and a signed measure
is called a positive set for
has nonnegative measure; that is, for every
Similarly, a set
is called a negative set for
Intuitively, a measurable set
is positive (resp.
is nonnegative (resp.
is a nonnegative measure, every element of
is a positive set for
In the light of Radon–Nikodym theorem, if
is a σ-finite positive measure such that
is a positive set for
Similarly, a negative set is a set where
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
is a measurable subset of a positive and negative set
The Hahn decomposition theorem states that for every measurable space
with a signed measure
into a positive and a negative set; such a partition
-null sets, and is called a Hahn decomposition of the signed measure
Given a Hahn decomposition
it is easy to show that
is a positive set if and only if
differs from a subset of
-null set; equivalently, if
The same is true for negative sets, if