Positive and negative sets

In measure theory, given a measurable space $$(X, \Sigma)$$ and a signed measure $$\mu$$ on it, a set $$A \in \Sigma$$ is called a  for $$\mu$$ if every $$\Sigma$$-measurable subset of $$A$$ has nonnegative measure; that is, for every $$E \subseteq A$$ that satisfies $$E \in \Sigma,$$ $$\mu(E) \geq 0$$ holds.

Similarly, a set $$A \in \Sigma$$ is called a  for $$\mu$$ if for every subset $$E \subseteq A$$ satisfying $$E \in \Sigma,$$ $$\mu(E) \leq 0$$ holds.

Intuitively, a measurable set $$A$$ is positive (resp. negative) for $$\mu$$ if $$\mu$$ is nonnegative (resp. nonpositive) everywhere on $$A.$$ Of course, if $$\mu$$ is a nonnegative measure, every element of $$\Sigma$$ is a positive set for $$\mu.$$

In the light of Radon–Nikodym theorem, if $$\nu$$ is a σ-finite positive measure such that $$|\mu| \ll \nu,$$ a set $$A$$ is a positive set for $$\mu$$ if and only if the Radon–Nikodym derivative $$d\mu/d\nu$$ is nonnegative $$\nu$$-almost everywhere on $$A.$$ Similarly, a negative set is a set where $$d\mu/d\nu \leq 0$$ $$\nu$$-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 $$A_1, A_2, \ldots$$ is a sequence of positive sets, then $$\bigcup_{n=1}^\infty A_n$$ 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 $$\mu$$-null set, for if $$E$$ is a measurable subset of a positive and negative set $$A,$$ then both $$\mu(E) \geq 0$$ and $$\mu(E) \leq 0$$ must hold, and therefore, $$\mu(E) = 0.$$

Hahn decomposition
The Hahn decomposition theorem states that for every measurable space $$(X, \Sigma)$$ with a signed measure $$\mu,$$ there is a partition of $$X$$ into a positive and a negative set; such a partition $$(P, N)$$ is unique up to $$\mu$$-null sets, and is called a Hahn decomposition of the signed measure $$\mu.$$

Given a Hahn decomposition $$(P, N)$$ of $$X,$$ it is easy to show that $$A \subseteq X$$ is a positive set if and only if $$A$$ differs from a subset of $$P$$ by a $$\mu$$-null set; equivalently, if $$A \setminus P$$ is $$\mu$$-null. The same is true for negative sets, if $$N$$ is used instead of $$P.$$