Theorem

Let F be a $ \sigma $-field, then

$ A_i\in\mathcal F\;\forall i=1,2,...\;\Rightarrow\;\bigcap_{i=1}^{\infty}A_i \in\mathcal F $


Proof

By definition of $ \sigma $-fields,

$ \begin{align} A_i\in\mathcal F\;\forall i\;&\Rightarrow\;A_i^C\in\mathcal F\;\forall i\\ &\Rightarrow\;\bigcup_{i=1}^{\infty}A_i^C\in\mathcal F \\ &\Rightarrow\;(\bigcup_{i=1}^{\infty}A_i^C)^C\in\mathcal F \\ &\Rightarrow\;\bigcap_{i=1}^{\infty}A_i\in\mathcal F \end{align} $


For the last implication, I am using an application of De Morgan's Law on countable Unions.
$ \blacksquare $


Back to Probability Spaces

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Property_of_sigma_fields_2&oldid=59168"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

Meet a recent graduate heading to Sweden for a Postdoctorate.

Christine Berkesch