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 (An)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 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.

gollark: Analogously, I would say you should probably not be required to have someone grafted to your circulatory system and stuff for 9 months if this would keep them from an otherwise lethal disease or something. You maybe *should* morally, but this is a different thing (and I don't think that really applies in the fetus case, as it isn't much of a "person").
gollark: Actually, I seem to have misread your angle, so it isn't entirely relevant. But regarding "I'll tell them what not to do with others bodies. And the child is another body. It's medically provable.", I would argue that you should not be *required* to put up with fairly substantial health risks/inconvenience because the fetus requires being attached to someone to survive.
gollark: No, before murdering someone you have to do a MRI scan to check brain development.
gollark: There is a difference between "body" and even "human body" and "person".
gollark: It's historically important, at least.
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