Difference between revisions of "Pre-measure/Properties in common with measure"
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{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
| − | * If {{M|A\subseteq B}} and {{M|\mu_0(A)<\infty}} then {{M|\mu_0(B-A)=\mu_0(B)-\mu(A)}} | + | * If {{M|A\subseteq B}} and {{M|\mu_0(A)<\infty}} then {{M|1=\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]]}} | ||
Latest revision as of 22:30, 30 March 2016
[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]
- 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 1] 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
Notes
- ↑ Sometimes stated as monotone (it is monotone in Measures, Integrals and Martingales in fact!)
References
