Positive and negative sets

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

[math]\displaystyle{ \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 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−P is μ-null. The same is true for negative sets, if N is used instead of P.