Sigma-additive set function: Difference between revisions

Revision as of 20:33, 31 July 2021

Latest revision as of 17:33, 18 July 2025

(26 intermediate revisions by 16 users not shown)

Line 1:

{{Short description|Mapping function}}

In [[mathematics]], an '''additive set function''' is a function mapping sets to numbers, with the property that its value on a [[Union (set theory)|union]] of two 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 funciton 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 infinitely many sets, that is, <math display="inline">\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n)</math>.

{{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 funcitonfunction]] 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 infinitely[[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.

== Additive (or finitely additive) set functions ==

Let <math>\mu</math> be a [[set function]] defined on an [[Field of sets|algebra of sets]] <math>\scriptstyle\mathcal{A}</math> with values in <math>[-\infty, \infty]</math> (see the [[extended real number line]]). The function <math>\mu</math> is called '''{{visible anchor|additive|additive, set function}}''' or '''{{visible anchor|finitely additive|finitely additive set function}}''', if, whenever <math>A</math> and <math>B</math> are [[disjoint set]]s in <math>\scriptstyle\mathcal{A},</math> one hasthen

<math display="block">\mu(A \cup B) = \mu(A) + \mu(B).</math>

A consequence of this is that an additive function cannot take both <math>- \infty</math> and <math>+ \infty</math> as values, for the expression <math>\infty - \infty</math> is undefined.

One can prove by [[mathematical induction]] that an additive function satisfies

<math display="block">\mu\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu\left(A_n\right)</math>

for any <math>A_1, A_2, \dotsldots, A_N</math> disjoint sets in <math display=inline>\mathcal{A}.</math>

== &sigma;-additive set functions ==

Suppose that <math>\scriptstyle\mathcal{A}</math> is a [[Sigma algebra|&sigma;-algebra]]. If for every [[sequence]] <math>A_1, A_2, \ldots, A_n, \ldots</math> of pairwise disjoint sets in <math>\scriptstyle\mathcal{A},</math>

<math display="block">\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n),</math>

holds then <math>\mu</math> is said to be {{em|countably additive}} or {{em|{{sigma}}-additive}}.

Every {{sigma}}-additive function is additive but not vice versa, as shown below.

== &tau;-additive set functions ==

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 ==

== Properties ==

Useful properties of an additive set function <math>\mu</math> include the following.

=== 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>. If <math>\mu(\varnothing) \neq 0,</math> then this equality can be satisfied only by plus or minus infinity.

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 ===

If <math>\mu</math> is non-negative and <math>A \subseteq B</math> then <math>\mu(A) \leq \mu(B).</math> That is, <math>\mu</math> is a ''monotone set function''. Similarly, If <math>\mu</math> is non-positive and <math>A \subseteq B</math> then <math>\mu(A) \geq \mu(B).</math>

=== Monotonicity ===

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

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\cdot \mu(A \cap B)</math>

If <math>\mu</math> is non-negative and <math>A \subseteq B</math> then <math>\mu(A) \leq \mu(B).</math> That is, <math>\mu</math> is a '''{{visible anchor|monotone set function}}'''. Similarly, If <math>\mu</math> is non-positive and <math>A \subseteq B</math> then <math>\mu(A) \geq \mu(B).</math>

The above property is called ''modularity'', and we have just proved that modularity is equivalent to additivity. However, there are related properties called [[Submodular set function|''submodularity'']] and [[Subadditive set function|''subadditivity'']], which are not equivalent.

=== 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"> \mu(A\cup B)+ \mu(A\cap B) = \mu(A) + \mu(B)</math>

The above property is called '''{{visible anchor|modularity}}''' and the argument below proves that additivity implies modularity.

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\cdot \mu(A \cap B).</math>

The above property is called ''modularity'', and we have just proved that modularity is equivalent to additivity. However, there arethe related properties calledof [[Submodular set function|''submodularity'']] and [[Subadditive set function|''subadditivity'']], which are not equivalent to each other.

Note that modularity has a different and unrelated meaning in the context of complex functions; see [[modular form]].

=== Set difference ===

If <math>A \subseteq B</math> and <math>\mu(B) - \mu(A)</math> is defined, then <math>\mu(B \setminus A) = \mu(B) - \mu(A).</math>

== Examples ==

An example of a {{sigma}}-additive function is the function <math>\mu</math> defined over the [[power set]] of the [[real number]]s, such that

<math display="block">\mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\

0 & \mbox{ if } 0 \notin A.

\end{cases}</math>

If <math>A_1, A_2, \ldots, A_n, \ldots</math> is a sequence of disjoint sets of real numbers, then either none of the sets contains 0, or precisely one of them does. In either case, the equality

<math display="block">\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n)</math>

holds.

See [[measureMeasure (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|last1=Bhaskara Rao|first1=K. P. S.|first2=M. |last2=Bhaskara Rao|title=Theory of charges: a study of finitely additive measures|date=1983|publisher=Academic Press|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 is a bounded subset of ''R''.)

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

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

An example of an additive function which is not &sigma;-additive is obtained by considering <math>\mu</math>, defined over the Lebesgue sets of the [[real number]]s <math>\R</math> by the formula

<math display="block">\mu(A) = \lim_{k\to\infty} \frac{1}{k} \cdot \lambda(A \cap (0,k)),</math>

where <math>\lambda</math> denotes the [[Lebesgue measure]] and ''<math>\lim''</math> the [[Banach limit]]. It satisfies <math>0 \leq \mu(A) \leq 1</math> and if <math>\sup A < \infty</math> then <math>\mu(A) = 0.</math>

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 [[groupGroup (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 ==

* {{annotated link|signed measure}}

* {{annotated link|Measure (mathematics)}}

* {{annotated link|Additive map}}

*[[Submodular set function]]

*[[Subadditive set function]]

* {{annotated link|σ-finite measure}}

* {{annotated link|Hahn–Kolmogorov theorem}}

* {{annotated link|Measure (mathematics)}}

* {{annotated link|σ-finite measure}}

* {{annotated link|signedSigned measure}}

*[[ {{annotated link|Submodular set function]]}}

*[[ {{annotated link|Subadditive set function]]}}

* {{annotated link|τ-additivity}}

* [[ba space]] – The set of bounded charges on a given sigma-algebra

{{PlanetMath attribution|id=3400|title=additive}}

== References ==

{{reflist|group=note}}