View source for Types of set algebras

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==Measure theory perspective==
{{Extra Maths}}In this table the class of sets {{M|\mathcal{A} }} is a collection of subsets from another set {{M|\Omega}}
{{/Type table}}
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 {{M|\cup}}, downwards with {{M|\cap}} (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 ''{{Sigma|algebra}}''
{| class="wikitable" border="1"
| style="text-align:center;" | <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 sigma-algebra|Borel {{Sigma|algebra}}]] on a topology to get a {{Sigma|algebra}}, the below diagram is more complete, at the cost of ease to remember
:{| class="wikitable" border="1"
| style="text-align:center;" | <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==
{{/Measure theory summary table}}


==Theorems used==
<references group="Theorem"/>

==Notes==
<references group="Note"/>

==References==
<references/>

{{Definition|Measure Theory|Set Theory|Topology}}