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

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{{Begin Inline Theorem}}
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<noinclude>{{Extra Maths}}</noinclude>{{Begin Inline Theorem}}
 
* '''Finitely additive:''' if {{M|1=A\cap B=\emptyset}} then {{M|1=\mu_0(A\udot B)=\mu_0(A)+\mu_0(B)}}
 
* '''Finitely additive:''' if {{M|1=A\cap B=\emptyset}} then {{M|1=\mu_0(A\udot B)=\mu_0(A)+\mu_0(B)}}
 
{{Begin Inline Proof}}
 
{{Begin Inline Proof}}
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{{Begin Inline Proof}}
 
{{Begin Inline Proof}}
 
{{Todo|Again - be bothered}}
 
{{Todo|Again - be bothered}}
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{{End Proof}}{{End Theorem}}
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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==
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<references/>
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{{Theorem Of|Measure Theory}}
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</noinclude>

Revision as of 15:07, 29 July 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} }
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  • Finitely additive: if A\cap B=\emptyset then \mu_0(A\udot B)=\mu_0(A)+\mu_0(B)

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

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

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

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  • 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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