- Source: Sigma-additive set function
In mathematics, an additive set function is a function
μ
{\textstyle \mu }
mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely,
μ
(
A
∪
B
)
=
μ
(
A
)
+
μ
(
B
)
.
{\textstyle \mu (A\cup B)=\mu (A)+\mu (B).}
If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is,
μ
(
⋃
n
=
1
∞
A
n
)
=
∑
n
=
1
∞
μ
(
A
n
)
.
{\textstyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}).}
Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity.
The term modular set function is equivalent to additive set function; see modularity below.
Additive (or finitely additive) set functions
Let
μ
{\displaystyle \mu }
be a set function defined on an algebra of sets
A
{\displaystyle \scriptstyle {\mathcal {A}}}
with values in
[
−
∞
,
∞
]
{\displaystyle [-\infty ,\infty ]}
(see the extended real number line). The function
μ
{\displaystyle \mu }
is called additive or finitely additive, if whenever
A
{\displaystyle A}
and
B
{\displaystyle B}
are disjoint sets in
A
,
{\displaystyle \scriptstyle {\mathcal {A}},}
then
μ
(
A
∪
B
)
=
μ
(
A
)
+
μ
(
B
)
.
{\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}
A consequence of this is that an additive function cannot take both
−
∞
{\displaystyle -\infty }
and
+
∞
{\displaystyle +\infty }
as values, for the expression
∞
−
∞
{\displaystyle \infty -\infty }
is undefined.
One can prove by mathematical induction that an additive function satisfies
μ
(
⋃
n
=
1
N
A
n
)
=
∑
n
=
1
N
μ
(
A
n
)
{\displaystyle \mu \left(\bigcup _{n=1}^{N}A_{n}\right)=\sum _{n=1}^{N}\mu \left(A_{n}\right)}
for any
A
1
,
A
2
,
…
,
A
N
{\displaystyle A_{1},A_{2},\ldots ,A_{N}}
disjoint sets in
A
.
{\textstyle {\mathcal {A}}.}
σ-additive set functions
Suppose that
A
{\displaystyle \scriptstyle {\mathcal {A}}}
is a σ-algebra. If for every sequence
A
1
,
A
2
,
…
,
A
n
,
…
{\displaystyle A_{1},A_{2},\ldots ,A_{n},\ldots }
of pairwise disjoint sets in
A
,
{\displaystyle \scriptstyle {\mathcal {A}},}
μ
(
⋃
n
=
1
∞
A
n
)
=
∑
n
=
1
∞
μ
(
A
n
)
,
{\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}),}
holds then
μ
{\displaystyle \mu }
is said to be countably additive or 𝜎-additive.
Every 𝜎-additive function is additive but not vice versa, as shown below.
τ-additive set functions
Suppose that in addition to a sigma algebra
A
,
{\textstyle {\mathcal {A}},}
we have a topology
τ
.
{\displaystyle \tau .}
If for every directed family of measurable open sets
G
⊆
A
∩
τ
,
{\textstyle {\mathcal {G}}\subseteq {\mathcal {A}}\cap \tau ,}
μ
(
⋃
G
)
=
sup
G
∈
G
μ
(
G
)
,
{\displaystyle \mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G),}
we say that
μ
{\displaystyle \mu }
is
τ
{\displaystyle \tau }
-additive. In particular, if
μ
{\displaystyle \mu }
is inner regular (with respect to compact sets) then it is τ-additive.
Properties
Useful properties of an additive set function
μ
{\displaystyle \mu }
include the following.
= Value of empty set
=Either
μ
(
∅
)
=
0
,
{\displaystyle \mu (\varnothing )=0,}
or
μ
{\displaystyle \mu }
assigns
∞
{\displaystyle \infty }
to all sets in its domain, or
μ
{\displaystyle \mu }
assigns
−
∞
{\displaystyle -\infty }
to all sets in its domain. Proof: additivity implies that for every set
A
,
{\displaystyle A,}
μ
(
A
)
=
μ
(
A
∪
∅
)
=
μ
(
A
)
+
μ
(
∅
)
.
{\displaystyle \mu (A)=\mu (A\cup \varnothing )=\mu (A)+\mu (\varnothing ).}
If
μ
(
∅
)
≠
0
,
{\displaystyle \mu (\varnothing )\neq 0,}
then this equality can be satisfied only by plus or minus infinity.
= Monotonicity
=If
μ
{\displaystyle \mu }
is non-negative and
A
⊆
B
{\displaystyle A\subseteq B}
then
μ
(
A
)
≤
μ
(
B
)
.
{\displaystyle \mu (A)\leq \mu (B).}
That is,
μ
{\displaystyle \mu }
is a monotone set function. Similarly, If
μ
{\displaystyle \mu }
is non-positive and
A
⊆
B
{\displaystyle A\subseteq B}
then
μ
(
A
)
≥
μ
(
B
)
.
{\displaystyle \mu (A)\geq \mu (B).}
= Modularity
=A set function
μ
{\displaystyle \mu }
on a family of sets
S
{\displaystyle {\mathcal {S}}}
is called a modular set function and a valuation if whenever
A
,
{\displaystyle A,}
B
,
{\displaystyle B,}
A
∪
B
,
{\displaystyle A\cup B,}
and
A
∩
B
{\displaystyle A\cap B}
are elements of
S
,
{\displaystyle {\mathcal {S}},}
then
ϕ
(
A
∪
B
)
+
ϕ
(
A
∩
B
)
=
ϕ
(
A
)
+
ϕ
(
B
)
{\displaystyle \phi (A\cup B)+\phi (A\cap B)=\phi (A)+\phi (B)}
The above property is called modularity and the argument below proves that additivity implies modularity.
