Consider, for simplicity, the Borel $\sigma$-algebra of the unit interval $[0,1]$.
Let $\{A_i\}$ be a family of Borel subsets which generate the sigma algebra (can be either countable or uncountable). I was wondering if there is some way of describing all subsets of the sigma-algebra only using $A_i$ and set operations.
Moreover, I am curious if we can linearly order all elements of the Borel sigma algebra in a way that any element in the ordering is either one of the generating elements or can be described as a countable union of elements that have appeared before it.
My motivation is, among other things, to prove following two statements in a straightforward way;
(1) The cardinality of the Borel $\sigma$-algebra equals the cardinality of $\mathbb{R}$.
(2) If a $\pi$ system $K$ is contained in a Dynkin system $D$, then the $\sigma$-algebra generated by $K$ is contained in $D$.
Yes. The usual construction is the following (from memory, someone please double check me):
Let $\mathcal{C}_0$ be some collection of sets, and for ordinals $\alpha > 0$, define $\mathcal{C}_\alpha$ recursively as the collection of all countable unions and complements of sets from $\bigcup_{\beta < \alpha} \mathcal{C}_\beta$. Then $\sigma(\mathcal{C}_0) = \mathcal{C}_{\omega_1}$, where $\omega_1$ is the first uncountable ordinal.
So in a sense, any Borel set can be produced by taking the sets $A_i$, taking countable unions and complements, and iterating uncountably many times.