Signed measure

Signed measure is a term from the mathematical branch of measure theory. It's like a measure on a set system, usually a σ -algebra defined function and differs from this only in that negative values ​​are allowed. The signed measure thus represents a generalization of the Maßbegriffs

• 4.1 design by Dimensions
• 4.2 Integral Induced signed Dimensions

Definition

Let be a non-empty set and a subset of the power set of with.

A lot of function or by means signed measure if

And for every disjoint family and

Applies. The latter property is also referred to as additivity. The convergence of the series is regarded as unconditional convergence to, that is their limit.

If the amount of system a σ - algebra, it will be referred to with. Particularly always is included.

Properties

Each ( pre) measure is a signed measure.

Continuity from above

Is a ring so is continuous from above, it is thus that for each sequence with monotonically decreasing, and

Applies. Is a σ - algebra, so the property is always fulfilled.

Absolute continuity

Be a degree and a signed measure, both of which are defined on the same σ - algebra, then that means absolutely continuous with respect to, symbolically, if for all

Holds, where the variation of (cf. Jordan decomposition ).

Jordan decomposition

In between Jordan decomposition theorem is shown that a signed measure on a σ - algebra defined as the difference of two ( unsigned ) dimensions and can be represented, so

Where and by

Are given. is also referred to as a positive or positive variation of part, corresponding to variation of a negative or partial negative.

When the decomposition is at least one of the measurements and finite.

Furthermore, is singular with respect to symbolic. Since the singularity of dimensions is a symmetric property is therefore also.

That is, by measure defined variations (including total variation ) of. It is true and

For a fixed degree is called the number of total variation of the measure.

The finite signed measure complete the set of finite dimensions to a normed vector space whose norm the total variation, also called variation norm is. This space is even a Banach space.

Construction of signed measures

Structure by means of measurements

Are and two ( pre) Dimensions on with so and are signed extent.

Integral Induced signed Dimensions

Signed extent also occur in the theory of integration, they are induced by an indefinite integral.

Let be a measure space and a measurable function. Is positive ( takes values ​​in a ) or quasiintegirierbar, so the integral exists with an indicator function and always. The figure with

Defines the indefinite integral.

• Is positive, so is a measure.
• Is integrable, then is a finite signed measure, ie for.
• Is quasi- integrable, so is a signed measure.

It usually used for the shorthand.

Applications

With signed measures can be, for example, distributions of positive and negative charges in a material model.