During my studies, I came across several different Stone spaces, e.g.:

(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;

(ii) The absolute $EX$ of $X$ (i.e. the Gleason cover of $X$ since $X$ is compact), which is the Stone space of the regular closed algebra of $X$;

(iii) The minimal basically-disconnected cover $\Lambda X$ of $X$, which is the Stone space of the $\sigma$-completion of the clopen algebra of $X$, etc…

One Stone space which still eludes me, however, is the Stone space $Y$ of the free $\sigma$-algebra on $\omega_1$ free generators. As a Stone space, it is certainly totally-disconnected compact Hausdorff, and, in fact, it can be shown that it is even basically-disconnected (such spaces are sometimes called quasi-Stonean).

(1) Is there a more precise characterisation of $Y$?

Also, it is known that the free $\sigma$-algebra on $\omega_1$ free generators is isomorphic to the Baire $\sigma$-algebra of $X$. But, as previously noted, $X$ is the Stone of the *free Boolean algebra* on $\omega_1$ free generators, not the Stone space $Y$ of the *free $\sigma$-algebra* on $\omega_1$ generators.

(2) What are the pros and cons of working with $X$ instead $Y$?

(3) How workable is $Y$, especially from a measure theory and integration point of view?