Difference between revisions of "Types of set algebras"

Revision as of 21:55, 14 June 2015

This diagram shows the relations between types of system:

Diagram showing the relationships
[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]^{\sigma\text{-}\cup} \ar[r]^{\Omega\in\mathcal{A}} & \text{algebra} \ar[ur]_{\sigma\text{-}\cup} & & \text{topology}\ar@{.>}[ul]_(.25){\text{Borel }\sigma\text{-algebra}} \\ \text{semiring} \ar[u]_{\cup\text{-closed}} & & & }\end{xy}[/math]

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	#

@@ Line 3: / Line 3: @@
 :{| class="wikitable" border="1"
 | style="text-align:center;" | <math>\begin{xy}\xymatrix{
-  & \text{Dynkin system} \ar[dr]^(0.6){\cap\text{-closed}} & \\
+  & & \text{Dynkin system} \ar[d]^{\cap\text{-closed}} & \\
-  & {\sigma\text{-ring}} \ar[r]^{\Omega\in\mathcal{A}} & {\sigma\text{-algebra}} \\
+  & {\sigma\text{-ring}} \ar[r]^{\Omega\in\mathcal{A}} & {\sigma\text{-algebra}} & \\
-\text{ring} \ar[ur]^{\sigma\text{-}\cup} \ar[r]^{\Omega\in\mathcal{A}} &  \text{algebra} \ar[ur]_{\sigma\text{-}\cup} & \\
+\text{ring} \ar[ur]^{\sigma\text{-}\cup} \ar[r]^{\Omega\in\mathcal{A}} &  \text{algebra} \ar[ur]_{\sigma\text{-}\cup} & & \text{topology}\ar@{.>}[ul]_(.25){\text{Borel }\sigma\text{-algebra}} \\
-\text{semiring} \ar[u]_{\cup\text{-closed}} & &
+\text{semiring} \ar[u]_{\cup\text{-closed}} & & &
 }\end{xy}</math>