Counting measure (measure theory)

from Wikipedia, the free encyclopedia

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 .


The count of a quantity is defined as follows:


Integral of the function on the interval with respect to the count over

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