Difference between revisions of "Pre-measure"

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{{Extra Maths}}
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==Definiton==
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==Definition==
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{{Extra Maths}} A ''pre-measure''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Measure|measure]] on an [[Algebra of sets|algebra]] rather than a [[Sigma-algebra|{{Sigma|algebra}}]], the properties are as follows:
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A pre-measure (often denoted {{M|\mu_0}}) is the precursor to a [[Measure|measure]] which are often denoted {{M|\mu}}
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* Here {{M|\mathcal{A} }} is an algebra of sets (a system of subsets of {{M|X}}) and {{M|\mu_0:\mathcal{A}\rightarrow[0,+\infty]}} such that:
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*# {{MM|1=\mu_0(\emptyset)=0}} - the measure of the empty set is {{M|0}}
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*# {{MM|1=\mu_0\left(\bigudot_{i=1}^nA_i\right)=\sum_{i=1}^n\mu_0(A_i)}}
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*#* Where {{M|1=(A_i)_{i=1}^n\subseteq\mathcal{A} }} is a [[Sequence|finite sequence]]
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*# {{MM|1=\mu_0\left(\bigudot_{n=1}^\infty A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)}} whenever {{MM|1=\bigudot_{n=1}^\infty A_n\in\mathcal{A} }}
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*#* Where {{M|1=(A_n)_{n=1}^\infty\subseteq\mathcal{A} }} is a sequence
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Notice that property 2 implies property 3 (as we can make a finite sequence infinite just by defining the terms after the finite ones as the emptyset) so often just the following is given<ref group="Note">Infact not stating 2 explicitly is the most common case. I only state it because it is intuitive to define it as a property we want. We are assured of closed-under-finite-union already, so we can measure over that. We then extend this to countably infinite.</ref>:
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* {{MM|1=\mu_0(\emptyset)=0}} - the measure of the empty set is {{M|0}} and
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* {{MM|1=\mu_0\left(\bigudot_{n=1}^\infty A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)}} whenever {{MM|1=\bigudot_{n=1}^\infty A_n\in\mathcal{A} }}
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'''Note: It could be worth reading the [[Pre-measure#Discussion|discussion]] section of this page now if the motivation for this definition isn't clear'''
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==Properties==
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Here {{M|(X,\mathcal{A},\mu_0)}} is a [[Pre-measure space|pre-measure space]], and {{M|A,B\in\mathcal{A} }}
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{{Begin Inline Theorem}}
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* '''Finitely additive:''' if {{M|1=A\cap B=\emptyset}} then {{M|1=\mu_0(A\udot B)=\mu_0(A)+\mu_0(B)}}
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{{Begin Inline Proof}}
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: Follows immediately from definition (property 2)
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{{End Proof}}{{End Theorem}}
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{{Begin Inline Theorem}}
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* '''Monotonic: '''<ref group="Note">Sometimes stated as ''monotone'' (it is ''monotone'' in ''Measures, Integrals and Martingales'' in fact!)</ref> if {{M|A\subseteq B}} then {{M|\mu_0(A)\le\mu_0(B)}}
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{{Begin Inline Proof}}
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{{Todo|Be bothered to write out}}
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{{End Proof}}{{End Theorem}}
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{{Begin Inline Theorem}}
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* If {{M|A\subseteq B}} and {{M|\mu_0(A)<\infty}} then {{M|\mu_0(B-A)=\mu_0(B)-\mu(A)}}
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{{Begin Inline Proof}}
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{{Todo|Be bothered, note the significance of the finite-ness of {{M|A}} - see [[Extended real value]]}}
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{{End Proof}}{{End Theorem}}
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{{Begin Inline Theorem}}
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* '''Strongly additive: ''' {{M|1=\mu_0(A\cup B)=\mu_0(A)+\mu_0(B)-\mu_0(A\cap B)}}
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{{Begin Inline Proof}}
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{{Todo|Be bothered}}
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{{End Proof}}{{End Theorem}}
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{{Begin Inline Theorem}}
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* '''Subadditive:''' {{M|\mu_0(A\cup B)\le\mu_0(A)+\mu_0(B)}}
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{{Begin Inline Proof}}
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{{Todo|Again - be bothered}}
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{{End Proof}}{{End Theorem}}
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==Discussion==
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An algebra of sets is closed under finite union anyway, so the {{M|2^\text{nd} }} part of the definition is fine. However there are ''infinite'' unions that are not closed on an ''algebra''. There are ''some'' infinite unions which are however in the union.
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===Example of infinite sequence not being in the algebra===
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Let {{M|\mathcal{I} }} be the set of all half-open-half-closed intervals of {{M|\mathbb{R} }}
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* That is to say that {{M|1=\mathcal{I}:=\{[a,b)\vert\ a,b\in\mathbb{R}\wedge b\ge a\} }} - with the convention that {{M|1=[a,a)=\emptyset}}
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And let {{M|1=\mathcal{A}:=\mathcal{A}(\mathcal{I})}} be the algebra generated by this set<ref group="Note">Not standard notation - made up for this example</ref>.
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* That is to say (informally, as this is just an example) {{M|\mathcal{A} }} contains everything which is the union of finitely many half-open-half-closed intervals.
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** Notice that:
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*** {{M|1=[a,b)^c=(-\infty,a)\cup[b,+\infty)}}
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*** {{M|1=[a,d)-[b,c)}} where {{M|1=a< b< c< d}} is {{M|1=[a,b)\cup[c,d)}}
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*** It's closed under union and so forth.
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** (It should be easy to convince yourself that this is an algebra)
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Then it is easy to see that all ''finite'' unions are in this system. However consider:
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* {{MM|1=\bigcup_{n=1}^\infty[0,1-\tfrac{1}{n})=[0,1]}} -which is not open, nor in {{M|\mathcal{A} }}!
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** But if this were finite: {{MM|1=\bigcup_{n=1}^m[0,1-\tfrac{1}{n})=[0,1-\tfrac{1}{m})\in\mathcal{A} }}
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* {{MM|1=\bigcap_{n=1}^\infty[-\tfrac{1}{n},\tfrac{1}{n}]=[0]}} which is again, not in {{M|\mathcal{A} }}
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** But of course if this were finite, then {{MM|1=\bigcap_{n=1}^m[-\tfrac{1}{n},\tfrac{1}{n}]=[-\tfrac{1}{m},\tfrac{1}{m})\in\mathcal{A} }}
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===Example of infinite sequence being in the algebra===
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This example constructs an infinite sequence from a finite one, a bit of a cheat but an example none the less!
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* Given a finite sequence: {{M|1=(B_i)_{i=1}^n\subseteq\mathcal{A} }} we define:
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** {{M|1=(A_m)_{m=1}^\infty}} with {{M|1=A_m=\left\{\begin{array}{lr}B_m & \text{if } m\le n\\ \emptyset & \text{otherwise}\end{array}\right.}}
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* Now {{M|1=\cup_{m=1}^\infty A_m=(\cup_{m=1}^n A_m)\bigcup(\cup_{m=n+1}^\infty A_m)=(\cup_{m=1}^n B_m)\bigcup(\cup_{m=n+1}^\infty\emptyset)=\cup_{m=1}^n B_m\in\mathcal{A} }} by definition (Algebras are closed under finite union)
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** Thus {{M|1=\cup_{m=1}^\infty A_m\in\mathcal{A} }} - despite being infinite!
  
