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

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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)}}
 
* '''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}}
 
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{{Begin Inline Proof}}
 
{{Todo|Again - be bothered}}
 
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==Notes==
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<references group="Note"/>
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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

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  • Finitely additive: if A∩B=∅ then μ0(A∪⋅B)=μ0(A)+μ0(B)

[Expand]

  • Monotonic: [Note 1] if A⊆B then μ0(A)≤μ0(B)

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  • If A⊆B and μ0(A)<∞ then

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  • Strongly additive: μ0(A∪B)=μ0(A)+μ0(B)−μ0(A∩B)

[Expand]

  • Subadditive: μ0(A∪B)≤μ0(A)+μ0(B)

Notes

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

References




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