Set function - Revision history

Citation bot: Add: volume, series. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Functions and mappings | #UCB_Category 14/160

2024-10-17T06:33:15Z

Add: volume, series. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Functions and mappings | #UCB_Category 14/160

← Previous revision		Revision as of 06:33, 17 October 2024
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	<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>		<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>
	* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>		* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>
	* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other. <math>\mu</math> is called a [[Equivalence (measure theory)#Supporting measure\|{{em\|{{visible anchor\|supporting measure}}}}]] of a measure <math>\nu</math> if <math>\mu</math> is [[sigma-finite\|<math>\sigma</math>-finite]] and they are equivalent.<ref>{{cite book \|last1=Kallenberg \|first1=Olav \|author-link1=Olav Kallenberg \|year=2017 \|title=Random Measures, Theory and Applications\|location= Switzerland \|publisher=Springer \|doi= 10.1007/978-3-319-41598-7\|isbn=978-3-319-41596-3\|page=21}}</ref>		* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other. <math>\mu</math> is called a [[Equivalence (measure theory)#Supporting measure\|{{em\|{{visible anchor\|supporting measure}}}}]] of a measure <math>\nu</math> if <math>\mu</math> is [[sigma-finite\|<math>\sigma</math>-finite]] and they are equivalent.<ref>{{cite book \|last1=Kallenberg \|first1=Olav \|author-link1=Olav Kallenberg \|year=2017 \|title=Random Measures, Theory and Applications\|series=Probability Theory and Stochastic Modelling \|volume=77 \|location= Switzerland \|publisher=Springer \|doi= 10.1007/978-3-319-41598-7\|isbn=978-3-319-41596-3\|page=21}}</ref>
	<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>		<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>
	</ul>		</ul>

David Eppstein: /* References */ sfn whitelist

2024-05-06T01:50:10Z

References: sfn whitelist

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	==References==		==References==
			{{sfn whitelist\|CITEREFDurrett2019\|CITEREFRoydenFitzpatrick2010}}

	* {{Durrett Probability Theory and Examples 5th Edition}} <!--{{sfn\|Durrett\|2019\|p=}}-->		* {{Durrett Probability Theory and Examples 5th Edition}} <!--{{sfn\|Durrett\|2019\|p=}}-->
	* {{Kolmogorov Fomin Elements of the Theory of Functions and Functional Analysis}} <!--{{sfn\|Kolmogorov\|Fomin\|1957\|p=}}-->		* {{Kolmogorov Fomin Elements of the Theory of Functions and Functional Analysis}} <!--{{sfn\|Kolmogorov\|Fomin\|1957\|p=}}-->

2600:4040:71C9:4800:6CF3:6ABC:3964:A56F: /* Definitions */Fixed sentence

2023-04-27T01:38:46Z

Definitions: Fixed sentence

← Previous revision		Revision as of 01:38, 27 April 2023
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	If <math>\mathcal{F}</math> is a [[family of sets]] over <math>\Omega</math> (meaning that <math>\mathcal{F} \subseteq \wp(\Omega)</math> where <math>\wp(\Omega)</math> denotes the [[powerset]]) then a {{em\|set function on <math>\mathcal{F}</math>}} is a function <math>\mu</math> with [[Domain of a function\|domain]] <math>\mathcal{F}</math> and [[codomain]] <math>[-\infty, \infty]</math> or, sometimes, the codomain is instead some [[vector space]], as with [[vector measure]]s, [[complex measure]]s, and [[projection-valued measure]]s.		If <math>\mathcal{F}</math> is a [[family of sets]] over <math>\Omega</math> (meaning that <math>\mathcal{F} \subseteq \wp(\Omega)</math> where <math>\wp(\Omega)</math> denotes the [[powerset]]) then a {{em\|set function on <math>\mathcal{F}</math>}} is a function <math>\mu</math> with [[Domain of a function\|domain]] <math>\mathcal{F}</math> and [[codomain]] <math>[-\infty, \infty]</math> or, sometimes, the codomain is instead some [[vector space]], as with [[vector measure]]s, [[complex measure]]s, and [[projection-valued measure]]s.
	The domain is a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below.		The domain of a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below.
	{{Families of sets}}		{{Families of sets}}

