Difference between revisions of "Measure"

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{{Extra Maths}}Not to be confused with [[Pre-measure]]
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{{Refactor notice}}
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==Definition==
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{{Extra Maths}}A (positive)<ref group="Note">What else is there? ''Measures, Integrals and Martingales'' mentions this</ref> measure<ref name="MIAM">Measures, Integrals and Martingales - Rene L. Schilling</ref> on a [[Measurable space|measurable space]] {{M|(X,\mathcal{A})}} (where recall {{M|X}} is a set and {{M|\mathcal{A} }} is a [[Sigma-algebra|{{sigma|algebra}}]] on that set) is a mapping:
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* {{MM|1=\mu:\mathcal{A}\rightarrow[0,\infty]}}
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That satisfies:
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# {{M|1=\mu(\emptyset)=0}}
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# For any finite [[Sequence|sequence]] of [[Pairwise disjoint|pairwise disjoint]] sets {{M|1=(A_i)_{i=1}^n\subseteq\mathcal{A} }} we have {{M|1=\mu\left(\udot_{i=1}^nA_i\right)=\sum^n_{i=1}\mu(A_i)}}
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# For any [[Countably infinite|countably infinite]] sequence of pairwise disjoint sets {{M|1=(A_n)_{n=1}^\infty\subseteq\mathcal{A} }} we have {{M|1=\mu\left(\udot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mu(A_n)}}
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==Terminology==
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{{Todo|Find references}}
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===Of sets===
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{| class="wikitable" border="1"
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! Term
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! Definition
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! Comment
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|-
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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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|
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|-
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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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|
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|}
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===Of measures===
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{| class="wikitable" border="1"
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|-
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! Term
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! Definition
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! Comment
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|-
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! Complete measure
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| {{M|\forall A\in\mathcal{A} }} we have <math>[\mu(A)=0\wedge B\subset A]\implies B\in \mathcal{A}</math>
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|
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|-
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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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|
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|-
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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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|
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|}
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==Contrast with pre-measure==
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'''Note:''' the family <math>A_n</math> must be pairwise disjoint
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{| class="wikitable" border="1"
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|-
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! Property
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! Measure
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! Pre-measure
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|-
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|
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| <math>\mu:\mathcal{A}\rightarrow[0,\infty]</math>
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| <math>\mu_0:R\rightarrow[0,\infty]</math>
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|-
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|
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| <math>\mu(\emptyset)=0</math>
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| <math>\mu_0(\emptyset)=0</math>
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|-
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| Finitely additive
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| <math>\mu\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu(A_i)</math>
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| <math>\mu_0\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu_0(A_i)</math>
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|-
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| Countably additive
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| <math>\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)</math>
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| If <math>\bigudot^\infty_{n=1}A_n\in R</math> then <math>\mu_0\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)</math>
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|}
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==Properties==
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Here {{M|(X,\mathcal{A},\mu)}} is a [[Measure space|measure space]], and {{M|A,B\in\mathcal{A} }}
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{{:Pre-measure/Properties in common with measure}}
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==Related theorems==
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* [[A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (with equiv. conditions)]]
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==See also==
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* [[Pre-measure]]
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
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{{Definition|Measure Theory}}
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=Old page=
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Not to be confused with [[Pre-measure]]
  
  

Revision as of 01:55, 26 July 2015

This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.

Definition

[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]A (positive)[Note 1] measure[1] on a measurable space [ilmath](X,\mathcal{A})[/ilmath] (where recall [ilmath]X[/ilmath] is a set and [ilmath]\mathcal{A} [/ilmath] is a [ilmath]\sigma[/ilmath]-algebra on that set) is a mapping:

  • [math]\mu:\mathcal{A}\rightarrow[0,\infty][/math]

That satisfies:

  1. [ilmath]\mu(\emptyset)=0[/ilmath]
  2. For any finite sequence of pairwise disjoint sets [ilmath](A_i)_{i=1}^n\subseteq\mathcal{A}[/ilmath] we have [ilmath]\mu\left(\udot_{i=1}^nA_i\right)=\sum^n_{i=1}\mu(A_i)[/ilmath]
  3. For any countably infinite sequence of pairwise disjoint sets [ilmath](A_n)_{n=1}^\infty\subseteq\mathcal{A}[/ilmath] we have [ilmath]\mu\left(\udot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mu(A_n)[/ilmath]

Terminology

TODO: Find references

Of sets

Term Definition Comment
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"
[ilmath]\sigma[/ilmath]-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]

Of measures

Term Definition Comment
Complete measure [ilmath]\forall A\in\mathcal{A} [/ilmath] we have [math][\mu(A)=0\wedge B\subset A]\implies B\in \mathcal{A}[/math]
Finite measure [ilmath]\mu[/ilmath] is a finite measure if every set [ilmath]\in\mathcal{A} [/ilmath] is finite.
[ilmath]\sigma[/ilmath]-finite measure [ilmath]\mu[/ilmath] is [ilmath]\sigma[/ilmath]-finite if every set [ilmath]\in\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-finite

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\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu(A_i)[/math] [math]\mu_0\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu_0(A_i)[/math]
Countably additive [math]\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)[/math] If [math]\bigudot^\infty_{n=1}A_n\in R[/math] then [math]\mu_0\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)[/math]

Properties

Here [ilmath](X,\mathcal{A},\mu)[/ilmath] is a measure space, and [ilmath]A,B\in\mathcal{A} [/ilmath]

  • Finitely additive: if [ilmath]A\cap B=\emptyset[/ilmath] then [ilmath]\mu_0(A\udot B)=\mu_0(A)+\mu_0(B)[/ilmath]

Follows immediately from definition (property 2)

  • Monotonic: [Note 2] if [ilmath]A\subseteq B[/ilmath] then [ilmath]\mu_0(A)\le\mu_0(B)[/ilmath]


TODO: Be bothered to write out

  • If [ilmath]A\subseteq B[/ilmath] and [ilmath]\mu_0(A)<\infty[/ilmath] then [ilmath]\mu_0(B-A)=\mu_0(B)-\mu(A)[/ilmath]


TODO: Be bothered, note the significance of the finite-ness of [ilmath]A[/ilmath] - see Extended real value

  • Strongly additive: [ilmath]\mu_0(A\cup B)=\mu_0(A)+\mu_0(B)-\mu_0(A\cap B)[/ilmath]


TODO: Be bothered

  • Subadditive: [ilmath]\mu_0(A\cup B)\le\mu_0(A)+\mu_0(B)[/ilmath]


TODO: Again - be bothered

Related theorems

See also

Notes

  1. ↑ What else is there? Measures, Integrals and Martingales mentions this
  2. ↑ Sometimes stated as monotone (it is monotone in Measures, Integrals and Martingales in fact!)

References

  1. ↑ Measures, Integrals and Martingales - Rene L. Schilling

Old page

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. That is:

  • [ilmath]\mu(\emptyset)=0[/ilmath]
  • [math]\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)[/math]
  • [math]\mu(S)\ge 0\ \forall S\in\mathcal{A}[/math]

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\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu(A_i)[/math] [math]\mu_0\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu_0(A_i)[/math]
Countably additive [math]\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)[/math] If [math]\bigudot^\infty_{n=1}A_n\in R[/math] then [math]\mu_0\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)[/math]

Terminology

These terms apply to pre-measures to, rather [ilmath]\mathcal{A} [/ilmath] you would use the ring the pre-measure is defined on.

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

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