Signed measure


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

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

such that \mu(\emptyset)=0 and μ is sigma additive, that is, it satisfies the equality

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

for any sequence A1, A2, ..., An, ... 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:XR 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.

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.

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:

  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 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: 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 real-valued functions on X, by the Riesz representation theorem.

See also

Notes

  1. ^ A charge need not be countably additive. A charge is additive: see reference Bhaskara Rao & Bhaskara Rao 1983 for a comprehensive introduction.

References


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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Measure (mathematics) — Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. In mathematical analysis …   Wikipedia

  • Radon measure — In mathematics (specifically, measure theory), a Radon measure, named after Johann Radon, is a measure on the σ algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular. Contents 1 Motivation 2 Definitions …   Wikipedia

  • Complex measure — In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. Contents 1 Definition… …   Wikipedia

  • Information theory and measure theory — Measures in information theory = Many of the formulas in information theory have separate versions for continuous and discrete cases, i.e. integrals for the continuous case and sums for the discrete case. These versions can often be generalized… …   Wikipedia

  • Support (measure theory) — In mathematics, the support (sometimes topological support or spectrum) of a measure μ on a measurable topological space ( X , Borel( X )) is a precise notion of where in the space X the measure lives . It is defined to be the largest (closed)… …   Wikipedia

  • Σ-finite measure — In mathematics, a positive (or signed) measure mu; defined on a sigma; algebra Sigma; of subsets of a set X is called finite, if mu;( X ) is a finite real number (rather than ∞). The measure mu; is called σ finite, if X is the countable union of… …   Wikipedia

  • Locally finite measure — In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.DefinitionLet ( X , T ) be a Hausdorff topological space and let Sigma; be a sigma; algebra on X that contains… …   Wikipedia

  • Vector measure — In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. Definitions and first consequencesGiven a field of sets (Omega, mathcal F) and a Banach space X, a finitely… …   Wikipedia

  • Oregon Ballot Measure 58 (1998) — Ballot Measure 58 was a citizen s initiative that was passed by the voters of the U.S. state of Oregon in the November 1998 General Election. The measure restored the right of adopted adults who were born in Oregon to access their original birth… …   Wikipedia

  • Pushforward measure — In mathematics, a pushforward measure (also push forward or push forward) is obtained by transferring ( pushing forward ) a measure from one measurable space to another using a measurable function.DefinitionGiven measurable spaces ( X 1, Sigma;1) …   Wikipedia