 Signed measure

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge,^{[1]} by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.
Contents
Definition
There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books signed measures are usually only allowed to take finite values, while undergraduate textbooks often 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), that is, a set X with a sigma algebra Σ on it, an extended signed measure is a function
such that and μ is sigma additive, that is, it satisfies the equality
for any sequence A_{1}, A_{2}, ..., A_{n}, ... of disjoint sets in Σ. One consequence is that any extended signed measure can take +∞ as value, or it can take −∞ as value, but both are not available. The expression ∞ − ∞ is undefined (see Extended real number line) and must be avoided.
A finite signed 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 vector space, while extended signed measures are not even closed under addition, which makes them rather hard to work with. On the other hand, measures are extended signed measures, but are not in general finite signed measures.
Examples
Consider a nonnegative measure ν 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.
Properties
What follows are two results which will imply that an extended signed measure is the difference of two nonnegative measures, and a finite signed measure is the difference of two finite nonnegative 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 μnull sets from P and N.
Consider then two nonnegative measures μ^{+} and μ^{} defined by
and
for all measurable sets E, that is, E in Σ.
One can check that both μ^{+} and μ^{} are nonnegative measures, with one taking only finite values, and are called the positive part and negative part of μ, respectively. One has that μ = μ^{+}  μ^{}. The measure μ = μ^{+} + μ^{} is called the variation of μ, and its maximum possible value, μ = μ(X), is called the total variation of μ.
This consequence of the Hahn decomposition theorem is called the Jordan decomposition. The measures μ^{+}, μ^{} and μ are independent of the choice of P and N in the Hahn decomposition theorem.
The space of signed measures
The sum of two finite signed measures is a finite signed measure, as is the product of a finite signed measure by a real number: they are closed under linear combination. It follows that the set of finite signed measures on a measure space (X, Σ) is a real vector space; this is in contrast to positive measures, which are only closed under conical combination, and thus form a convex cone but not a vector space. Furthermore, the total variation defines a norm in respect to which the space of finite signed measures becomes a Banach space.
If X is a compact separable space, then the space of finite signed Baire measures is the dual of the real Banach space of all continuous realvalued functions on X, by the Riesz representation theorem.
See also
 Complex measure
 Spectral measure
 Vector measure
 Riesz representation theorem
 Total variation
Notes
 ^ A charge need not be countably additive. A charge is additive: see reference Bhaskara Rao & Bhaskara Rao 1983 for a comprehensive introduction.
References
 Bartle, Robert G. (1966), The Elements of Integration, New YorkLondonSydney: John Wiley and Sons, pp. X+129, Zbl 0146.28201
 Bhaskara Rao, K. P. S.; Bhaskara Rao, M. (1983), Theory of Charges: A Study of Finitely Additive Measures, Pure and Applied Mathematics, 109, London: Academic Press, pp. x + 315, ISBN 0120957809, Zbl 0516.28001, http://books.google.it/books?id=mTNQvfe54CoC&printsec=frontcover#v=onepage&q&f=false
 Cohn, Donald L. (1997) [1980], Measure theory (reprint ed.), Boston–Basel–Stuttgart: Birkhäuser Verlag, pp. IX+373, ISBN 3764330031., Zbl 0436.28001, http://books.google.it/books?id=vRxV2FwJvoAC&printsec=frontcover&dq=Measure+theory+Cohn&cd=1#v=onepage&q&f=false
 Diestel, J. E.; Uhl, J. J. Jr. (1977), Vector measures, Mathematical Surveys and Monographs, 15, Providence, R.I.: American Mathematical Society, ISBN 0821815156, Zbl 0369.46039, http://books.google.it/books?id=NCm4E2By8DQC&printsec=frontcover#v=onepage&q&f=false
 Dunford, Nelson; Schwartz, Jacob T. (1959), Linear Operators. Part I: General Theory. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. Part III: Spectral Operators., Pure and Applied Mathematics, 6, New York and London: Interscience Publishers, pp. XIV+858, ISBN 0471608483, Zbl 0084.104
 Dunford, Nelson; Schwartz, Jacob T. (1963), Linear Operators. Part I: General Theory. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. Part III: Spectral Operators., Pure and Applied Mathematics, 7, New York and London: Interscience Publishers, pp. IX+8591923, ISBN 0471608475, Zbl 0128.34803
 Dunford, Nelson; Schwartz, Jacob T. (1971), Linear Operators. Part I: General Theory. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. Part III: Spectral Operators., Pure and Applied Mathematics, 8, New York and London: Interscience Publishers, pp. XIX+1925–2592, ISBN 0471608467, Zbl 0243.47001
This article incorporates material from the following PlanetMath articles: Signed measure, Hahn decomposition theorem, and Jordan decomposition. Their content is licensed under the GFDL.
Categories: Integral calculus
 Measures (measure theory)
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