Counting measure

The counting measure is a special measure in mathematics, the amounts assigns the number of its elements. Formally, define the counting measure on a measurement space, with an arbitrary set and its power set is. In the first case this results in a finite, in the second case, a σ - finite extent.

Definition

The counting measure of a set is defined as follows:

Examples

Over the integers, i.e. the measurement space, corresponds to the counting of the Figure

Herein, the characteristic function of the amount.

With the help of Zählmaßes on each finite sum or infinite, absolutely convergent series can be represented as a Lebesgue integral. In particular, for each image: