Positive and negative sets

In measure theory, given a measurable space (X,Σ) and a signed measure μ on it, a set A∈Σ is called a positive set for μ if every Σ-measurable subset of A has nonnegative measure; that is, for every E⊆A that satisfies E∈Σ, μ(E)≥0 holds.[1]

Similarly, a set A∈Σ is called a negative set for μ if for every subset E⊆A satisfying E∈Σ, μ(E)≤0 holds.[1]

Intuitively, a measurable set A is positive (resp. negative) for μ if μ is nonnegative (resp. nonpositive) everywhere on A. Of course, if μ is a nonnegative measure, every element of Σ is a positive set for μ.

In the light of Radon–Nikodym theorem, if ν is a σ-finite positive measure such that |μ|≪ν, a set A is a positive set for μ if and only if the Radon–Nikodym derivative dμ/dν is nonnegative ν-almost everywhere on A. Similarly, a negative set is a set where dμ/dν≤0 ν-almost everywhere.

## Contents

* 1 Properties
* 2 Hahn decomposition
* 3 See also
* 4 References

## 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 A1,A2,… is a sequence of positive sets, then ⋃n=1∞An 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-null set, for if E is a measurable subset of a positive and negative set A, then both μ(E)≥0 and μ(E)≤0 must hold, and therefore, μ(E)=0.

## Hahn decomposition

The Hahn decomposition theorem states that for every measurable space (X,Σ) with a signed measure μ, there is a partition of X into a positive and a negative set; such a partition (P,N) is unique up to μ-null sets, and is called a Hahn decomposition of the signed measure μ.[1]

Given a Hahn decomposition (P,N) of X, it is easy to show that A⊆X is a positive set if and only if A differs from a subset of P by a μ-null set; equivalently, if A∖P is μ-null. The same is true for negative sets, if N is used instead of P.

## See also

* Set function - Function from sets to numbers

## References

1. ↑ 1.0 1.1 1.2 Folland, Gerald B (1984). Real Analysis (1st ed.). John Wiley & Sons. p. 81. ISBN 0-471-80958-6.

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