[Math] What are some good intuitions for understanding Souslin’s operation $\mathcal{A}$

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What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of non-empty finite sequences in $\mathbb{N}$ and let $\mathcal{E} \subseteq P(X)$ be a family of subsets of a given set $X$. A Souslin scheme is an assignment $E \colon S \to \mathcal{E}, s \mapsto E_s$ and one defines its kernel to be
$$
\mathcal{A}E = \mathcal{A}_s E_s = \bigcup_{\sigma \in \mathbb{N}^{\mathbb N}}\bigcap_{n=1}^\infty E_{\sigma{\upharpoonright}n} \subseteq X
$$
where $\sigma{\upharpoonright}n = \langle \sigma(0),\dots,\sigma(n-1)\rangle \in \mathbb{N}^n$. The collection of all subsets of $X$ obtained from $\mathcal{E}$ in this fashion is denoted by $\mathcal{A}(\mathcal E)$.

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.

Best Answer

This is taken from T. Jech's "Set Theory" book Ch. Lemma 11.6 provides equivalent definitions of analytic sets in a polish space X, most of them require projection of a product of 2 spaces (Baire space and X). In order to "stay within the boundries" of the space X, you need the Suslin operation A. At the end of the chapter Jech mentions that "Suslin’s discovery of an error in a proof in Lebesgue’s article led to a construction of an analytic non-Borel set and introduction of the operation A."

Alternatively if you take the recursive definition of the analytic sets as $\Sigma^1_1(x)$ sets (x - a real) then simplifying the recursive function/formula (Turing machine) defined by a given $\Sigma^1_1(x)$ set you realize why adding a second order quantifier ($\Sigma^1$) is equal to a projection of a closed sets. For the recursive definition of the projective hierarchy see D. Marker Descriptive Set Theory.

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