Sigma-additive set function: Difference between revisions

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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>

In [[mathematics]], '''additivity''' (specifically '''finite''' '''additivity''')Additivity and '''sigma -additivity''' (alsoare calledparticularly '''countableimportant additivity''')properties of a [[function (mathematics)|function]] (often a [[Measure (mathematics)|measuremeasures]]). defined on [[subset]]s of a given [[Set (mathematics)|set]]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,; andthat 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 function defined on an [[field of sets|algebra of sets]] <math>\scriptstyle\mathcal{A}</math> with values in [&minus;&infin;, +&infin;] (see the [[extended real number line]]). The function <math>\mu</math> is called '''additive''', or '''finitely additive''', if, whenever ''A'' and ''B'' are [[disjoint set]]s in <math>\scriptstyle\mathcal{A}</math>, one has

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

:<math> \mu(A \cup B) = \mu(A) + \mu(B). \, </math>

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> then

(A consequence of this is that an additive function cannot take both &minus;&infin; and +&infin; as values, for the expression &infin;&nbsp;&minus;&nbsp;&infin; is undefined.)

<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, \ldots, A_N</math> disjoint sets in <math display=inline>\mathcal{A}.</math>

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

for any <math>A_1,A_2,\dots,A_N</math> disjoint sets in <math>\scriptstyle\mathcal{A}</math>.

==&sigma;-additive set functions==

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

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

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

we say that &mu; is '''countably additive''' or '''&sigma;-additive'''. <br />

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

Any &sigma;-additive function is additive but not vice versa, as shown below.

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>\scriptstyle\mathcal{A}</math>, we have a [[Topological space|topology]] &tau;. If for any [[Directed set|directed]] family of measurable open sets <math>\scriptstyle\mathcal{G}</math>&sube;<math>\scriptstyle\mathcal{A}</math>&cap;&tau;,

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

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

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>

== Properties ==

<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==

=== Basic properties ===

Useful properties of an additive function &mu; include the following:

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

# Either &mu;(&empty;) = 0, or &mu; assigns ∞ to all sets in its ___domain, or &mu; assigns &minus;∞ to all sets in its ___domain.

# If &mu; is non-negative and ''A'' &sube; ''B'', then &mu;(''A'') &le; &mu;(''B'').

===Value of empty set===

# If ''A'' &sube; ''B'' and &mu;(''B'') &minus; &mu;(''A'') is defined, then &mu;(''B'' \ ''A'') = &mu;(''B'') &minus; &mu;(''A'').

# Given ''A'' and ''B'', &mu;(''A'' &cup; ''B'') + &mu;(''A'' &cap; ''B'') = &mu;(''A'') + &mu;(''B'').

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 '''{{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>

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

However, the related properties of [[Submodular set function|''submodularity'']] and [[Subadditive set function|''subadditivity'']] 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 &mu; defined over the [[power set]] of the [[real number]]s, such that

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> \mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\

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

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

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

where ''λ'' denotes the [[Lebesgue measure]] and ''lim'' the [[Banach limit]].

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 =\left [n,n + 1\right)</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''<sub>''n''A_n)</submath>) 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]].

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 ==

* [[signed measure]]

==See also==

* [[measure (mathematics)]]

* [[additive map]]

* {{annotated link|Additive map}}

* [[subadditive function]]

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

* [[σ-finite measure]]

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

* [[Hahn–Kolmogorov theorem]]

* {{annotated link|σ-finite measure}}

* [[τ-additivity]]

* {{annotated link|Signed 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==

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[[Category:Measure theory]]