Sigma-additive set function: Difference between revisions

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{{Short description|Mapping function}}

{{mcn|date=April 2024}}

In [[mathematics]], an '''additive set function''' is a [[function (mathematics)|function]] <math display>\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.

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Suppose that in addition to a sigma algebra <math display=inline>\mathcal{A},</math> we have a [[Topological space|topology]] <math>\tau.</math> If for every [[Directed set|directed]] family of measurable [[open set]]s <math display=inline>\mathcal{G} \subseteq \mathcal{A} \cap \tau,</math>

<math display=block>\mu\left(\bigcup \mathcal{G} \right) = \sup_{G\in\mathcal{G}} \mu(G),</math>

we say that <math>\mu</math> is <math>\tau</math>-additive. In particular, if <math>\mu</math> is [[Inner regular measure|inner regular]] (with respect to compact sets) then it is &<math>\tau;</math>-additive.<ref name=Fremlin>D. H. Fremlin ''Measure Theory, Volume 4'', Torres Fremlin, 2003.</ref>

==Properties==

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===Value of empty set===

Either <math>\mu(\varnothing) = 0,</math> or <math>\mu</math> assigns <math>\infty</math> to all sets in its ___domain, or <math>\mu</math> assigns <math>- \infty</math> to all sets in its ___domain. ''Proof'': additivity implies that for every set <math>A,</math> <math>\mu(A) = \mu(A \cup \varnothing) = \mu(A) + \mu( \varnothing).</math> (it's possible in the edge case of an empty ___domain that the only choice for <math>A</math> is the [[empty set]] itself, but that still works). If <math>\mu(\varnothing) \neq 0,</math> then this equality can be satisfied only by plus or minus infinity.

===Monotonicity===

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===Modularity{{Anchor|modularity}}===

{{See also|Valuation (geometry)}}

{{See also|Valuation (measure theory)}}

A [[set function]] <math>\mu</math> on a [[family of sets]] <math>\mathcal{S}</math> is called a '''{{visible anchor|modular set function}}''' and a '''[[Valuation (geometry)|{{visible anchor|valuation}}]]''' if whenever <math>A,</math> <math>B,</math> <math>A\cup B,</math> and <math>A\cap B</math> are elements of <math>\mathcal{S},</math> then

<math display="block"> \phimu(A\cup B)+ \phimu(A\cap B) = \phimu(A) + \phimu(B)</math>

The above property is called '''{{visible anchor|modularity}}''' and the argument below proves that modularityadditivity isimplies equivalent to additivitymodularity.

Given <math>A</math> and <math>B,</math> <math>\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B).</math> ''Proof'': write <math>A = (A \cap B) \cup (A \setminus B)</math> and <math>B = (A \cap B) \cup (B \setminus A)</math> and <math>A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A),</math> where all sets in the union are disjoint. Additivity implies that both sides of the equality equal <math>\mu(A \setminus B) + \mu(B \setminus A) + 2\mu(A \cap B).</math>

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See [[Measure (mathematics)|measure]] and [[signed measure]] for more examples of {{sigma}}-additive functions.

A ''charge'' is defined to be a finitely additive set function that maps <math>\varnothing</math> to <math>0.</math><ref>{{Cite book|lastlast1=Bhaskara Rao|firstfirst1=K. P. S.|urlfirst2=https://wwwM.worldcat.org/oclc/21196971 |last2=Bhaskara Rao|title=Theory of charges: a study of finitely additive measures|date=1983|publisher=Academic Press|others=M. Bhaskara Rao|isbn=0-12-095780-9|___location=London|pages=35|oclc=21196971}}</ref> (Cf. [[ba space]] for information about ''bounded'' charges, where we say a charge is ''bounded'' to mean its range itsis a bounded subset of ''R''.)

===An additive function which is not &sigma;-additive===

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One can check that this function is additive by using the linearity of the limit. That this function is not &sigma;-additive follows by considering the sequence of disjoint sets

<math display=block>A_n = [n,n + 1)</math>

for <math>n = 0, 1, 2, \ldots</math> The union of these sets is the [[positive reals]], and <math>\mu</math> applied to the union is then one, while <math>\mu</math> applied to any of the individual sets is zero, so the sum of <math>\mu(A_n)</math> is also zero, which proves the counterexample.

==Generalizations==

One may define additive functions with values in any additive [[monoid]] (for example any [[Group (mathematics)|group]] or more commonly a [[vector space]]). For sigma-additivity, one needs in addition that the concept of [[limit of a sequence]] be defined on that set. For example, [[spectral measure]]s are sigma-additive functions with values in a [[Banach algebra]]. Another example, also from [[quantum mechanics]], is the [[positive operator-valued measure]].

==See also==