Difference between revisions of "Measure"

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| If <math>\bigudot^\infty_{n=1}A_n\in R</math> then <math>\mu_0(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)</math>
 
| If <math>\bigudot^\infty_{n=1}A_n\in R</math> then <math>\mu_0(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)</math>
 
|}
 
|}
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==Terminology==
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===Complete measure===
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A measure is complete if for {{M|A\in\mathcal{A} }} we have <math>[\mu(A)=0\wedge B\subset A]\implies B\in \mathcal{A}</math>
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===Finite===
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A set {{M|A\in\mathcal{A} }} is finite if {{M|\mu(A)<\infty}} - we say "{{M|A}} has finite measure"
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====Finite measure====
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{{M|\mu}} is a finite measure if every set {{M|\in\mathcal{A} }} is finite.
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===Sigma-finite===
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A set {{M|A\in\mathcal{A} }} is {{sigma|finite}} if <math>\exists(A_n)_{n=1}^\infty:[A\subseteq\cup^\infty_{n=1}A_n\wedge(\forall A_n,\ \mu(A_n)<\infty)]</math>
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====Sigma-finite measure====
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{{M|\mu}} is {{sigma|finite}} if every set {{M|\in\mathcal{A} }} is {{sigma|finite}}
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===Total===
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If {{M|\mathcal{A} }} is a [[Sigma-algebra|{{sigma|algebra}}]] rather than a ring (that is {{M|X\in\mathcal{A} }} where {{M|X}} is the space) then we use
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====Totally finite measure====
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If {{M|X}} is finite
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====Totally sigma-finite measure====
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If {{M|X}} is {{Sigma|finite}}
  
 
==Examples==
 
==Examples==

Revision as of 14:09, 18 March 2015

[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]Not to be confused with Pre-measure


Definition

A [ilmath]\sigma[/ilmath]-ring [ilmath]\mathcal{A} [/ilmath] and a countably additive, extended real valued. non-negative set function [math]\mu:\mathcal{A}\rightarrow[0,\infty][/math] is a measure.

Contrast with pre-measure

Note: the family [math]A_n[/math] must be pairwise disjoint

Property Measure Pre-measure
[math]\mu:\mathcal{A}\rightarrow[0,\infty][/math] [math]\mu_0:R\rightarrow[0,\infty][/math]
[math]\mu(\emptyset)=0[/math] [math]\mu_0(\emptyset)=0[/math]
Finitely additive [math]\mu(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu(A_i)[/math] [math]\mu_0(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu_0(A_i)[/math]
Countably additive [math]\mu(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu(A_n)[/math] If [math]\bigudot^\infty_{n=1}A_n\in R[/math] then [math]\mu_0(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)[/math]

Terminology

Complete measure

A measure is complete if for [ilmath]A\in\mathcal{A} [/ilmath] we have [math][\mu(A)=0\wedge B\subset A]\implies B\in \mathcal{A}[/math]

Finite

A set [ilmath]A\in\mathcal{A} [/ilmath] is finite if [ilmath]\mu(A)<\infty[/ilmath] - we say "[ilmath]A[/ilmath] has finite measure"

Finite measure

[ilmath]\mu[/ilmath] is a finite measure if every set [ilmath]\in\mathcal{A} [/ilmath] is finite.

Sigma-finite

A set [ilmath]A\in\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-finite if [math]\exists(A_n)_{n=1}^\infty:[A\subseteq\cup^\infty_{n=1}A_n\wedge(\forall A_n,\ \mu(A_n)<\infty)][/math]

Sigma-finite measure

[ilmath]\mu[/ilmath] is [ilmath]\sigma[/ilmath]-finite if every set [ilmath]\in\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-finite

Total

If [ilmath]\mathcal{A} [/ilmath] is a [ilmath]\sigma[/ilmath]-algebra rather than a ring (that is [ilmath]X\in\mathcal{A} [/ilmath] where [ilmath]X[/ilmath] is the space) then we use

Totally finite measure

If [ilmath]X[/ilmath] is finite

Totally sigma-finite measure

If [ilmath]X[/ilmath] is [ilmath]\sigma[/ilmath]-finite

Examples

Trivial measures

Given the Measurable space [ilmath](X,\mathcal{A})[/ilmath] we can define:

[math]\mu:\mathcal{A}\rightarrow\{0,+\infty\}[/math] by [math]\mu(A)=\left\{\begin{array}{lr} 0 & \text{if }A=\emptyset \\ +\infty & \text{otherwise} \end{array}\right.[/math]

Another trivial measure is:

[math]v:\mathcal{A}\rightarrow\{0\}[/math] by [math]v(A)=0[/math] for all [math]A\in\mathcal{A}[/math]

See also