Types of set algebras
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Contents
Measure theory perspective
[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]In this table the class of sets [ilmath]\mathcal{A} [/ilmath] is a collection of subsets from another set [ilmath]\Omega[/ilmath]
| System Type | Definition | Deductions |
|---|---|---|
| Ring[1][2] |
| |
| [ilmath]\sigma[/ilmath]-ring[1][2] |
|
|
| Algebra[1][2] |
|
|
| [ilmath]\sigma[/ilmath]-algebra[1][2] |
|
|
| Semiring[1] |
TODO: Page 3 in[1] | |
| Dynkin system[1][3] |
|
|
These types are all related and I have a nice diagram to remember this which uses arrow directions to 'encode' the difference. In my diagram upwards arrows indicate something to do with union, with [ilmath]\cup[/ilmath], downwards with [ilmath]\cap[/ilmath] (think "make bigger"=up, which is union and "going down" being cap. A rightward slant means "sigma-whatever-the-vertical-direction-is" which means closed under countable whatever. Lastly, a horizontal arrow indicates membership, right means "contains entire set" and that's all that is used. Lastly:
- All paths lead to [ilmath]\sigma[/ilmath]-algebra
| [math]\begin{xy}\xymatrix{ & & \text{Dynkin system} \ar[d]^{\cap\text{-closed}} \\ & {\sigma\text{-ring}} \ar[r]^{\Omega\in\mathcal{A}} & {\sigma\text{-algebra}} \\ \text{ring} \ar[ur]^(.4){\sigma\text{-}\cup} \ar[r]^{\Omega\in\mathcal{A}} & \text{algebra} \ar[ur]^(.4){\sigma\text{-}\cup} & \\ \text{semiring} \ar[u]_{\cup\text{-closed}} & & }\end{xy}[/math] |
| Alec's 'super' diagram |
|---|
Notice in addition the nice symmetry of the diagram (the line of symmetry would be from top left to bottom right), it doesn't preserve arrow directions, and obviously not names, but shape.
Overall this is a very easy diagram to remember. I remember ring easily (it's what you'd need to "do probability" on, unions and set-subtractions, and the empty set (required to have subtractions anyway)). This lets me build the rest. The only not obvious ones are Dynkin-systems and semirings
Relationship between all types
This of course isn't the entire picture. In addition we can use the Borel [ilmath]\sigma[/ilmath]-algebra on a topology to get a [ilmath]\sigma[/ilmath]-algebra, the below diagram is more complete, at the cost of ease to remember
[math]\begin{xy}\xymatrix{ & & \text{Dynkin system} \ar[d]^{\cap\text{-closed}} & \\ & {\sigma\text{-ring}} \ar[r]^{\Omega\in\mathcal{A}} & {\sigma\text{-algebra}} & \\ \text{ring} \ar[ur]^(.4){\sigma\text{-}\cup} \ar[r]^{\Omega\in\mathcal{A}} & \text{algebra} \ar[ur]^(.4){\sigma\text{-}\cup} & & \text{topology}\ar@{.>}[ul]_(.25){\text{Borel }\sigma\text{-algebra}} \\ \text{semiring} \ar[u]_{\cup\text{-closed}} & & & }\end{xy}[/math] Diagram showing ALL the relationships
Other Notes
| Closed under | |||||||
|---|---|---|---|---|---|---|---|
| Type | [ilmath]\sigma\in\mathcal{A} [/ilmath] | [ilmath]\bigcap[/ilmath] | [ilmath]\sigma[/ilmath]-[ilmath]\bigcap[/ilmath] | [ilmath]\bigcup[/ilmath] | [ilmath]\sigma[/ilmath]-[ilmath]\bigcup[/ilmath] | [ilmath]-[/ilmath] | [ilmath]C[/ilmath] |
| Semi-Ring | |||||||
| Ring | |||||||
| [ilmath]\sigma[/ilmath]-Ring | |||||||
| Algebra | |||||||
| Dynkin system | |||||||
| [ilmath]\sigma[/ilmath]-Algebra | # | # | X | X | # | ||
Theorems used
- ↑ 1.0 1.1 Using Class of sets closed under set-subtraction properties we know that if [ilmath]\mathcal{A} [/ilmath] is closed under Set subtraction then:
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed
- [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[ilmath]\implies[/ilmath][ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed
- ↑ Using Class of sets closed under complements properties we see that if [ilmath]\mathcal{A} [/ilmath] is closed under complements then:
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed
Notes
- ↑ Closed under finite Set subtraction
- ↑ Closed under finite Union
- ↑ As given [ilmath]A\in\mathcal{A} [/ilmath] we must have [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath]
- ↑ closed under finite or countably infinite union
- ↑ Note that [ilmath]A-B=A\cap B^c=(A^c\cup B)^c[/ilmath] - or that [ilmath]A-B=(A^c\cup B)^c[/ilmath] - so we see that being closed under union and complement means we have closure under set subtraction.
- ↑ As we are closed under set subtraction we see [ilmath]A-A=\emptyset[/ilmath] so [ilmath]\emptyset\in\mathcal{A} [/ilmath]
- ↑ As we are closed under set subtraction we see that [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath], so [ilmath]\emptyset\in\mathcal{A} [/ilmath] - but we are also closed under complements, so [ilmath]\emptyset^c\in\mathcal{A} [/ilmath] and [ilmath]\emptyset^c=\Omega\in\mathcal{A}[/ilmath]
- ↑ Trivial - satisfies the definitions
- ↑ As [ilmath]\Omega^c=\emptyset[/ilmath] by being closed of complements, [ilmath]\emptyset\in\mathcal{A} [/ilmath]
