Positive and negative sets

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

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

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

In the light of Radon–Nikodym theorem, if ${\displaystyle \nu }$ is a σ-finite positive measure such that ${\displaystyle |\mu |\ll \nu ,}$ a set ${\displaystyle A}$ is a positive set for ${\displaystyle \mu }$ if and only if the Radon–Nikodym derivative ${\displaystyle d\mu /d\nu }$ is nonnegative ${\displaystyle \nu }$-almost everywhere on ${\displaystyle A.}$ Similarly, a negative set is a set where ${\displaystyle d\mu /d\nu \leq 0}$ ${\displaystyle \nu }$-almost everywhere.