Editing Sigma-additive set function (section)

{{Short description|Mapping function}}
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In [[mathematics]], an '''additive set function''' is a [[function (mathematics)|function]] <math display=inline>\mu</math> mapping sets to numbers, with the property that its value on a [[Union (set theory)|union]] of two [[disjoint set|disjoint]] sets equals the sum of its values on these sets, namely, <math display=inline>\mu(A \cup B) = \mu(A) + \mu(B).</math> 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 '''&sigma;-additive set function''' is a function that has the additivity property even for [[countably infinite]] many sets, that is, <math display=inline>\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n).</math>

Additivity and sigma-additivity are particularly important properties of [[Measure (mathematics)|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 &sigma;-additivity; that is, &sigma;-additivity implies additivity.

The term '''[[#modular set function|modular set function]]''' is equivalent to additive set function; see [[Sigma-additive set function#modularity|modularity]] below.