Difference between revisions of "Pre-measure/Properties in common with measure"

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m (Reverted edits by JessicaBelinda133 (talk) to last revision by Alec)
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{{End Proof}}{{End Theorem}}
 
{{End Proof}}{{End Theorem}}
 
{{Begin Inline Theorem}}
 
{{Begin Inline Theorem}}
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* '''Monotonic: 'esand {{M|\mu_0(A)<\infty}} then {{M|\mu_0(B-A)=\mu_0(B)-\mu(A)}}
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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)}}
 
{{Begin Inline Proof}}
 
{{Begin Inline Proof}}
 
{{Todo|Be bothered, note the significance of the finite-ness of {{M|A}} - see [[Extended real value]]}}
 
{{Todo|Be bothered, note the significance of the finite-ness of {{M|A}} - see [[Extended real value]]}}
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{{Begin Inline Theorem}}
 
{{Begin Inline Theorem}}
 
* '''Subadditive:''' {{M|\mu_0(A\cup B)\le\mu_0(A)+\mu_0(B)}}
 
* '''Subadditive:''' {{M|\mu_0(A\cup B)\le\mu_0(A)+\mu_0(B)}}
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{{Begin Inliyes
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{{Begin Inline Proof}}
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{{Todo|Again - be bothered}}
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{{End Proof}}{{End Theorem}}<noinclude>
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==Notes==
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<references group="Note"/>
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==References==
 
<references/>
 
<references/>
 
{{Theorem Of|Measure Theory}}
 
{{Theorem Of|Measure Theory}}
 
</noinclude>
 
</noinclude>

Revision as of 16:30, 23 August 2015

\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}} }\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }
[Expand]

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

[Expand]

  • Monotonic: [Note 1] if A\subseteq B then \mu_0(A)\le\mu_0(B)

[Expand]

  • If A\subseteq B and \mu_0(A)<\infty then

[Expand]

  • Strongly additive: \mu_0(A\cup B)=\mu_0(A)+\mu_0(B)-\mu_0(A\cap B)

[Expand]

  • Subadditive: \mu_0(A\cup B)\le\mu_0(A)+\mu_0(B)

Notes

  1. Jump up ↑ Sometimes stated as monotone (it is monotone in Measures, Integrals and Martingales in fact!)

References




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