Difference between revisions of "Pre-measure/Properties in common with measure"
From Maths
m (Fixing typo) |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 5: | Line 5: | ||
{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
| − | * '''Monotonic: ' | + | * '''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)}} |
| + | {{Begin Inline Proof}} | ||
| + | {{Todo|Be bothered to write out}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | {{Begin Inline Theorem}} | ||
| + | * 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]]}} | ||
| Line 16: | Line 21: | ||
{{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)}} | ||
| − | {{Begin | + | {{Begin Inline Proof}} |
| + | {{Todo|Again - be bothered}} | ||
| + | {{End Proof}}{{End Theorem}}<noinclude> | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
| + | ==References== | ||
<references/> | <references/> | ||
{{Theorem Of|Measure Theory}} | {{Theorem Of|Measure Theory}} | ||
</noinclude> | </noinclude> | ||
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
