# Index of properties

## Index

Note:

• Things are indexed by the adjective in the property, for example: [ilmath]\sigma[/ilmath]-finite is under "finite".
• The specific case contains extra information, so [ilmath]\sigma[/ilmath]-finite is under finite, but specifically [ilmath]\sigma[/ilmath]-finite
• The word "under" is ignored in the index
Closed (general) CLOSED To say something is closed under means one cannot leave it through the stated property, eg "the integers are closed under addition
[ilmath]\backslash[/ilmath]-closed[1] CLOSED_backslash To say [ilmath]\mathcal{A} [/ilmath] is [ilmath]\backslash[/ilmath]-closed uses [ilmath]\backslash[/ilmath] to denote set subtraction[Note 1], this means $\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]$
[ilmath]\cap[/ilmath]-closed[1] CLOSED_cap If [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed then $\forall A,B\in\mathcal{A}[A\cap B\in\mathcal{A}]$ - [ilmath]\mathcal{A} [/ilmath] is closed under finite intersection
[ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed[1] CLOSED_cap_sigma closed under countably infinite intersection. $\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cap_{n=1}^\infty A_n\in\mathcal{A}]$
closed under complement[1] CLOSED_complement If [ilmath]\mathcal{A} [/ilmath] is closed under complement then $\forall A\in\mathcal{A}[A^c\in\mathcal{A}]$
[ilmath]\cup[/ilmath]-closed[1] CLOSED_cup If [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed then $\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]$ - [ilmath]\mathcal{A} [/ilmath] is closed under finite union
[ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[1] CLOSED_cup_sigma closed under countably infinite union. $\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cup_{n=1}^\infty A_n\in\mathcal{A}]$
[ilmath]\backslash[/ilmath]-closed CLOSED_division See CLOSED_backslash

## Notes

1. This is because [ilmath]-[/ilmath]-closed is not a good way to write this

## References

1. Probability Theory - A comprehensive course - Second Edition - Achim Klenke