Sigma-additive set function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an additive set function is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 μ
 
 
 {\textstyle \mu }
 
 mapping sets to numbers, with the property that its value on a <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of two <a href="/facts/Disjoint_set/nLr8XRVd">disjoint</a> 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 <a href="/facts/Set_function/Dy46B9EJ">set function</a> 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 <a href="/facts/Countably_infinite/Wj4DmWPv">countably infinite</a> 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 <a href="/facts/Measure_(mathematics)/yCq7zlI4">measures</a>. They are abstractions of how intuitive properties of size (<a href="/facts/Length/EwFyAqGH">length</a>, <a href="/facts/Area/KsNG1G9t">area</a>, <a href="/facts/Volume/aeAXCBUd">volume</a>) 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.

Sigma-additive set function open-in-new

Sigma-additive set function