Exercise 2.2 (Intersection of $\sigma$-algebras is a $\sigma$-algebra)

Question

Let $(\mathcal{G}_\alpha)_{\alpha \in A}$ be an arbitrary family of $\sigma$-algebras defined on an abstract space $\Omega$. Show hat $\mathcal{H}=\bigcup_{\alpha\in A}\mathcal{G}_\alpha$ is also a $\sigma$-algebra.

Toğrul Topal · Accepted Answer

\begin{enumerate}
	\item 	extit{Empty set}
ewline
	The empty set is contained in $\mathcal{H}$ since for all $\alpha \in I$ we have $\emptyset \in \mathcal{G}_\alpha$ by definition.
	
	\item 	extit{Complement}
ewline
	Pick an arbitrary $E \in \bigcap_{\alpha \in I} \mathcal{G}_\alpha$; in other words, for all $\alpha \in I: E \in \mathcal{G}_\alpha$. By definition of the $\sigma$-algebra, we also must have for all $\alpha \in I: E^c \in \mathcal{G}_\alpha$. Thus, $E^c \in  \bigcap_{\alpha \in I} \mathcal{G}_\alpha$.
	
	\item 	extit{Countably finite unions}
ewline
	Pick an arbitrary countable sequence $\left(Eight)_{n\in\mathbb{N}}$ of measurable sets in $\bigcap_{\alpha \in I} \mathcal{G}_\alpha$. Again, since by definition of the $\sigma$-algebra for all $\alpha \in I$ and $\forall n \in \mathbb{N}: E_n \in \mathcal{G}_\alpha$, we also have for all $\alpha \in I: \bigcup_{n=1}^{\infty} E_n \in \mathcal{G}_\alpha$.
\end{enumerate}

Exercise 2.2 (Intersection of $\sigma$-algebras is a $\sigma$-algebra)

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Exercise 2.2 (Intersection of $\sigma$-algebras is a $\sigma$-algebra)

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