like it at all~~. ~~Office supervising~~ is ~~exactly where her main income arrives from~~. ~~The favorite pastime for him~~ and ~~his children~~ is ~~style~~ and he'll be starting ~~something else alongside~~ with it. ~~For many years he~~'~~s been residing in Alaska and~~ [http://~~galab-work~~.cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl=1489804 accurate psychic predictions~~] ~~he doesn~~'~~t plan~~ on ~~changing it~~.<br><br>~~My web~~-~~site~~ :: [~~http~~://si.~~dgmensa.org~~/xe/~~index~~.~~php?document_srl~~=~~48014&mid~~=~~c0102 best psychics~~] ~~tarot readings (~~[~~http:~~/~~/www~~.~~herandkingscounty~~.~~com/content/information~~-~~and~~-~~facts~~-~~you~~-~~must~~-~~know~~-~~about~~-~~hobbies click the following article~~])		In [[mathematics]], more precisely in [[measure theory]], an '''atom''' is a measurable set which has positive measure and contains no set of smaller but positive measure. A measure which has no atoms is called '''non-atomic''' or '''atomless'''.

			==Definition==
			Given a [[measurable space]] <math>(X, \Sigma)</math> and a [[measure (mathematics)\|measure]] <math>\mu</math> on that space, a set <math>A</math> in <math>\Sigma</math> is called an '''atom''' if

			: <math> \mu (A) >0\, </math>

			and for any measurable subset <math>B</math> of <math>A</math> with

			: <math> \mu(A) > \mu (B) \, </math>

			one has <math> \mu(B)=0.</math>

			== Examples==
			* Consider the set ''X''={1, 2, ..., 9, 10} and let the sigma-algebra <math>\Sigma</math> be the [[power set]] of ''X''. Define the measure <math>\mu</math> of a set to be its [[cardinality]], that is, the number of elements in the set. Then, each of the [[singleton (mathematics)\|singleton]]s {''i''}, for ''i''=1,2, ..., 9, 10 is an atom.
			* Consider the [[Lebesgue measure]] on the [[real line]]. This measure has no atoms.

			== Non-atomic measures==
			A measure which has no atoms is called '''non-atomic'''. In other words, a measure is non-atomic if for any measurable set <math>A</math> with <math> \mu (A) >0</math> there exists a measurable subset ''B'' of ''A'' such that

			: <math> \mu(A) > \mu (B) > 0. \, </math>

			A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set ''A'' with <math> \mu (A) >0</math> one can construct a decreasing sequence of measurable sets

			:<math>A=A_1\supset A_2 \supset A_3 \supset \cdots</math>

			such that

			:<math>\mu(A)=\mu(A_1) > \mu(A_2) > \mu(A_3) > \cdots > 0. </math>

			This may not be true for measures having atoms; see the first example above.

			It turns out that non-atomic measures actually have a [[Continuum (theory)\|continuum]] of values. It can be proved that if μ is a non-atomic measure and ''A'' is a measurable set with <math>\mu (A) >0,</math> then for any real number ''b'' satisfying

			: <math>\mu (A) \geq b \geq0\, </math>

			there exists a measurable subset ''B'' of ''A'' such that

			: <math>\mu(B)=b.\,</math>

			This theorem is due to [[Wacław Sierpiński]].<ref>{{cite journal \|first=W. \|last=Sierpinski \|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3125.pdf \|title=Sur les fonctions d'ensemble additives et continues \|journal=Fundamenta Mathematicae \|volume=3 \|issue= \|pages=240–246 \|year=1922\|language=French }}</ref><ref>{{Cite book \|last=Fryszkowski \|first=Andrzej \|title=Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications) \|year=2005 \|publisher=Springer \|location=New York \|isbn=1-4020-2498-3 \|page=39}}</ref>
			It is reminiscent of the [[intermediate value theorem]] for continuous functions.

			'''Sketch of proof''' of Sierpiński's theorem on non-atomic measures. A slightly stronger statement, which however makes the proof easier, is that if <math>(X,\Sigma, \mu)</math> is a non-atomic measure space and <math>\mu(X)=c</math>, there exists a function <math>S:[0, c]\to\Sigma</math> that is monotone with respect to inclusion, and a right-inverse to <math>\mu:\Sigma\to[0,\,c]</math>. That is, there exists a one-parameter family of measurable sets S(t) such that for all <math>0\leq t \leq t'\leq c</math>
			:<math>S(t)\subset S(t'),</math>
			:<math>\mu\left (S(t)\right)=t.</math>
			The proof easily follows from [[Zorn's lemma]] applied to the set of all monotone partial sections to <math>\mu</math> :
			:<math>\Gamma:=\{S:D\to\Sigma\; :\; D\subset[0,\,c],\, S\; \mathrm{ monotone }, \forall t\in D\; (\mu\left (S(t)\right)=t)\},</math>
			ordered by inclusion of graphs, <math>\mathrm{graph}(S)\subset \mathrm{graph}(S').</math> It's then standard to show that every chain in <math>\Gamma</math> has an upper bound in <math>\Gamma</math>, and that any maximal element of <math>\Gamma</math> has domain <math>[0,c],</math> proving the claim.

			== See also ==
			* [[Atom (order theory)]] — an analogous concept in order theory
			* [[Dirac delta function]]
			* [[Elementary event]], also known as an '''atomic event'''

			== Notes==
			<references />

			==References==
			* {{Cite book \| author=Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. \| authorlink= \| coauthors= \| title=Real analysis \| year=1997 \| publisher=Prentice-Hall \| location=Upper Saddle River, N.J. \| isbn=0-13-458886-X \| page=108}}
			* {{Cite book \| author=Butnariu, Dan; Klement, E. P. \| authorlink= \| coauthors= \| title=Triangular norm-based measures and games with fuzzy coalitions \| year=1993 \| publisher=Kluwer Academic \| location=Dordrecht \| isbn=0-7923-2369-6 \| page=87}}

			[[Category:Measure theory]]

en>Xqbot: r2.7.3) (Robot: Adding de:Komplexes Maß

2012-07-10T21:16:35Z

r2.7.3) (Robot: Adding de:Komplexes Maß

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