Positive and negative sets


In measure theory, given a measurable space 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 ≥ 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 ≤ 0.
Intuitively, a measurable set A is positive for μ if μ is nonnegative 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 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 ≥ 0 and μ ≤ 0 must hold, and therefore, μ = 0.

Hahn decomposition

The Hahn decomposition theorem states that for every measurable space with a signed measure μ, there is a partition of X into a positive and a negative set; such a partition is unique up to μ-null sets, and is called a Hahn decomposition of the signed measure μ.
Given a Hahn decomposition 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.