- Positive and negative sets
In
measure theory , given a measurable space ("X",Σ) and asigned 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 μif 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.Hahn decomposition
The
Hahn decomposition theorem states that for every measurable space ("X",Σ) with a signed measure μ, there is a partition of "X" into a positive and a negative set; such a partition ("P","N") is uniqueup 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".
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