Does some axiom independent of ZFC (e.g., PD) imply that a correspondence with a projective graph have universally measurable selectors?

Question

I'm looking at a situation in which I have, say, two Borel spaces, X,Y, and a correspondence F from one to the other whose graph is projective (i.e., the projective hierarchy), or perhaps even analytically measurable (i.e., in the sigma-algebra generated by the analytic sets). I want there to be a universally measurable (or even P-measurable, for some fixed Borel measure P on X) selector of F. I don't think this follows under ZFC, but if one assumes, e.g., projective determinacy, can it be shown? Please note I am not really a hardcore logic person, rather an analysist, so please reply in a language an analysist can understand, thanks!

Answer 1

Yes! For terminology, suppose $X$ and $Y$ are Polish spaces and $F\subseteq X\times Y$. A set $G\subseteq F$ uniformizes $F$ if $G$ is the graph of a function $g\colon \pi_X(F)\to Y$, called the uniformizing function. Note that if $U\subseteq Y$ is open, then $g^{-1}(U) = \pi_X(G\cap (X\times U))$, so if $G$ is $\mathbf{\Sigma}^1_{n}$ in the projective heirarchy, then $g$ has the property that the preimage of every open set is $\mathbf{\Sigma}^1_{n}$ (since $\mathbf{\Sigma}^1_{n}$ is closed under projection).

It's a theorem of Moschovakis (Theorem 1 in Uniformization in a playful universe, Bull. Amer. Math. Soc. 77 (1971), 731-736) that under Projective Determinacy (PD), every projective set $F$ is uniformized by a projective set $G$, and thus has a uniformizing function $g$ with the property that the preimage of every open set is a projective set. Since PD also implies that projective sets are universally measurable, $g$ is universally measurable.

Uniformization results are treated extensively in Moschovakis's book Descriptive Set Theory, which is freely available on the author's webpage. The uniformization theorem is Theorem 6C.5, and the theorem that projective sets are universally measurable is Exercise 6A.18.

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Here's some additional information. Caveat: I'm far an expert in this stuff. But if I've made any errors here, I'm sure someone (Noah Schweber, maybe?) will correct them.

The von Neumann selection theorem (Exercise 4E.9 in Moschovakis) is a theorem of ZFC: Every analytic ($\mathbf{\Sigma}^1_1$) set has a uniformizing function that is universally measurable.

This is the best possible result of this kind in ZFC: It is consistent with ZFC (it follows from $V = L$) that there is a function $f\colon X\to Y$ between Polish spaces which is not Lebesgue measurable, but whose graph $F\subseteq X\times Y$ is $\mathbf{\Pi}^1_1$. Note that $f$ is the unique uniformizing function of $F$, so $F$ is a $\mathbf{\Pi}^1_1$ set with no Lebesgue measurable uniformizing function. See Exercise 5A.7 in Moschovakis.

In the positive direction, if you only need to uniformize sets which are $\mathbf{\Sigma}^1_2$ (this class contains sets you call analytically measurable: $\sigma(\mathbf{\Sigma}^1_1)$, which in turn contains all $\mathbf{\Pi}^1_1$ and $\mathbf{\Sigma}^1_1$ sets), you can get away with assuming a lot less than PD. It's a theorem of ZFC that every $\mathbf{\Sigma}^1_2$ set can be uniformized by a $\mathbf{\Sigma}^1_2$ set, (this is the Kondo uniformization theorem, Theorem 4E.4 in Moschovakis). And Analytic Determinacy (determinacy restricted to $\mathbf{\Sigma}^1_1$ sets) implies that every $\mathbf{\Sigma}^1_2$ set is universally measurable (Exercise 6G.12 in Moschovakis). Thus, assuming Analytic Determinacy, every $\mathbf{\Sigma}^1_2$ set has a universally measurable uniformizing function. Analytic Determinacy has much lower consistency strength than Projective Determinacy - for example, it follows from the existence of a measurable cardinal.

The book Classical Descriptive Set Theory by Kechris is another good reference for this material. Moschovakis uniformization is Corollary 39.9 on p. 339, universal measurability of projective sets is Theorem 38.17 on p. 326, the von Neumann selection theorem is Theorem 18.1 on p. 120, the Kondo uniformization theorem is Corollary 38.7 on p. 323, and the universal measurability of $\mathbf{\Sigma}^1_2$ sets is Theorem 36.20 on p. 307.