Exercise 1.4.15 (Recursive description of a generated $\sigma$-algebra)

Let $X$ be a set, and let $\mathcal{ℱ}\subseteq \mathcal{𝒫}(X)$ be a collection of sets in $X$. Let ${\omega }_1$ be the first uncountable ordinal. For each countable ordinal $\alpha \in {\omega }_1$ define via transfinite induction

(i)

${\mathcal{ℱ}}_{\alpha }:=\mathcal{ℱ}$

(ii)

For each countable ordinal $\alpha =\beta +1$, we define ${\mathcal{ℱ}}_{\alpha }$ to be the collection of all sets that are either the union of an at most countable number of sets in ${\mathcal{ℱ}}_{\alpha }$ or the complement of such a union.

(iii)

For each countable limit ordinal $\alpha =\underset{\beta <\alpha }{\sup <!--\; FUNCTION\; APPLICATION\; -->}\beta$, we define ${\mathcal{ℱ}}_{\alpha }:={\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{\beta <\alpha }{\mathcal{ℱ}}_{\beta }$.

Show that $⟨\mathcal{ℱ}⟩={\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{\alpha \in {\omega }_1}{\mathcal{ℱ}}_{\alpha }$.