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In measure theory, given a 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 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''''n'')''n'' is a sequence of positive sets, then : 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. 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「Positive and negative sets」の詳細全文を読む スポンサード リンク
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