Complex measure
A complex measure is a kind of generalization of the measure from the mathematical branch of measure theory . Like the measure, it is a function that represents a system of sets , usually a σ-algebra . The complex measure, however, allows the complex numbers as a range of values .
definition
Let be a nonempty set and a subset of the power set of with .
A set function of in the complex numbers is called a complex measure if
and for each disjoint family with and
holds, where the series must absolutely converge , that is . The latter property is also referred to as - additivity .
In most applications the system of sets is a σ-algebra , then it is always contained in.
properties
Every finite (pre) measure is a complex measure if one embeds the real image area of the measure in the complex numbers .
For a complex dimension, real and imaginary parts are obviously signed dimensions . Since every signed measure can be written as the difference between two positive measures ( Hahn-Jordan decomposition ), every complex measure can be written as a linear combination of four positive measures.
See also
literature
- Walter Rudin: Real and Complex Analysis. Oldenbourg Wissenschaftsverlag, Munich 1999, ISBN 3-486-24789-1 , chap. 6th