In**** measure theory****,**** given a measurable**** space (X, *Sig**ma**)*** and a signed**** measure *mu*** on**** it****,**** a set is**** called**** a **** for *mu*** if**** every *Sig**ma**-mea**su**ra**bl**e*** subset**** of**** A has nonnegative measure;**** that**** is****,**** for every **** that**** satisfies **** **** holds.**** Similarly,**** a set is**** called**** a **** for *mu*** if**** for every subset**** **** satisfying**** **** **** 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 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 \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 is a sequence of positive sets, then 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 and 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 is a positive set if and only if A differs from a subset of P by a \mu-null set; equivalently, if is \mu-null. The same is true for negative sets, if N is used instead of P.