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A pre-measure is a [[Ring of sets|ring]] {{M|R}} together with an [[Extended real value|extended real valued]], non-negative [[Set function|set]] [[Function|function]] <math>\mu_0:R\rightarrow[0,\infty]</math> that is countably [[Additive function|additive]] with <math>\mu_0(\emptyset)=0</math>
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So it makes sense that when given an infinite sequence whose union happens to be in {{M|\mathcal{A} }} - which is the definition of the domain of our pre-measure, we use our pre-measure to measure it!
  
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To sum up:
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==See also==
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* <math>\mu_0:R\rightarrow [0,\infty]</math>
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* [[Measure]]
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* <math>\mu_0(\emptyset)=0</math>
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* [[Sigma-algebra|{{Sigma|algebra}}]]
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* <math>\mu_0(\bigudot_{n\in\mathbb{N}}A_n)=\sum_{n\in\mathbb{N}}\mu_0(A_n)</math> where the <math>A_n</math> are pairwise disjoint (as implied by the <math>\bigudot</math>)
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Notice that given a finite family of pairwise disjoint sets <math>\{F_i\}_{i=1}^n</math> we can define the countably infinite family of sets <math>\{A_n\}_{n\in\mathbb{N}}</math> by: <math>A_i=\left\{\begin{array}{lr}F_i&\text{if }i\le n\\ \emptyset & \text{otherwise} \end{array}\right.</math> as clearly <math>\bigudot_{i\in\mathbb{N}}A_i=\bigudot_{i=1}^nF_n</math>
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==Notes==
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<references group="Note"/>
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Immediately one should think "but <math>\bigudot^\infty_{n=1}A_n</math> is not always in {{M|R}}" this is true - but it is sometimes. When it is <math>\in R</math> that is when it is defined.
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==References==
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<references/>
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Note that for any finite sequence we can make it infinite by just bolting <math>\emptyset</math> on indefinitely.
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{{Definition|Measure Theory}}
 
{{Definition|Measure Theory}}
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==References==
 

Revision as of 15:43, 21 July 2015

Definiton

[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 pre-measure[1] is a measure on an algebra rather than a [ilmath]\sigma[/ilmath]-algebra, the properties are as follows:

