# 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: ${\ displaystyle A \ subseteq \ Omega}$

${\ displaystyle \ mu (A) = {\ begin {cases} \ vert A \ vert & {\ text {if}} A {\ text {is finite,}} \\ + \ infty & {\ text {, if}} A {\ text {is infinite.}} \ end {cases}}}$

## 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

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