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 . ${\displaystyle (\Omega ,{\mathfrak {P}}(\Omega ))}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathfrak {P}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

definition

The count of a quantity is defined as follows: ${\displaystyle A\subseteq \Omega }$

${\displaystyle \mu (A)={\begin{cases}\vert A\vert &{\text{, falls }}A{\text{ endlich ist,}}\\+\infty &{\text{, falls }}A{\text{ unendlich ist.}}\end{cases}}}$

Examples

Integral of the function on the interval with respect to the count over ${\displaystyle x\mapsto x^{2}}$ ${\displaystyle [-10,10]}$ ${\displaystyle \mathbb {N} }$

Above the natural numbers , i.e. the measuring space , the counting measure corresponds to the figure ${\displaystyle (\mathbb {N} ,{\mathfrak {P}}(\mathbb {N} ))}$

${\displaystyle \mu \colon {\mathfrak {P}}(\mathbb {N} )\to [0,\infty ]{\text{, }}A\mapsto \sum _{k\in \mathbb {N} }\chi _{A}(k).}$

Here denotes the characteristic function of the set . ${\displaystyle \chi _{A}}$ ${\displaystyle A\subseteq \mathbb {N} }$

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 : ${\displaystyle \mathbb {N} }$ ${\displaystyle f\colon \mathbb {N} \to \mathbb {R} }$

${\displaystyle \sum _{k=1}^{\infty }f(k)}$ converges absolute can be integrated with respect to the counting measure ${\displaystyle \Longleftrightarrow }$ ${\displaystyle f}$ ${\displaystyle {\mathfrak {P}}(\mathbb {N} ).}$

In this case

${\displaystyle \int _{\mathbb {N} }f\,\mathrm {d} \mu =\sum _{k=1}^{\infty }f(k)}$.

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.