additivity Definition and 1 Threads

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.

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