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 positive set 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 negative set 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 $P.$

See also

• Set function – Function from sets to numbers