Can it be that universal measurability is preserved by projections?

Question

I am currently discovering descriptive set theory—with much pleasure! It is something of a surprise to me that, while the Borel hierarchy is indexed by $\omega_1$, the projective hierarchy is only indexed by $\mathbf{N}$, and that classical descriptive set theory stops there…

In particular, it would seem natural to “merge” the respective ideas behind Borel and projective hierarchies, to consider the smallest algebra of subsets of $\mathbf{N}^{\mathbf{N}}$ stable under complementation, countable unions and projections. Has this been done; and if yes, what is it called? I wonder in particular if it would be consistent with ZFC for every set in this algebra to be universally measurable? (and, maybe, that being universally measurable is preserved by projections…?). Clearly this cannot be a theorem of ZFC, since assuming $V=L$ proves existence of a $\mathbf{\Delta}^1_2$ set not being universally measurable; but I have read nowhere that my question would be contradictory with ZFC…

(Also, subsidiary question: the notation for Borel and projective hierarchies, with respective exponents $0$ and $1$, suggest that that there would be a “$k$-th order hierarchy” for $k \geq 2$, giving rise to $\mathbf{\Sigma}^k_n$, $\mathbf{\Pi}^k_n$ and $\mathbf{\Delta}^k_n$ sets… But I have nowhere read such a definition! So, does it exist something continuing the Borel and projective hierarchies in this way?…).

Answer 1

I'm not an expert, so please correct me if I'm wrong:

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We can indeed continue the projective hierarchy beyond its finite levels. And like the Borel hierarchy, we can do this "from below" as follows:

• $\bf\Sigma^1_1$ is the class of analytic sets.

• $X$ is ${\bf \Pi^1_\alpha}$ iff $\omega^\omega\setminus X$ is ${\bf\Sigma^1_\alpha}$.

• For $\alpha>0$, a set $X$ is ${\bf \Sigma^1_\alpha}$ iff $$X=\bigcup_{i\in\omega}Y_i$$ for some family of sets $(Y_i)_{i\in\omega}$ such that each $Y_i$ is the projection of some $Z_i$ with $Z_i\in{\bf \Pi^1_{\beta_i}}$ for some $\beta_i<\alpha$.

(Note that this last clause means that we don't simply take unions of pointclasses at limit stages; we could do that instead if desired.)

As with the Borel hierarchy, the regularity of $\omega_1$ implies that this hierarchy stablizes at level $\omega_1$. Call the sets that appear in this hierarchy the transfinitely projective sets (I don't know if there is a standard term for them). It's not hard to show that, just like the projective sets, each transfinitely projective set $X$ is an element of the inner model $L(\mathbb{R})$. Under large cardinals, $L(\mathbb{R})$ satisfies ZF + DC + AD (indeed a bit more), and so in fact assuming large cardinals, every transfinitely projective set is "tame" for the usual meanings of the word "tame."

In particular, the answer to your question is yes. (Well, the first question, anyways - obviously this doesn't address the issue of whether in universal measurability could be preserved by projections consistently with ZFC. I suspect the answer to that question is negative, although I don't see it at the moment.)

It's worth noting that pointclasses of the form $\mathcal{P}(\omega^\omega)\cap\mathcal{M}$ for some "tame" (assuming large cardinals) inner model $\mathcal{M}$ are extensively studied. These, per the above, reach far beyond the transfinitely projective. I think this explains why the transfinitely projective sets aren't as well-studied: my impression is that if we want to go past the finite levels of the projective hierarchy we generally want to go way past that to one of these more set-theoretic pointclasses. (Relatedly, see this old Mathoverflow question.)