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 .
(
Ω
,
P
(
Ω
)
)
{\ displaystyle (\ Omega, {\ mathfrak {P}} (\ Omega))}
Ω
{\ displaystyle \ Omega}
P
(
Ω
)
{\ displaystyle {\ mathfrak {P}} (\ Omega)}
Ω
{\ displaystyle \ Omega}
Ω
{\ displaystyle \ Omega}
definition
The count of a quantity is defined as follows:
A.
⊆
Ω
{\ displaystyle A \ subseteq \ Omega}
μ
(
A.
)
=
{
|
A.
|
if
A.
is finite
+
∞
if
A.
is infinite.
{\ displaystyle \ mu (A) = {\ begin {cases} \ vert A \ vert & {\ text {if}} A {\ text {is finite,}} \\ + \ infty & {\ text {, if}} A {\ text {is infinite.}} \ end {cases}}}
Examples
Above the natural numbers , i.e. the measuring space , the counting measure corresponds to the figure
(
N
,
P
(
N
)
)
{\ displaystyle (\ mathbb {N}, {\ mathfrak {P}} (\ mathbb {N}))}
μ
:
P
(
N
)
→
[
0
,
∞
]
,
A.
↦
∑
k
∈
N
χ
A.
(
k
)
.
{\ 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 .
χ
A.
{\ displaystyle \ chi _ {A}}
A.
⊆
N
{\ 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 :
N
{\ displaystyle \ mathbb {N}}
f
:
N
→
R.
{\ displaystyle f \ colon \ mathbb {N} \ to \ mathbb {R}}
∑
k
=
1
∞
f
(
k
)
{\ displaystyle \ sum _ {k = 1} ^ {\ infty} f (k)}
converges absolute can be integrated with respect to the counting measure
⟺
{\ displaystyle \ Longleftrightarrow}
f
{\ displaystyle f}
P
(
N
)
.
{\ displaystyle {\ mathfrak {P}} (\ mathbb {N}).}
In this case
∫
N
f
d
μ
=
∑
k
=
1
∞
f
(
k
)
{\ displaystyle \ int _ {\ mathbb {N}} f \, \ mathrm {d} \ mu = \ sum _ {k = 1} ^ {\ infty} f (k)}
.
literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">