In mathematics, an additive set function is a function μ \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}},}
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