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- Positive_and_negative_sets abstract "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.".
- Positive_and_negative_sets wikiPageID "3375352".
- Positive_and_negative_sets wikiPageRevisionID "544225637".
- Positive_and_negative_sets hasPhotoCollection Positive_and_negative_sets.
- Positive_and_negative_sets subject Category:Measure_theory.
- Positive_and_negative_sets comment "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.".
- Positive_and_negative_sets label "Insieme positivo e insieme negativo".
- Positive_and_negative_sets label "Positive and negative sets".
- Positive_and_negative_sets sameAs Insieme_positivo_e_insieme_negativo.
- Positive_and_negative_sets sameAs m.098b16.
- Positive_and_negative_sets sameAs Q3799160.
- Positive_and_negative_sets sameAs Q3799160.
- Positive_and_negative_sets wasDerivedFrom Positive_and_negative_sets?oldid=544225637.
- Positive_and_negative_sets isPrimaryTopicOf Positive_and_negative_sets.