View source for Positive and negative sets

{{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''&minus;''P'' is μ-null. The same is true for negative sets, if ''N'' is used instead of ''P''.

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[[Category:Measure theory]]