How to think of a sub-sigma algebra as a collection of random variables? Where $\mathcal{H}$ is a $\sigma$-algebra on $\Omega$, section 9.1 here discusses thinking of a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{H}$ "as the collection of all numerical random variables that are $\mathcal{G}$-measurable".
This answer gives some intuition for $\sigma$-algebras generated by random variables ("restricting the world's probabilism"), which aligns with the lecture notes (the first link). Even so, I don't see where any functions (random variables) are in a set of sets ($\mathcal{G}$).
Can I have some pointers to further reading?
 A: $\newcommand{\classG}{\mathcal{G}}$
To facilitate discussion, let's first settle the probability space to be $(\Omega, \mathcal{H}, P)$, and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{H}$.  The complete sentence that the author wrote is:

Cinlar
advocates for thinking of the sub-$\sigma$-algebra $\mathcal{G}$ both as a collection of events (i.e., measurable subsets
of $\Omega$), and as the collection of all numerical random variables that are $\classG$-measurable.

The first part of the quote is obvious:  by definition $\classG$ (in fact any subclass $\mathcal{A}$ of $\mathcal{H}$) is a collection of events.
Mechanically speaking, the second part of the quote does not make sense, because $\classG$ is a class of sets, instead of random variables (i.e., functions defined on $\Omega$).  However, such "thinking" does make sense in view of the following isomorphism $\tau$ between $\classG$ and $\mathscr{I}$, which is defined as the class of indicator functions of events in $\classG$: for every $G \in \classG$, define $\tau(G) = I_G(\omega), \omega \in \Omega$.  Now $\mathscr{I} = \{I_G: G \in \classG\}$ is a collection of numerical random variables that are $\classG$-measurable -- intuitively, for each $\omega \in \Omega$, knowing the value of $I_G(\omega)$ is equivalent to knowing if the event $G$ occurs, that is, if $\omega \in G$.  In addition, $\tau$ is obviously a bijective between $\classG$ and $\mathscr{I}$. This justifies the second part of the quote.
For more enlightening discussion on sub-$\sigma$-algebra and its heuristic relation to partial information, I recommend you reading the Subfields subsection in Section 4 of Probability and Measure by Patrick Billingsley.
