What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
Recall the definition: Let $S = \mathbb{N^{
I feel comfortable with the fundamental properties of the $\mathcal{A}$-operation (and their proofs). To list a few of the basic facts I think I understand:
- it subsumes countable unions and intersections;
- idempotence: if $\mathcal{E} \subseteq P(X)$ is any class of subsets then $\mathcal{A}(\mathcal{E}) = \mathcal{A}(\mathcal{A}(\mathcal{E}))$;
if $\emptyset, X \in \mathcal{A}(\mathcal{E})$ and $X \setminus E \in \mathcal{A}(\mathcal{E})$ for all $E \in \mathcal{E}$ then $\sigma(\mathcal{E}) \subseteq \mathcal{A}(\mathcal{E})$.
In particular if $\mathcal{E} \subset P(\mathbb{R})$ is the family of closed intervals with rational endpoints then $\mathcal{A}(\mathcal{E})$ contains the $\sigma$-algebra Borel sets (and in fact the containment is strict).
- if $(X,\Sigma,\mu)$ is a measure space obtained from Carathéodory's construction on some outer measure on $X$ then $\Sigma$ is closed under the $\mathcal{A}$-operation: $\mathcal{A}\Sigma = \Sigma$.
- the kernel of a Souslin scheme can be interpreted as the image $R[\mathbb{N^N}]$ of a relation $R \subseteq \mathbb{N}^\mathbb{N} \times X$, in particular if $X$ is Polish then the $\mathcal{A}$-operation on closed sets gives us the analytic sets.
- If $\langle E_s : s \in \mathbb{N}^{\lt \mathbb{N}}\rangle$ is a regular Souslin scheme of closed sets with vanishing diameter then its associated relation $R \subset \mathbb{N}^\mathbb{N} \times X$ is the graph of a continuous function $f\colon D \to X$ defined on some closed subset $D$ of $\mathbb{N^N}$.
- etc.
The point of this list is just to mention that I think that I've done my share of the manipulations with trees and $\mathbb{N}^\mathbb{N}$ that come along with $\mathcal{A}$, but I still have the feeling that something fundamental escapes me.
After looking at the two 1917 Comptes Rendus papers Sur une définition des ensembles mesurables $B$ sans nombres transfinis by Souslin and Sur la classification de M. Baire by Lusin, I also think I understand that part of the inspiration was the continued fraction representation of real numbers.
Given the importance of the $\mathcal{A}$-operation (entire books were written on its uses, e.g. C.A. Rogers et al., Analytic Sets where there is a wealth of applications) it would be nice to have some good intuitions that allow me to have a firmer grasp of what is going on.
Somehow it seems that the $\mathcal{A}$-operation is mostly presented as a technical device having an enormous range of applications, but this doesn't seem to do justice to the concept.