Signed measure - Citizendia

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with A negative number is a Number that is less than zero, such as −2 Some authors may call it a "charge".

1 Definition
2 Examples
3 Properties
4 The space of signed measures
5 See also
6 References

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

$\mu:\Sigma\to \mathbb {R}\cup\{\infty,-\infty\}$

which is sigma additive, that is, satisfies the equality

$\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n)$

for any sequence A₁, A₂, . In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, additivity and sigma additivity of a function defined on Subsets of a given set are abstractions of the intuitive properties In Mathematics, a sequence is an ordered list of objects (or events . . , A_n, . . . of disjoint sets in Σ. In Mathematics, two sets are said to be disjoint if they have no element in common Notice that an extended signed measure can either take +∞ as value but not −∞, or vice-versa, since the expression ∞−∞ is undefined (see Extended real number line), and thus must be avoided. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced

A finite signed measure is defined in the same way, except that it is only allowed to take (finite) real values.

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

$\int_X \! |f(x)| \, d\nu (x) < \infty.$

Then, a finite signed measure is given by

$\mu (A) = \int_A \! f(x) \, d\nu (x)$

for all A in Σ. In Mathematics, measurable functions are Well-behaved functions between measurable spaces.

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

$\int_X \! f^-(x) \, d\nu (x) < \infty,$

where f⁻(x) = max(−f(x), 0) is the negative part of f. In Mathematics, the positive part of a real or extended real -valued function is defined by the formula f^+(x = \max(f(x0

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 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. In Mathematics, the Hahn decomposition theorem, named after the Austrian Mathematician Hans Hahn, states that given a measurable space In Measure theory, given a measurable space ( X,&Sigma and a Signed measure &mu on it a set A &isin &Sigma is called a positive set

More, this decomposition is unique up to adding to/subtracting from P and N μ-null sets. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Mathematics, a null set is a set that is negligible in some sense.

Consider then two nonnegative measures μ⁺ and μ^- defined by

$\mu^+(E) = \mu(P\cap E)$

and

$\mu^-(E)=-\mu(N\cap E)$

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. It follows that the set of finite signed measures on a measure space (X, Σ) is a real vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Furthermore, the total variation defines a norm in respect to which the space of finite signed measures becomes a Banach space. In Mathematics, the total variation of a real-valued function &fnof defined on an interval &sub R is a measure of the In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

If X is a compact 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 representation theorem. There are several well-known theorems in Functional analysis known as the Riesz representation theorem.

References

Donald L. In Mathematics, or more specifically in Measure theory, a complex measure is a generalisation of the concept of measure by letting it have complex In Mathematics, particularly Functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint In Mathematics, a vector measure is a function defined on a Family of sets and taking vector values satisfying certain properties There are several well-known theorems in Functional analysis known as the Riesz representation theorem. Cohn, Measure theory, Birkhäuser, 1997. ISBN 3-7643-3003-1.
Robert G. Bartle, "The Elements of Integration", 1966.

This article incorporates material from the following PlanetMath articles: Signed measure, Hahn decomposition theorem, and Jordan decomposition. PlanetMath is a free, collaborative online Mathematics Encyclopedia. Their content is licensed under the GFDL.

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Contents

Definition

Examples

Properties

The space of signed measures

See also

References