Mgkrupa: /* Inner measures, outer measures, and other properties */

2023-03-07T23:34:13Z

Inner measures, outer measures, and other properties

← Previous revision		Revision as of 23:34, 7 March 2023
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	<li>an [[#outer measure\|{{em\|outer measure}}]] if <math>\mu</math> is non-negative, [[#countably subadditive\|countably subadditive]], has a [[#null empty set\|null empty set]], and has the [[power set]] <math>\wp(\Omega)</math> as its domain.</li>		<li>an [[#outer measure\|{{em\|outer measure}}]] if <math>\mu</math> is non-negative, [[#countably subadditive\|countably subadditive]], has a [[#null empty set\|null empty set]], and has the [[power set]] <math>\wp(\Omega)</math> as its domain.</li>
	<li>an [[Inner measure\|{{em\|{{visible anchor\|inner measure}}}}]] if <math>\mu</math> is non-negative, [[#superadditive\|superadditive]], [[#continuous from above\|continuous from above]], has a [[#null empty set\|null empty set]], has the [[power set]] <math>\wp(\Omega)</math> as its domain, and [[#infinity is approached from below\|<math>+\infty</math> is approached from below]].</li>		<li>an [[Inner measure\|{{em\|{{visible anchor\|inner measure}}}}]] if <math>\mu</math> is non-negative, [[#superadditive\|superadditive]], [[#continuous from above\|continuous from above]], has a [[#null empty set\|null empty set]], has the [[power set]] <math>\wp(\Omega)</math> as its domain, and [[#infinity is approached from below\|<math>+\infty</math> is approached from below]].</li>
			<li>[[Atomic measure\|{{em\|atomic}}]] if every measurable set of positive measure contains an [[Atom (measure theory)\|atom]].</li>
	</ul>		</ul>

Mgkrupa at 18:53, 7 March 2023

2023-03-07T18:53:21Z

← Previous revision		Revision as of 18:53, 7 March 2023
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	[[Category:Basic concepts in set theory]]		[[Category:Basic concepts in set theory]]
	[[Category:Functions and mappings]]		[[Category:Functions and mappings]]
			[[Category:Measure theory]]
	[[Category:Measures (measure theory)]]		[[Category:Measures (measure theory)]]

Mgkrupa: /* Further reading */

2023-02-02T05:02:47Z

Mgkrupa: /* Infinite-dimensional space */

2023-02-02T04:51:33Z

Infinite-dimensional space

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	====Infinite-dimensional space====		====Infinite-dimensional space====

	{{See also\|Gaussian measure#Infinite-dimensional spaces\|Abstract Wiener space\|Feldman–Hájek theorem}}		{{See also\|Gaussian measure#Infinite-dimensional spaces\|Abstract Wiener space\|Feldman–Hájek theorem\|Radonifying function}}

	As detailed in the article on [[infinite-dimensional Lebesgue measure]], the only locally finite and translation-invariant [[Borel measure]] on an infinite-dimensional [[Separable space\|separable]] [[normed space]] is the [[trivial measure]]. However, it is possible to define [[Gaussian measure]]s on infinite-dimensional [[topological vector space]]s. The [[structure theorem for Gaussian measures]] shows that the [[abstract Wiener space]] construction is essentially the only way to obtain a strictly positive Gaussian measure on a [[Separable space\|separable]] [[Banach space]].		As detailed in the article on [[infinite-dimensional Lebesgue measure]], the only locally finite and translation-invariant [[Borel measure]] on an infinite-dimensional [[Separable space\|separable]] [[normed space]] is the [[trivial measure]]. However, it is possible to define [[Gaussian measure]]s on infinite-dimensional [[topological vector space]]s. The [[structure theorem for Gaussian measures]] shows that the [[abstract Wiener space]] construction is essentially the only way to obtain a strictly positive Gaussian measure on a [[Separable space\|separable]] [[Banach space]].

Mgkrupa: /* Infinite-dimensional space */

2023-02-02T04:17:41Z

Infinite-dimensional space

← Previous revision		Revision as of 04:17, 2 February 2023
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	====Infinite-dimensional space====		====Infinite-dimensional space====

	{{See also\|Gaussian measure#Infinite-dimensional spaces\|Abstract Wiener space}}		{{See also\|Gaussian measure#Infinite-dimensional spaces\|Abstract Wiener space\|Feldman–Hájek theorem}}

	As detailed in the article on [[infinite-dimensional Lebesgue measure]], the only locally finite and translation-invariant [[Borel measure]] on an infinite-dimensional [[Separable space\|separable]] [[normed space]] is the [[trivial measure]]. However, it is possible to define [[Gaussian measure]]s on infinite-dimensional [[topological vector space]]s. The [[structure theorem for Gaussian measures]] shows that the [[abstract Wiener space]] construction is essentially the only way to obtain a strictly positive Gaussian measure on a [[Separable space\|separable]] [[Banach space]].		As detailed in the article on [[infinite-dimensional Lebesgue measure]], the only locally finite and translation-invariant [[Borel measure]] on an infinite-dimensional [[Separable space\|separable]] [[normed space]] is the [[trivial measure]]. However, it is possible to define [[Gaussian measure]]s on infinite-dimensional [[topological vector space]]s. The [[structure theorem for Gaussian measures]] shows that the [[abstract Wiener space]] construction is essentially the only way to obtain a strictly positive Gaussian measure on a [[Separable space\|separable]] [[Banach space]].

