Signed measure

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In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values.

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[edit] 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

\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, ..., An, ... 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.

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.

[edit] 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 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.

[edit] 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:

  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.

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 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.

[edit] 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.

[edit] See also

[edit] 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.

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