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Concept
Signed measure
Summary
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values.
There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures".
Given a measurable space (that is, a set with a σ-algebra on it), an extended signed measure is a set function
such that and is σ-additive – that is, it satisfies the equality
for any sequence of disjoint sets in
The series on the right must converge absolutely when the value of the left-hand side is finite. One consequence is that an extended signed measure can take or as a value, but not both. The expression is undefined and must be avoided.
A finite signed measure (a.k.a. real measure) is defined in the same way, except that it is only allowed to take real values. That is, it cannot take or
Finite signed measures form a real vector space, while extended signed measures do not because they are not closed under addition. On the other hand, measures are extended signed measures, but are not in general finite signed measures.
Consider a non-negative measure on the space (X, Σ) and a measurable function f: X → R such that
Then, a finite signed measure is given by
for all A in Σ.
This signed measure takes only finite values. To allow it to take +∞ as a value, one needs to replace the assumption about f being absolutely integrable with the more relaxed condition
where f−(x) = max(−f(x), 0) is the negative part of f.
What follows are two results which will imply that an extended signed measure is the difference of two non-negative measures, and a finite signed measure is the difference of two finite non-negative measures.
Official source
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