Sigma-additive set function: Difference between revisions
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{{Short description|Mapping function}} |
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In [[mathematics]], additivity (specifically finite additivity) and sigma additivity (also called countable additivity) of a [[function (mathematics)|function]] (often a [[Measure (mathematics)|measure]]) defined on [[subset]]s of a given [[Set (mathematics)|set]] 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. |
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{{mcn|date=April 2024}} |
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In [[mathematics]], an '''additive set function''' is a [[function (mathematics)|function]] <math display=inline>\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 '''σ-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> |
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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 σ-additivity; that is, σ-additivity implies additivity. |
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== Additive (or finitely additive) set functions == |
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Let ''<math>\mu</math>'' be a function defined on an [[field of sets|algebra of sets]] <math>\scriptstyle\mathcal{A}</math> with values in [−∞, +∞] (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 |
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The term '''[[#modular set function|modular set function]]''' is equivalent to additive set function; see [[Sigma-additive set function#modularity|modularity]] below. |
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:<math> \mu(A \cup B) = \mu(A) + \mu(B). \, </math> |
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==Additive (or finitely additive) set functions== |
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(A consequence of this is that an additive function cannot take both −∞ and +∞ as values, for the expression ∞ − ∞ is undefined.)° |
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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 |
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<math display=block>\mu(A \cup B) = \mu(A) + \mu(B).</math> |
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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. |
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One can prove by [[mathematical induction]] that an additive function satisfies |
One can prove by [[mathematical induction]] that an additive function satisfies |
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<math display=block>\mu\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu\left(A_n\right)</math> |
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for any <math>A_1, A_2, \ldots, A_N</math> disjoint sets in <math display=inline>\mathcal{A}.</math> |
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: <math>\mu\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu(A_n)</math> |
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for any <math>A_1,A_2,\dots,A_N</math> disjoint sets in <math>\scriptstyle\mathcal{A}</math>. |
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==σ-additive set functions== |
==σ-additive set functions== |
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Suppose that <math>\scriptstyle\mathcal{A}</math> is a [[sigma algebra|σ-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 |
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Suppose that <math>\scriptstyle\mathcal{A}</math> is a [[Sigma algebra|σ-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> |
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:<math> \mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n)</math>{{math|,}} |
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<math display=block>\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n),</math> |
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we say that μ is countably additive or σ-additive. <br /> |
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holds then <math>\mu</math> is said to be {{em|countably additive}} or {{em|{{sigma}}-additive}}. |
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Any σ-additive function is additive but not vice versa, as shown below. |
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Every {{sigma}}-additive function is additive but not vice versa, as shown below. |
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==τ-additive set functions== |
==τ-additive set functions== |
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Suppose that in addition to a sigma algebra <math>\scriptstyle\mathcal{A}</math>, we have a [[Topological space|topology]] τ. If for any [[Directed set|directed]] family of measurable [[open set]]s <math>\scriptstyle\mathcal{G}</math> ⊆ <math>\scriptstyle\mathcal{A}</math> ∩ τ, |
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:<math> \mu\left(\bigcup \mathcal{G} \right) = \sup_{G\in\mathcal{G}} \mu(G)</math>{{math|,}} |
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we say that μ is τ-additive. In particular, if μ is [[Inner regular measure|inner regular]] (with respect to compact sets) then it is τ-additive.<ref name=Fremlin>D.H. Fremlin ''Measure Theory, Volume 4'', Torres Fremlin, 2003.</ref> |
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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> |
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== Properties == |
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<math display=block>\mu\left(\bigcup \mathcal{G} \right) = \sup_{G\in\mathcal{G}} \mu(G),</math> |
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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 τ-additive.<ref name=Fremlin>D. H. Fremlin ''Measure Theory, Volume 4'', Torres Fremlin, 2003.</ref> |
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==Properties== |
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=== Basic properties === |
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Useful properties of an additive function μ include the following: |
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Useful properties of an additive set function <math>\mu</math> include the following. |
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# Either μ(∅) = 0, or μ assigns ∞ to all sets in its domain, or μ assigns −∞ to all sets in its domain. |
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# If μ is non-negative and ''A'' ⊆ ''B'', then μ(''A'') ≤ μ(''B''). |
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===Value of empty set=== |
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# If ''A'' ⊆ ''B'' and μ(''B'') − μ(''A'') is defined, then μ(''B'' \ ''A'') = μ(''B'') − μ(''A''). |
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# Given ''A'' and ''B'', μ(''A'' ∪ ''B'') - μ(''A'' ∩ ''B'') = μ(''A'') + μ(''B''). |
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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. |
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===Monotonicity=== |
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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> |
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===Modularity{{Anchor|modularity}}=== |
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{{See also|Valuation (geometry)}} |
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{{See also|Valuation (measure theory)}} |
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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 |
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<math display=block> \phi(A\cup B)+ \phi(A\cap B) = \phi(A) + \phi(B)</math> |
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The above property is called '''{{visible anchor|modularity}}''' and the argument below proves that additivity implies modularity. |
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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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However, the related properties of [[Submodular set function|''submodularity'']] and [[Subadditive set function|''subadditivity'']] are not equivalent to each other. |
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Note that modularity has a different and unrelated meaning in the context of complex functions; see [[modular form]]. |
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===Set difference=== |
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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> |
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==Examples== |
==Examples== |
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An example of a σ-additive function is the function μ defined over the [[power set]] of the [[real number]]s, such that |
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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 |
