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Concept
Sigma-additive set function
Summary
In mathematics, an additive set function is a function 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, 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,
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.
Let be a set function defined on an algebra of sets with values in (see the extended real number line). The function is called or , if whenever and are disjoint sets in then
A consequence of this is that an additive function cannot take both and as values, for the expression is undefined.
One can prove by mathematical induction that an additive function satisfies
for any disjoint sets in
Suppose that is a σ-algebra. If for every sequence of pairwise disjoint sets in
holds then is said to be or .
Every sigma-additive function is additive but not vice versa, as shown below.
Suppose that in addition to a sigma algebra we have a topology If for every directed family of measurable open sets
we say that is -additive. In particular, if is inner regular (with respect to compact sets) then it is τ-additive.
Useful properties of an additive set function include the following.
Official source
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