Positive and negative sets explained

(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$$\backslash bigcup\_^\backslash infty\; A\_n$$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.