$\sigma$-algebra closed under countable intersection

$\sigma$-algebra closed under countable intersection

theorem

A 20231205121417-sigma_algebra is closed under countable intersection because of De Morgan’s law.

Proof

For a $\sigma$-algebra $\mathcal{S}$ and sets $A,B\in \mathcal{S}$

$$\begin{array}{rl}A\cap B& =((A\cap B{)}^C{)}^C\\ & =(A^C\cup B^C{)}^C\in \mathcal{S}\end{array}$$

The last line is because the $\sigma$-algebra is closed under union and complement.