The definition for a Borel $\sigma$-algebra is hard for me to grasp. I understand that it can be thought of as the "smallest" $\sigma$-algebra of some set from this post.
For concreteness, if we take the example of a Borel $\sigma$-algebra on $(0,1]$, what does it look like? More generally, what does $\mathbf{B}(\mathbb{R})$ look like?
A picture or image would be appreciated.
This is a very large set. In what follows, I'm basically going to quote a discussion that can be found in the Notes and References Section of Folland's real analysis book (Section 1.6)
Start with some subsets, $\mathcal{E}$ of $\mathbb{R}$. Then proceed inductively. Let $\mathcal{E}_1 = \mathcal{E} \cup \{E^c : E \in \mathcal{E}\}$. Keep going by taking compliements and "unioning" it with the original set. Do this "more than countably infinite times." The result is:
$\sigma(\mathcal{E}) = \cup_{\alpha \in \Omega}\mathcal{E}_{\alpha}$ where $\Omega$ is the set of countable ordinals
(Prop 1.23 in my edition of the source quoted above)
Just to add to my note, I think the reason we work with sigma algebra in the way you mention in your post is to avoid dealing with this "transfinite induction." The non-constructive definition is often easier (conceptually) to work with.
