Difference between revisions of "Sigma-algebra"
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That is it is closed under [[Complement|complement]] and [[Countable|countable]] [[Union|union]] | That is it is closed under [[Complement|complement]] and [[Countable|countable]] [[Union|union]] | ||
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| + | ==First theorems== | ||
| + | {{Begin Theorem}} | ||
| + | The intersection of {{Sigma|algebras}} is a {{Sigma|algebra}} | ||
| + | {{Begin Proof}} | ||
| + | {{Todo|Proof - see PTACC page 5, also in Halmos AND in that other book}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
==See also== | ==See also== | ||
Revision as of 13:59, 16 June 2015
A Sigma-algebra of sets, or [ilmath]\sigma[/ilmath]-algebra is very similar to a [ilmath]\sigma[/ilmath]-ring of sets.
Like how ring of sets and algebra of sets differ, the same applies to [ilmath]\sigma[/ilmath]-ring compared to [ilmath]\sigma[/ilmath]-algebra
Definition
A non empty class of sets [ilmath]S[/ilmath] is a [ilmath]\sigma[/ilmath]-algebra if[1]
- if [math]A\in S[/math] then [math]A^c\in S[/math]
- if [math]\{A_n\}_{n=1}^\infty\subset S[/math] then [math]\cup^\infty_{n=1}A_n\in S[/math]
That is it is closed under complement and countable union
First theorems
The intersection of [ilmath]\sigma[/ilmath]-algebras is a [ilmath]\sigma[/ilmath]-algebra
TODO: Proof - see PTACC page 5, also in Halmos AND in that other book
See also
References
- ↑ Halmos - Measure Theory - page 28 - Springer - Graduate Texts in Mathematics - 18
