Exercise 1.4.13 (Intersection of $\sigma$-algebras)

We copy-paste the proof of the Exercise 1.4.6. Verify the criteria from the Definition 1.4.12 ($\sigma$-algebra).

(i)

Empty set
The empty set is contained in ${\bigcap <!--\; FUNCTION\; APPLICATION\; -->}_{\alpha \in I}{B}_{\alpha }$ since for all $\alpha \in I$ we have $\varnothing \in B_{\alpha }$ by definition.

(ii)

Complement
Pick an arbitrary $E\in {\bigcap <!--\; FUNCTION\; APPLICATION\; -->}_{\alpha \in I}{B}_{\alpha }$; in other words, for all $\alpha \in I:E\in B_{\alpha }$. By definition of the $\sigma$-algebra, we also must have for all $\alpha \in I:E^c\in B_{\alpha }$. Thus, $E^c\in {\bigcap <!--\; FUNCTION\; APPLICATION\; -->}_{\alpha \in I}{B}_{\alpha }$.

(iii)

Countably finite unions
Pick an arbitrary countable sequence ${\left(E\right)}_{n\in \mathbb{N}}$ of measurable sets in ${\bigcap <!--\; FUNCTION\; APPLICATION\; -->}_{\alpha \in I}{B}_{\alpha }$. Again, since by definition of the $\sigma$-algebra for all $\alpha \in I$ and $\forall <!--\; FUNCTION\; APPLICATION\; -->n\in \mathbb{N}:E_n\in B_{\alpha }$, we also have for all $\alpha \in I:{\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{E}_n\in B_{\alpha }$.

Now let $C$ be another $\sigma$-algebra that is coarser than all $B_{\alpha }$, i.e., $\forall <!--\; FUNCTION\; APPLICATION\; -->\alpha \in I:C\subseteq B_{\alpha }$. Then $C$ is also coarser than their intersection $C\subseteq {\bigcap <!--\; FUNCTION\; APPLICATION\; -->}_{\alpha \in I}{B}_{\alpha }$ by the set-theoretic laws.