Complex measure

A complex measure is a kind of generalization of the measure from the mathematical branch of measure theory. It's like the measure of a function that depends on a lot of system, usually a σ -algebra maps. The complex degree can however as a range of values ​​to the complex numbers.

Definition

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

A lot of function in the complex numbers is called a complex measure, if

And for every disjoint family and

Holds, where the absolute number has to converge, ie. The latter property is also referred to as additivity.

In most applications, the quantity system is a σ - algebra, then is always included.

Properties

Every finite ( pre) measure is a complex measure, if one embeds the real image of the measurement range in the complex numbers.

A complex measurement are obvious real and imaginary dimensions signed. Since every signed measure can be written as the difference of two positive dimensions ( Hahn- Jordan decomposition ), each complex measure can be written as a linear combination of four positive dimensions.