In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign.
Definition 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 ( X , Σ ) {\displaystyle (X,\Sigma )} (that is, a set X {\displaystyle X} with a σ-algebra Σ {\displaystyle \Sigma } on it), an extended signed measure is a set function
μ : Σ → R ∪ { ∞ , − ∞ } {\displaystyle \mu :\Sigma \to \mathbb {R} \cup \{\infty ,-\infty \}}
such that μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0} and μ {\displaystyle \mu } is σ-additive – that is, it satisfies the equality
μ ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ ( A n ) {\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n})}
for any sequence A 1 , A 2 , … , A n , … {\displaystyle A_{1},A_{2},\ldots ,A_{n},\ldots } of disjoint sets in Σ . {\displaystyle \Sigma .}
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 + ∞ {\displaystyle +\infty } or − ∞ {\displaystyle -\infty } as a value, but not both. The expression ∞ − ∞ {\displaystyle \infty -\infty } 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 + ∞ {\displaystyle +\infty } or − ∞ . {\displaystyle -\infty .}
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.
Examples Consider a non-negative measure ν {\displaystyle \nu } on the space (X, Σ) and a measurable function f: X → R such that
∫ X | f ( x ) | d ν ( x ) < ∞ . {\displaystyle \int _{X}\!|f(x)|\,d\nu (x)<\infty .}
Then, a finite signed measure is given by
μ ( A ) = ∫ A f ( x ) d ν ( x ) {\displaystyle \mu (A)=\int _{A}\!f(x)\,d\nu (x)}
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
∫ X f − ( x ) d ν ( x ) < ∞ , {\displaystyle \int _{X}\!f^{-}(x)\,d\nu (x)<\infty ,}
where f−(x) = max(−f(x), 0) is the negative part of f.
Properties 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. The Hahn decomposition theorem states that given a signed measure μ, there exist two measurable sets P and N such that:
P∪N = X and P∩N = ∅; μ(E) ≥ 0 for each E in Σ such that E ⊆ P — in other words, P is a positive set; μ(E) ≤ 0 for each E in Σ such that E ⊆ N — that is, N is a negative set. Moreover, this decomposition is unique up to adding to/subtracting μ-null sets from P and N. Consider then two non-negative measures μ+ and μ− defined by
μ + ( E ) = μ ( P ∩ E ) {\displaystyle \mu ^{+}(E)=\mu (P\cap E)}
and
μ − ( E ) = − μ ( N ∩ E ) {\displaystyle \mu ^{-}(E)=-\mu (N\cap E)}
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