- Source: Counting measure
- Ukuran pencacahan
- Fraktal
- Jumlah partai efektif
- Ekonomi feminis
- Daftar masalah matematika yang belum terpecahkan
- Pedometer
- Counting measure
- Measure (mathematics)
- Ergodicity
- Σ-finite measure
- Hölder's inequality
- Probability mass function
- Japanese counter word
- Pip (counting)
- Random measure
- Equivalence (measure theory)
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity
∞
{\displaystyle \infty }
if the subset is infinite.
The counting measure can be defined on any measurable space (that is, any set
X
{\displaystyle X}
along with a sigma-algebra) but is mostly used on countable sets.
In formal notation, we can turn any set
X
{\displaystyle X}
into a measurable space by taking the power set of
X
{\displaystyle X}
as the sigma-algebra
Σ
;
{\displaystyle \Sigma ;}
that is, all subsets of
X
{\displaystyle X}
are measurable sets.
Then the counting measure
μ
{\displaystyle \mu }
on this measurable space
(
X
,
Σ
)
{\displaystyle (X,\Sigma )}
is the positive measure
Σ
→
[
0
,
+
∞
]
{\displaystyle \Sigma \to [0,+\infty ]}
defined by
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}}}
for all
A
∈
Σ
,
{\displaystyle A\in \Sigma ,}
where
|
A
|
{\displaystyle \vert A\vert }
denotes the cardinality of the set
A
.
{\displaystyle A.}
The counting measure on
(
X
,
Σ
)
{\displaystyle (X,\Sigma )}
is σ-finite if and only if the space
X
{\displaystyle X}
is countable.
Integration on
N
{\displaystyle \mathbb {N} }
with counting measure
Take the measure space
(
N
,
2
N
,
μ
)
{\displaystyle (\mathbb {N} ,2^{\mathbb {N} },\mu )}
, where
2
N
{\displaystyle 2^{\mathbb {N} }}
is the set of all subsets of the naturals and
μ
{\displaystyle \mu }
the counting measure. Take any measurable
f
:
N
→
[
0
,
∞
]
{\displaystyle f:\mathbb {N} \to [0,\infty ]}
. As it is defined on
N
{\displaystyle \mathbb {N} }
,
f
{\displaystyle f}
can be represented pointwise as
f
(
x
)
=
∑
n
=
1
∞
f
(
n
)
1
{
n
}
(
x
)
=
lim
M
→
∞
∑
n
=
1
M
f
(
n
)
1
{
n
}
(
x
)
⏟
ϕ
M
(
x
)
=
lim
M
→
∞
ϕ
M
(
x
)
{\displaystyle f(x)=\sum _{n=1}^{\infty }f(n)1_{\{n\}}(x)=\lim _{M\to \infty }\underbrace {\ \sum _{n=1}^{M}f(n)1_{\{n\}}(x)\ } _{\phi _{M}(x)}=\lim _{M\to \infty }\phi _{M}(x)}
Each
ϕ
M
{\displaystyle \phi _{M}}
is measurable. Moreover
ϕ
M
+
1
(
x
)
=
ϕ
M
(
x
)
+
f
(
M
+
1
)
⋅
1
{
M
+
1
}
(
x
)
≥
ϕ
M
(
x
)
{\displaystyle \phi _{M+1}(x)=\phi _{M}(x)+f(M+1)\cdot 1_{\{M+1\}}(x)\geq \phi _{M}(x)}
. Still further, as each
ϕ
M
{\displaystyle \phi _{M}}
is a simple function
∫
N
ϕ
M
d
μ
=
∫
N
(
∑
n
=
1
M
f
(
n
)
1
{
n
}
(
x
)
)
d
μ
=
∑
n
=
1
M
f
(
n
)
μ
(
{
n
}
)
=
∑
n
=
1
M
f
(
n
)
⋅
1
=
∑
n
=
1
M
f
(
n
)
{\displaystyle \int _{\mathbb {N} }\phi _{M}d\mu =\int _{\mathbb {N} }\left(\sum _{n=1}^{M}f(n)1_{\{n\}}(x)\right)d\mu =\sum _{n=1}^{M}f(n)\mu (\{n\})=\sum _{n=1}^{M}f(n)\cdot 1=\sum _{n=1}^{M}f(n)}
Hence by the monotone convergence theorem
∫
N
f
d
μ
=
lim
M
→
∞
∫
N
ϕ
M
d
μ
=
lim
M
→
∞
∑
n
=
1
M
f
(
n
)
=
∑
n
=
1
∞
f
(
n
)
{\displaystyle \int _{\mathbb {N} }fd\mu =\lim _{M\to \infty }\int _{\mathbb {N} }\phi _{M}d\mu =\lim _{M\to \infty }\sum _{n=1}^{M}f(n)=\sum _{n=1}^{\infty }f(n)}
Discussion
The counting measure is a special case of a more general construction. With the notation as above, any function
f
:
X
→
[
0
,
∞
)
{\displaystyle f:X\to [0,\infty )}
defines a measure
μ
{\displaystyle \mu }
on
(
X
,
Σ
)
{\displaystyle (X,\Sigma )}
via
μ
(
A
)
:=
∑
a
∈
A
f
(
a
)
for all
A
⊆
X
,
{\displaystyle \mu (A):=\sum _{a\in A}f(a)\quad {\text{ for all }}A\subseteq X,}
where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is,
∑
y
∈
Y
⊆
R
y
:=
sup
F
⊆
Y
,
|
F
|
<
∞
{
∑
y
∈
F
y
}
.
{\displaystyle \sum _{y\,\in \,Y\!\ \subseteq \,\mathbb {R} }y\ :=\ \sup _{F\subseteq Y,\,|F|<\infty }\left\{\sum _{y\in F}y\right\}.}
Taking
f
(
x
)
=
1
{\displaystyle f(x)=1}
for all
x
∈
X
{\displaystyle x\in X}
gives the counting measure.
See also
Pip (counting) – Easily countable items
Random counting measure
Set function – Function from sets to numbers