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

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.

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Don't worry about bumping it with small edits to draw attention. If you don't get an answer within the next 26 hours I'll put a bounty on it. –  Rudy the Reindeer Oct 29 '12 at 20:42
Thank you, that's very kind of you! I just received a message that I earned that privilege, so I will be able to put one myself. I hope it will work out. –  geezer Oct 29 '12 at 20:56
I suggest that you save your reputation and let me do this for you. : ) –  Rudy the Reindeer Oct 29 '12 at 21:05
Let me know if you still need a bounty but looks as if not. : ) –  Rudy the Reindeer Oct 30 '12 at 5:59
No answer = ok. –  Rudy the Reindeer Dec 3 '12 at 8:49

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.