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 . ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {C}}\subseteq 2^{\Omega }}$ ${\displaystyle \Omega }$ ${\displaystyle \emptyset \in {\mathcal {C}}}$

A set function of in the complex numbers is called a complex measure if ${\displaystyle \nu }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathbb {C} }$

${\displaystyle \nu (\emptyset )=0}$

and for each disjoint family with and ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A_{i}\in {\mathcal {C}}}$ ${\displaystyle \textstyle \bigcup _{i\in \mathbb {N} }A_{i}\in {\mathcal {C}}}$

${\displaystyle \nu \left(\bigcup _{i\in \mathbb {N} }A_{i}\right)=\sum _{i\in \mathbb {N} }\nu (A_{i})}$

holds, where the series must absolutely converge , that is . The latter property is also referred to as - additivity . ${\displaystyle \textstyle \sum _{i\in \mathbb {N} }\nu (A_{i})}$ ${\displaystyle \textstyle \sum _{i\in \mathbb {N} }|\nu (A_{i})|<\infty }$ ${\displaystyle \sigma }$

In most applications the system of sets is a σ-algebra , then it is always contained in. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \textstyle \bigcup _{i\in \mathbb {N} }A_{i}}$ ${\displaystyle {\mathcal {C}}}$

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

• Signed measure

literature

• Walter Rudin: Real and Complex Analysis. Oldenbourg Wissenschaftsverlag, Munich 1999, ISBN 3-486-24789-1 , chap. 6th