Counting measure (measure theory)
The counting measure is a special measure in mathematics that assigns the number of elements to quantities . Formally, the counting measure can be defined in a measuring space , with any set and its power set . If there is a finite set , then a finite measure arises . It is a σ-finite measure if and only if it is countable .
definition
The count of a quantity is defined as follows:
Examples
Above the natural numbers , i.e. the measuring space , the counting measure corresponds to the figure
Here denotes the characteristic function of the set .
With the help of the counting measure , every finite sum or infinite, absolutely convergent series can be represented as a Lebesgue integral . In particular, the following applies to each figure :
- converges absolute can be integrated with respect to the counting measure
In this case
- .
literature
- Christian Hesse : Applied probability theory. Vieweg, Braunschweig et al. 2003, ISBN 3-528-03183-2 , p. 31.
- Jürgen Elstrodt : Measure and integration theory. 4th, corrected edition. Springer, Berlin et al. 2005, ISBN 3-540-21390-2 , p. 29.