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 $


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Basic linear algebra uncovers and clarifies very important geometry and algebra.

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