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:<math> \mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\ |
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<math display=block>\mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\ |
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0 & \mbox{ if } 0 \notin A. |
0 & \mbox{ if } 0 \notin A. |
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\end{cases}</math> |
\end{cases}</math> |
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If <math>A_1,A_2,\ |
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 |
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<math display=block>\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n)</math> |
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holds. |
holds. |
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See [[ |
See [[Measure (mathematics)|measure]] and [[signed measure]] for more examples of {{sigma}}-additive functions. |
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A ''charge'' is defined to be a finitely additive set function that maps <math>\varnothing</math> to <math>0.</math><ref>{{Cite book|last=Bhaskara Rao|first=K. P. S.|first2=M. |last2=Bhaskara Rao|url=https://www.worldcat.org/oclc/21196971|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''.) |
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===An additive function which is not σ-additive=== |
===An additive function which is not σ-additive=== |
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An example of an additive function which is not σ-additive is obtained by considering μ, defined over the Lebesgue sets of the [[real number]]s by the formula |
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An example of an additive function which is not σ-additive is obtained by considering <math>\mu</math>, defined over the Lebesgue sets of the [[real number]]s <math>\R</math> by the formula |
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:<math> \mu(A)=\lim_{k\to\infty} \frac{1}{k} \cdot \lambda\left(A \cap \left(0,k\right)\right),</math> |
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<math display=block>\mu(A) = \lim_{k\to\infty} \frac{1}{k} \cdot \lambda(A \cap (0,k)),</math> |
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where ''λ'' denotes the [[Lebesgue measure]] and ''lim'' the [[Banach limit]]. |
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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> |
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One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets |
One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets |
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<math display=block>A_n = [n,n + 1)</math> |
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for |
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. |
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==Generalizations== |
==Generalizations== |
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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]]. |
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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]]. |
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== See also == |
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* [[signed measure]] |
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==See also== |
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* [[measure (mathematics)]] |
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* [[additive map]] |
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* {{annotated link|Additive map}} |
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* [[subadditive function]] |
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* {{annotated link|Hahn–Kolmogorov theorem}} |
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* [[σ-finite measure]] |
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* {{annotated link|Measure (mathematics)}} |
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* [[Hahn–Kolmogorov theorem]] |
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* {{annotated link|σ-finite measure}} |
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* [[τ-additivity]] |
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* {{annotated link|Signed measure}} |
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* {{annotated link|Submodular set function}} |
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* {{annotated link|Subadditive set function}} |
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* {{annotated link|τ-additivity}} |
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* [[ba space]] – The set of bounded charges on a given sigma-algebra |
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{{PlanetMath attribution|id=3400|title=additive}} |
{{PlanetMath attribution|id=3400|title=additive}} |
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==References== |
==References== |
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{{Reflist}} |
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{{reflist|group=note}} |
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{{reflist}} |
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[[Category:Measure theory]] |
[[Category:Measure theory]] |
Latest revision as of 13:22, 4 October 2024
This article needs additional citations for verification. (April 2024) |
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.
Additive (or finitely additive) set functions
[edit]Let be a set function defined on an algebra of sets with values in (see the extended real number line). The function is called additive or finitely additive, 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
σ-additive set functions
[edit]Suppose that is a σ-algebra. If for every sequence of pairwise disjoint sets in holds then is said to be countably additive or 𝜎-additive. Every 𝜎-additive function is additive but not vice versa, as shown below.
τ-additive set functions
[edit]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.[1]
Properties
[edit]Useful properties of an additive set function include the following.
Value of empty set
[edit]Either or assigns to all sets in its domain, or assigns to all sets in its domain. Proof: additivity implies that for every set If then this equality can be satisfied only by plus or minus infinity.
Monotonicity
[edit]If is non-negative and then That is, is a monotone set function. Similarly, If is non-positive and then
Modularity
[edit]A set function on a family of sets is called a modular set function and a valuation if whenever and are elements of then The above property is called modularity and the argument below proves that additivity implies modularity.
Given and Proof: write and and where all sets in the union are disjoint. Additivity implies that both sides of the equality equal
However, the related properties of submodularity and 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
[edit]If and is defined, then
Examples
[edit]An example of a 𝜎-additive function is the function defined over the power set of the real numbers, such that
If 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 holds.
See measure and signed measure for more examples of 𝜎-additive functions.
A charge is defined to be a finitely additive set function that maps to [2] (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 σ-additive
[edit]An example of an additive function which is not σ-additive is obtained by considering , defined over the Lebesgue sets of the real numbers by the formula where denotes the Lebesgue measure and the Banach limit. It satisfies and if then
One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets for The union of these sets is the positive reals, and applied to the union is then one, while applied to any of the individual sets is zero, so the sum of is also zero, which proves the counterexample.
Generalizations
[edit]One may define additive functions with values in any additive monoid (for example any 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 measures are sigma-additive functions with values in a Banach algebra. Another example, also from quantum mechanics, is the positive operator-valued measure.
See also
[edit]- Additive map – Z-module homomorphism
- Hahn–Kolmogorov theorem – Theorem extending pre-measures to measures
- Measure (mathematics) – Generalization of mass, length, area and volume
- σ-finite measure – Concept in measure theory
- Signed measure – Generalized notion of measure in mathematics
- Submodular set function – Set-to-real map with diminishing returns
- Subadditive set function
- τ-additivity
- ba space – The set of bounded charges on a given sigma-algebra
This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
References
[edit]- ^ D. H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.
- ^ Bhaskara Rao, K. P. S.; Bhaskara Rao, M. (1983). Theory of charges: a study of finitely additive measures. London: Academic Press. p. 35. ISBN 0-12-095780-9. OCLC 21196971.