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". For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... 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 physical sizes). ... A negative number is a number that is less than zero, such as −3. ...

Definition

Given a measure space (X, Σ), that is, a set X with a sigma algebra Σ on it, a signed measure is a function In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

which is sigma additive, that is, satisfies the equality In mathematics, a measure is a function that assigns a number, e. ...

for any sequence A₁, A₂, ..., A_n, ... of disjoint sets in Σ. Notice that a signed measure can either take +∞ as value but not −∞, or viceversa, since the expression ∞−∞ is undefined (see Extended real number line), and thus must be avoided. For other senses of this word, see sequence (disambiguation). ... In mathematics, two sets are said to be disjoint if they have no element in common. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

To avoid confusion, from here on ordinary measures, that is, measures with non-negative values, will be called nonnegative measures, in contrast with signed measures which can take negative values. In this article it will be assumed, for the sake of simplicity, that the value -∞ is not taken by any of the signed measures considered - the other case is dealt with similarly.

Examples

Consider a nonnegative measure ν on the space (X, Σ) and a measurable function f:X→ R such that In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

Then, a 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. In mathematics, the positive part of a real or extended real-valued function is defined by the formula Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x-axis, and letting take the value zero there. ...

Properties

What follows are two results which will imply that a signed measure is the difference of two nonnegative measures, and as such, that signed measures are really no more complicated than ordinary nonnegative measures.

The Hahn decomposition theorem states that given a signed measure μ, there exist two measurable sets P and N such that: The Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space (X,Σ) and a signed measure μ defined on the σ-algebra Σ, there exist two sets P and N in Σ such that: P∪N = X and P∩N = ∅. For each E in Σ such that E ⊆ P...

1. P∪N = X and P∩N = ∅;
2. μ(E) ≥ 0 for each E in Σ such that E ⊆ P — in other words, P is a positive set;
3. μ(E) ≤ 0 for each E in Σ such that E ⊆ N — that is, N is a negative set.

More, this decomposition is unique up to adding to/subtracting from P and N μ-null sets. In measure theory, given a measurable space (X,Σ) and a signed measure μ on it, a set A ∈ Σ is called a positive set for μ if every Σ-measurable subset of A has nonnegative measure; that is, for every E ⊆ A that satisfies E ∈ Σ, one has μ(E) ≥ 0. ... Look up Up to on Wiktionary, the free dictionary 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 measure theory, a null set is a set that is negligible for the purposes of the measure in question. ...

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 the second 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-valued signed measures is a signed measure, as is the product of a finite-valued signed measure by a real number. It follows that the set of finite-valued signed measures on a measure space (X, Σ) is a real vector space. Furthermore, the total variation defines a norm in respect to which the space of measures becomes a Banach space. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the total variation of a real-valued function f on the bounded interval [a, b] is the supremum running over all partitions P = { x1, ..., xn } of the interval [a, b]. In effect, the total variation is the vertical component of the arc-length of the graph of f. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

See also

• Complex measure
• Spectral measure
• Vector measure
• Riesz representation theorem

In mathematics, or more specifically in measure theory, a complex measure is a generalisation of the concept of measure by letting it have complex values. ... In mathematics, projection-valued measures are used to express results in spectral theory. ... 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. ...

References

• Donald L. 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. Their content is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...