  • Here [ilmath]\mathcal{A} [/ilmath] is an algebra of sets (a system of subsets of [ilmath]X[/ilmath]) and [ilmath]\mu_0:\mathcal{A}\rightarrow[0,+\infty][/ilmath] such that:
    1. [math]\mu_0(\emptyset)=0[/math] - the measure of the empty set is [ilmath]0[/ilmath]
    2. [math]\mu_0\left(\bigudot_{i=1}^nA_i\right)=\sum_{i=1}^n\mu_0(A_i)[/math]
      • Where [ilmath](A_i)_{i=1}^n\subseteq\mathcal{A}[/ilmath] is a finite sequence
    3. [math]\mu_0\left(\bigudot_{n=1}^\infty A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)[/math] whenever [math]\bigudot_{n=1}^\infty A_n\in\mathcal{A}[/math]
      • Where [ilmath](A_n)_{n=1}^\infty\subseteq\mathcal{A}[/ilmath] is a sequence

Notice that property 2 implies property 3 (as we can make a finite sequence infinite just by defining the terms after the finite ones as the emptyset) so often just the following is given[Note 1]:

  • [math]\mu_0(\emptyset)=0[/math] - the measure of the empty set is [ilmath]0[/ilmath] and
  • [math]\mu_0\left(\bigudot_{n=1}^\infty A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)[/math] whenever [math]\bigudot_{n=1}^\infty A_n\in\mathcal{A}[/math]

Note: It could be worth reading the discussion section of this page now if the motivation for this definition isn't clear

Properties

Here [ilmath](X,\mathcal{A},\mu_0)[/ilmath] is a pre-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


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

Discussion

An algebra of sets is closed under finite union anyway, so the [ilmath]2^\text{nd} [/ilmath] part of the definition is fine. However there are infinite unions that are not closed on an algebra. There are some infinite unions which are however in the union.

Example of infinite sequence not being in the algebra

Let [ilmath]\mathcal{I} [/ilmath] be the set of all half-open-half-closed intervals of [ilmath]\mathbb{R} [/ilmath]

  • That is to say that [ilmath]\mathcal{I}:=\{[a,b)\vert\ a,b\in\mathbb{R}\wedge b\ge a\}[/ilmath] - with the convention that [ilmath][a,a)=\emptyset[/ilmath]

And let [ilmath]\mathcal{A}:=\mathcal{A}(\mathcal{I})[/ilmath] be the algebra generated by this set[Note 3].

  • That is to say (informally, as this is just an example) [ilmath]\mathcal{A} [/ilmath] contains everything which is the union of finitely many half-open-half-closed intervals.
    • Notice that:
      • [ilmath][a,b)^c=(-\infty,a)\cup[b,+\infty)[/ilmath]
      • [ilmath][a,d)-[b,c)[/ilmath] where [ilmath]a< b< c< d[/ilmath] is [ilmath][a,b)\cup[c,d)[/ilmath]
      • It's closed under union and so forth.
    • (It should be easy to convince yourself that this is an algebra)

Then it is easy to see that all finite unions are in this system. However consider:

  • [math]\bigcup_{n=1}^\infty[0,1-\tfrac{1}{n})=[0,1][/math] -which is not open, nor in [ilmath]\mathcal{A} [/ilmath]!
    • But if this were finite: [math]\bigcup_{n=1}^m[0,1-\tfrac{1}{n})=[0,1-\tfrac{1}{m})\in\mathcal{A}[/math]
  • [math]\bigcap_{n=1}^\infty[-\tfrac{1}{n},\tfrac{1}{n}]=[0][/math] which is again, not in [ilmath]\mathcal{A} [/ilmath]
    • But of course if this were finite, then [math]\bigcap_{n=1}^m[-\tfrac{1}{n},\tfrac{1}{n}]=[-\tfrac{1}{m},\tfrac{1}{m})\in\mathcal{A}[/math]

Example of infinite sequence being in the algebra

This example constructs an infinite sequence from a finite one, a bit of a cheat but an example none the less!

  • Given a finite sequence: [ilmath](B_i)_{i=1}^n\subseteq\mathcal{A}[/ilmath] we define:
    • [ilmath](A_m)_{m=1}^\infty[/ilmath] with [ilmath]A_m=\left\{\begin{array}{lr}B_m & \text{if } m\le n\\ \emptyset & \text{otherwise}\end{array}\right.[/ilmath]
  • Now [ilmath]\cup_{m=1}^\infty A_m=(\cup_{m=1}^n A_m)\bigcup(\cup_{m=n+1}^\infty A_m)=(\cup_{m=1}^n B_m)\bigcup(\cup_{m=n+1}^\infty\emptyset)=\cup_{m=1}^n B_m\in\mathcal{A}[/ilmath] by definition (Algebras are closed under finite union)
    • Thus [ilmath]\cup_{m=1}^\infty A_m\in\mathcal{A}[/ilmath] - despite being infinite!

So it makes sense that when given an infinite sequence whose union happens to be in [ilmath]\mathcal{A} [/ilmath] - which is the definition of the domain of our pre-measure, we use our pre-measure to measure it!

See also

Notes

  1. ↑ Infact not stating 2 explicitly is the most common case. I only state it because it is intuitive to define it as a property we want. We are assured of closed-under-finite-union already, so we can measure over that. We then extend this to countably infinite.
  2. ↑ Sometimes stated as monotone (it is monotone in Measures, Integrals and Martingales in fact!)
  3. ↑ Not standard notation - made up for this example

References

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