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.

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 ${\displaystyle A_{1},A_{2},\ldots }$ is a sequence of positive sets, then $${\displaystyle \bigcup _{n=1}^{\infty }A_{n}}$$ 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 ${\displaystyle \mu }$-null set, for if ${\displaystyle E}$ is a measurable subset of a positive and negative set ${\displaystyle A,}$ then both ${\displaystyle \mu (E)\geq 0}$ and ${\displaystyle \mu (E)\leq 0}$ must hold, and therefore, ${\displaystyle \mu (E)=0.}$

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

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

• Set function – Function from sets to numbers