Given
A
{\displaystyle A}
and
B
,
{\displaystyle B,}
μ
(
A
∪
B
)
+
μ
(
A
∩
B
)
=
μ
(
A
)
+
μ
(
B
)
.
{\displaystyle \mu (A\cup B)+\mu (A\cap B)=\mu (A)+\mu (B).}
Proof: write
A
=
(
A
∩
B
)
∪
(
A
∖
B
)
{\displaystyle A=(A\cap B)\cup (A\setminus B)}
and
B
=
(
A
∩
B
)
∪
(
B
∖
A
)
{\displaystyle B=(A\cap B)\cup (B\setminus A)}
and
A
∪
B
=
(
A
∩
B
)
∪
(
A
∖
B
)
∪
(
B
∖
A
)
,
{\displaystyle A\cup B=(A\cap B)\cup (A\setminus B)\cup (B\setminus A),}
where all sets in the union are disjoint. Additivity implies that both sides of the equality equal
μ
(
A
∖
B
)
+
μ
(
B
∖
A
)
+
2
μ
(
A
∩
B
)
.
{\displaystyle \mu (A\setminus B)+\mu (B\setminus A)+2\mu (A\cap B).}
However, the related properties of submodularity and subadditivity are not equivalent to each other.
Note that modularity has a different and unrelated meaning in the context of complex functions; see modular form.
= Set difference
=If
A
⊆
B
{\displaystyle A\subseteq B}
and
μ
(
B
)
−
μ
(
A
)
{\displaystyle \mu (B)-\mu (A)}
is defined, then
μ
(
B
∖
A
)
=
μ
(
B
)
−
μ
(
A
)
.
{\displaystyle \mu (B\setminus A)=\mu (B)-\mu (A).}
Examples
An example of a 𝜎-additive function is the function
μ
{\displaystyle \mu }
defined over the power set of the real numbers, such that
μ
(
A
)
=
{
1
if
0
∈
A
0
if
0
∉
A
.
{\displaystyle \mu (A)={\begin{cases}1&{\mbox{ if }}0\in A\\0&{\mbox{ if }}0\notin A.\end{cases}}}
If
A
1
,
A
2
,
…
,
A
n
,
…
{\displaystyle A_{1},A_{2},\ldots ,A_{n},\ldots }
is a sequence of disjoint sets of real numbers, then either none of the sets contains 0, or precisely one of them does. In either case, the equality
μ
(
⋃
n
=
1
∞
A
n
)
=
∑
n
=
1
∞
μ
(
A
n
)
{\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n})}
holds.
See measure and signed measure for more examples of 𝜎-additive functions.
A charge is defined to be a finitely additive set function that maps
∅
{\displaystyle \varnothing }
to
0.
{\displaystyle 0.}
(Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range is a bounded subset of R.)
= An additive function which is not σ-additive
=An example of an additive function which is not σ-additive is obtained by considering
μ
{\displaystyle \mu }
, defined over the Lebesgue sets of the real numbers
R
{\displaystyle \mathbb {R} }
by the formula
μ
(
A
)
=
lim
k
→
∞
1
k
⋅
λ
(
A
∩
(
0
,
k
)
)
,
{\displaystyle \mu (A)=\lim _{k\to \infty }{\frac {1}{k}}\cdot \lambda (A\cap (0,k)),}
where
λ
{\displaystyle \lambda }
denotes the Lebesgue measure and
lim
{\displaystyle \lim }
the Banach limit. It satisfies
0
≤
μ
(
A
)
≤
1
{\displaystyle 0\leq \mu (A)\leq 1}
and if
sup
A
<
∞
{\displaystyle \sup A<\infty }
then
μ
(
A
)
=
0.
{\displaystyle \mu (A)=0.}
One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets
A
n
=
[
n
,
n
+
1
)
{\displaystyle A_{n}=[n,n+1)}
for
n
=
0
,
1
,
2
,
…
{\displaystyle n=0,1,2,\ldots }
The union of these sets is the positive reals, and
μ
{\displaystyle \mu }
applied to the union is then one, while
μ
{\displaystyle \mu }
applied to any of the individual sets is zero, so the sum of
μ
(
A
n
)
{\displaystyle \mu (A_{n})}
is also zero, which proves the counterexample.
Generalizations
One may define additive functions with values in any additive monoid (for example any group or more commonly a vector space). For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on that set. For example, spectral measures are sigma-additive functions with values in a Banach algebra. Another example, also from quantum mechanics, is the positive operator-valued measure.
See also
Additive map – Z-module homomorphism
Hahn–Kolmogorov theorem – Theorem extending pre-measures to measuresPages displaying short descriptions of redirect targets
Measure (mathematics) – Generalization of mass, length, area and volume
σ-finite measure – Concept in measure theory
Signed measure – Generalized notion of measure in mathematics
Submodular set function – Set-to-real map with diminishing returns
Subadditive set function
τ-additivity
ba space – The set of bounded charges on a given sigma-algebra
This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
References
Kata Kunci Pencarian:
- Sigma-additive set function
- Σ-algebra
- Measure (mathematics)
- Set function
- Additive
- Normal distribution
- Arithmetic function
- Vitali set
- Sigma (disambiguation)
- Generalized additive model
