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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hahn decomposition theorem ： ウィキペディア英語版 Hahn decomposition theorem In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space (''X'',Σ) and a signed measure ''μ'' defined on the σ-algebra Σ, there exist two measurable sets ''P'' and ''N'' in Σ such that: #''P'' ∪ ''N'' = ''X'' and ''P'' ∩ ''N'' = ∅. #For each ''E'' in Σ such that ''E'' ⊆ ''P'' one has ''μ''(''E'') ≥ 0; that is, ''P'' is a positive set for ''μ''. #For each ''E'' in Σ such that ''E'' ⊆ ''N'' one has ''μ''(''E'') ≤ 0; that is, ''N'' is a negative set for ''μ''. Moreover, this decomposition is essentially unique, in the sense that for any other pair (''P''', ''N''') of measurable sets fulfilling the above three conditions, the symmetric differences ''P'' Δ ''P''' and ''N'' Δ ''N''' are ''μ''-null sets in the strong sense that every measurable subset of them has zero measure. The pair (''P'',''N'') is called a ''Hahn decomposition'' of the signed measure ''μ''. ==Jordan measure decomposition== A consequence of the Hahn decomposition theorem is the ''Jordan decomposition theorem'', which states that every signed measure ''μ'' has a ''unique'' decomposition into a difference μ = μ+ − μ− of two positive measures ''μ''+ and ''μ''−, at least one of which is finite, such that μ+(E) = 0 if E ⊆ N and μ−(E) = 0 if E ⊆ P for any Hahn decomposition (P,N) of μ. ''μ''+ and ''μ''− are called the ''positive'' and ''negative part'' of ''μ'', respectively. The pair (''μ''+, ''μ''−) is called a ''Jordan decomposition'' (or sometimes ''Hahn–Jordan decomposition'') of ''μ''. The two measures can be defined as :$\backslash mu^+(E):=\backslash mu(E\backslash cap\; P)\backslash ,$ and :$\backslash mu^-(E):=-\backslash mu(E\backslash cap\; N)\backslash ,$ for every ''E'' in Σ and any Hahn decomposition (P,N) of μ. Note that the Jordan decomposition is unique, while the Hahn decomposition is only essentially unique. The Jordan decomposition has the following corollary: Given a Jordan decomposition (μ+, μ−) of a finite signed measure μ, :$$ \mu^+(E) = \sup_ \mu(B) and :$$ \mu^-(E) = -\inf_ \mu(B) for any E in Σ. Also, if μ = ν+ − ν− for a pair of finite non-negative measures (ν+, ν−), then :$$ \nu^+ \geq \mu^+ \text \nu^- \geq \mu^- . The last expression means that the Jordan decomposition is the ''minimal'' decomposition of μ into a difference of non-negative measures. This is the ''minimality property'' of the Jordan decomposition. Proof of the Jordan decomposition: For an elementary proof of the existence, uniqueness, and minimality of the Jordan measure decomposition see (Fischer (2012) ). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hahn decomposition theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.