Mgkrupa: /* Relationships between set functions */

2023-02-02T03:54:36Z

Relationships between set functions

← Previous revision		Revision as of 03:54, 2 February 2023
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	<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>		<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>
	* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>		* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>
	* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other.		* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other. <math>\mu</math> is called a [[Equivalence (measure theory)#Supporting measure\|{{em\|{{visible anchor\|supporting measure}}}}]] of a measure <math>\nu</math> if <math>\mu</math> is [[sigma-finite\|<math>\sigma</math>-finite]] and they are equivalent.<ref>{{cite book \|last1=Kallenberg \|first1=Olav \|author-link1=Olav Kallenberg \|year=2017 \|title=Random Measures, Theory and Applications\|location= Switzerland \|publisher=Springer \|doi= 10.1007/978-3-319-41598-7\|isbn=978-3-319-41596-3\|page=21}}</ref>
	<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>		<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>
	</ul>		</ul>

Mgkrupa: /* Relationships between set functions */

2023-02-02T03:51:39Z

Relationships between set functions

← Previous revision		Revision as of 03:51, 2 February 2023
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	<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>		<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>
	* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>		* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>
			* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other.
	<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>		<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>
	</ul>		</ul>

← Previous revision		Revision as of 06:33, 17 October 2024
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	<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>		<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>
	* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>		* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>
	* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other. <math>\mu</math> is called a [[Equivalence (measure theory)#Supporting measure\|{{em\|{{visible anchor\|supporting measure}}}}]] of a measure <math>\nu</math> if <math>\mu</math> is [[sigma-finite\|<math>\sigma</math>-finite]] and they are equivalent.<ref>{{cite book \|last1=Kallenberg \|first1=Olav \|author-link1=Olav Kallenberg \|year=2017 \|title=Random Measures, Theory and Applications\|location= Switzerland \|publisher=Springer \|doi= 10.1007/978-3-319-41598-7\|isbn=978-3-319-41596-3\|page=21}}</ref>		* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other. <math>\mu</math> is called a [[Equivalence (measure theory)#Supporting measure\|{{em\|{{visible anchor\|supporting measure}}}}]] of a measure <math>\nu</math> if <math>\mu</math> is [[sigma-finite\|<math>\sigma</math>-finite]] and they are equivalent.<ref>{{cite book \|last1=Kallenberg \|first1=Olav \|author-link1=Olav Kallenberg \|year=2017 \|title=Random Measures, Theory and Applications\|series=Probability Theory and Stochastic Modelling \|volume=77 \|location= Switzerland \|publisher=Springer \|doi= 10.1007/978-3-319-41598-7\|isbn=978-3-319-41596-3\|page=21}}</ref>
	<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>		<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>
	</ul>		</ul>

← Previous revision		Revision as of 05:02, 2 February 2023
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	{{Measure theory}}		{{Measure theory}}
			{{Analysis in topological vector spaces}}

	[[Category:Basic concepts in set theory]]		[[Category:Basic concepts in set theory]]

← Previous revision		Revision as of 03:54, 2 February 2023
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	<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>		<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>
	* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>		* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>
	* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other.		* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other. <math>\mu</math> is called a [[Equivalence (measure theory)#Supporting measure\|{{em\|{{visible anchor\|supporting measure}}}}]] of a measure <math>\nu</math> if <math>\mu</math> is [[sigma-finite\|<math>\sigma</math>-finite]] and they are equivalent.<ref>{{cite book \|last1=Kallenberg \|first1=Olav \|author-link1=Olav Kallenberg \|year=2017 \|title=Random Measures, Theory and Applications\|location= Switzerland \|publisher=Springer \|doi= 10.1007/978-3-319-41598-7\|isbn=978-3-319-41596-3\|page=21}}</ref>
	<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>		<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>
	</ul>		</ul>

← Previous revision		Revision as of 03:51, 2 February 2023
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	<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>		<li><math>\mu</math> is said to be [[Absolute continuity (measure theory)\|{{em\|{{visible anchor\|absolutely continuous}} with respect to <math>\nu</math>}}]] or [[Domination (measure theory)\|{{em\|dominated by <math>\nu</math>}}]], written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math>
	* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>		* If <math>\mu</math> and <math>\nu</math> are [[σ-finite measure\|<math>\sigma</math>-finite measure]]s on the same measurable space and if <math>\mu \ll \nu,</math> then the [[Radon–Nikodym derivative]] <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li>
			* <math>\mu</math> and <math>\nu</math> are called [[Equivalence (measure theory)\|{{em\|{{visible anchor\|equivalent}}}}]] if each one is [[#absolutely continuous\|absolutely continuous]] with respect to the other.
	<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>		<li><math>\mu</math> and <math>\nu</math> are [[Singular measure\|{{em\|{{visible anchor\|singular}}}}]], written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li>
	</ul>		</ul>