In measure theory, given a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} and a signed measure μ {\displaystyle \mu } on it, a set A ∈ Σ {\displaystyle A\in \Sigma } is called a positive set for μ {\displaystyle \mu } if every Σ {\displaystyle \Sigma } -measurable subset of A {\displaystyle A} has nonnegative measure; that is, for every E ⊆ A {\displaystyle E\subseteq A} that satisfies E ∈ Σ , {\displaystyle E\in \Sigma ,} μ ( E ) ≥ 0 {\displaystyle \mu (E)\geq 0} holds.
Similarly, a set A ∈ Σ {\displaystyle A\in \Sigma } is called a negative set for μ {\displaystyle \mu } if for every subset E ⊆ A {\displaystyle E\subseteq A} satisfying E ∈ Σ , {\displaystyle E\in \Sigma ,} μ ( E ) ≤ 0 {\displaystyle \mu (E)\leq 0} holds.
Intuitively, a measurable set A {\displaystyle A} is positive (resp. negative) for μ {\displaystyle \mu } if μ {\displaystyle \mu } is nonnegative (resp. nonpositive) everywhere on A . {\displaystyle A.} Of course, if μ {\displaystyle \mu } is a nonnegative measure, every element of Σ {\displaystyle \Sigma } is a positive set for μ . {\displaystyle \mu .}
In the light of Radon–Nikodym theorem, if ν {\displaystyle \nu } is a σ-finite positive measure such that | μ | ≪ ν , {\displaystyle |\mu |\ll \nu ,} a set A {\displaystyle A} is a positive set for μ {\displaystyle \mu } if and only if the Radon–Nikodym derivative d μ / d ν {\displaystyle d\mu /d\nu } is nonnegative ν {\displaystyle \nu } -almost everywhere on A . {\displaystyle A.} Similarly, a negative set is a set where d μ / d ν ≤ 0 {\displaystyle d\mu /d\nu \leq 0} ν {\displaystyle \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 , … {\displaystyle A_{1},A_{2},\ldots } is a sequence of positive sets, then ⋃ n = 1 ∞ A n {\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 μ {\displaystyle \mu } -null set, for if E {\displaystyle E} is a measurable subset of a positive and negative set A , {\displaystyle A,} then both μ ( E ) ≥ 0 {\displaystyle \mu (E)\geq 0} and μ ( E ) ≤ 0 {\displaystyle \mu (E)\leq 0} must hold, and therefore, μ ( E ) = 0. {\displaystyle \mu (E)=0.}
Hahn decomposition
The Hahn decomposition theorem states that for every measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} with a signed measure μ , {\displaystyle \mu ,} there is a partition of X {\displaystyle X} into a positive and a negative set; such a partition ( P , N ) {\displaystyle (P,N)} is unique up to μ {\displaystyle \mu } -null sets, and is called a Hahn decomposition of the signed measure μ . {\displaystyle \mu .}
Given a Hahn decomposition ( P , N ) {\displaystyle (P,N)} of X , {\displaystyle X,} it is easy to show that A ⊆ X {\displaystyle A\subseteq X} is a positive set if and only if A {\displaystyle A} differs from a subset of P {\displaystyle P} by a μ {\displaystyle \mu } -null set; equivalently, if A ∖ P {\displaystyle A\setminus P} is μ {\displaystyle \mu } -null. The same is true for negative sets, if N {\displaystyle N} is used instead of P . {\displaystyle P.}
See also
- Set function – Function from sets to numbers