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← Older revision		Revision as of 01:35, 15 March 2013
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	~~Myrtle Benny is how I~~'m [~~http://www.neweracinema.com/tube/blog/74189 home std test~~] ~~known as and I feel comfy when individuals use the complete name~~. ~~For many years he~~'~~s been working as~~ a ~~meter reader and it~~'~~s [http://www~~.~~ninfeta~~.~~tv/blog/99493 www~~.~~ninfeta~~.tv] ~~something he truly enjoy~~. ~~One~~ of the ~~issues he enjoys most~~ is ~~ice skating but he~~ is ~~struggling to discover time for it~~. ~~North Dakota~~ is ~~her std home test birth place but she at home std testing will have to transfer one working day~~ or ~~another~~.<br><br>~~Also visit my over~~ the ~~counter std test weblog; over~~ the ~~[http://Www~~.~~Wisegeek.com/what-are-some-types~~-of~~-bacteria~~.~~htm counter std~~] ~~test~~ ([~~http://www~~.~~uknewsx.com/what-you-want-~~to-do-~~when-confronted-with-candida-albicans/ visit the up coming internet site~~])		{{Unreferenced\|date=December 2009}}
			In [[measure theory]], given a [[sigma-algebra\|measurable space]] (''X'',Σ) and a [[signed measure]] μ on it, a set ''A'' ∈ Σ is called a '''positive set''' for μ if every Σ-measurable subset of ''A'' has nonnegative measure; that is, for every ''E'' ⊆ ''A'' that satisfies ''E'' ∈ Σ, one has μ(''E'') ≥ 0.

			Similarly, a set ''A'' ∈ Σ is called a '''negative set''' for μ if for every subset ''E'' of ''A'' satisfying ''E'' ∈ Σ, one has μ(''E'') ≤ 0.

			Intuitively, a measurable set ''A'' is positive (resp. negative) for μ if μ is nonnegative (resp. nonpositive) everywhere on ''A''. Of course, if μ is a [[measure (mathematics)\|nonnegative measure]], every element of Σ is a positive set for μ.

			In the light of [[Radon–Nikodym theorem]], if ν is a σ-finite positive measure such that \|μ\| << ν, a set ''A'' is a positive set for μ [[if and only if]] the Radon–Nikodym derivative dμ/dν is nonnegative ν-almost everywhere on ''A''. Similarly, a negative set is a set where dμ/dν ≤ 0 ν-almost everywhere.

			==Properties==
			It follows from the definition that every measurable subset of a positive or negative set is also positive or negative. Also, the union of a sequence of positive or negative sets is also positive or negative; more formally, if (''A''<sub>''n''</sub>)<sub>''n''</sub> is a sequence of positive sets, then
			:<math>\bigcup_{n=1}^\infty A_n</math>
			is also a positive set; the same is true if the word "positive" is replaced by "negative".

			A set which is both positive and negative is a μ-[[null set]], for if ''E'' is a measurable subset of a positive and negative set ''A'', then both μ(''E'') ≥ 0 and μ(''E'') ≤ 0 must hold, and therefore, μ(''E'') = 0.

			==Hahn decomposition==
			The [[Hahn decomposition theorem]] states that for every measurable space (''X'',Σ) with a signed measure μ, there is a [[partition of a set\|partition]] of ''X'' into a positive and a negative set; such a partition (''P'',''N'') is unique [[up to]] μ-null sets, and is called a ''Hahn decomposition'' of the signed measure μ.

			Given a Hahn decomposition (''P'',''N'') of ''X'', it is easy to show that ''A'' ⊆ ''X'' is a positive set if and only if ''A'' differs from a subset of ''P'' by a μ-null set; equivalently, if ''A''−''P'' is μ-null. The same is true for negative sets, if ''N'' is used instead of ''P''.

			{{DEFAULTSORT:Positive And Negative Sets}}
			[[Category:Measure theory]]

en>Cerebellum: formatting

2011-02-19T01:16:28Z

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