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, i.e., to acquire sign.

Definition edit

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[1] 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.

Examples edit

Consider a non-negative measure   on the space (X, Σ) and a measurable function f: XR 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 edit

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:

  1. PN = X and PN = ∅;
  2. μ(E) ≥ 0 for each E in Σ such that EP — in other words, P is a positive set;
  3. μ(E) ≤ 0 for each E in Σ such that EN — 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

 

and

 

for all measurable sets E, that is, E in Σ.

One can check that both μ+ and μ are non-negative 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 edit

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 – that is, they are closed under linear combinations. It follows that the set of finite signed measures on a measurable space (X, Σ) is a real vector space; this is in contrast to positive measures, which are only closed under conical combinations, 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. This space has even more structure, in that it can be shown to be a Dedekind complete Banach lattice and in so doing the Radon–Nikodym theorem can be shown to be a special case of the Freudenthal spectral theorem.

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 real-valued functions on X, by the Riesz–Markov–Kakutani representation theorem.

See also edit

Notes edit

  1. ^ See the article "Extended real number line" for more information.

References edit

  • Bartle, Robert G. (1966), The Elements of Integration, New York: John Wiley and Sons, 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, London: Academic Press, ISBN 0-12-095780-9, Zbl 0516.28001
  • Cohn, Donald L. (1997) [1980], Measure theory, Boston: Birkhäuser Verlag, ISBN 3-7643-3003-1, Zbl 0436.28001
  • Diestel, J. E.; Uhl, J. J. Jr. (1977), Vector measures, Mathematical Surveys and Monographs, vol. 15, Providence, R.I.: American Mathematical Society, ISBN 0-8218-1515-6, Zbl 0369.46039
  • 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, vol. 6, New York and London: Interscience Publishers, pp. XIV+858, ISBN 0-471-60848-3, Zbl 0084.10402
  • 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, vol. 7, New York and London: Interscience Publishers, pp. IX+859–1923, ISBN 0-471-60847-5, 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, vol. 8, New York and London: Interscience Publishers, pp. XIX+1925–2592, ISBN 0-471-60846-7, Zbl 0243.47001
  • Zaanen, Adriaan C. (1996), Introduction to Operator Theory in Riesz spaces, Springer Publishing, ISBN 3-540-61989-5

This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: Signed measure, Hahn decomposition theorem, Jordan decomposition.