A L\'opez-Escobar theorem for metric structures, and the topological Vaught conjecture

We show that a version of L'opez-Escobar’s theorem holds in the setting of logic for metric structures. More precisely, let $\mathbb{U}$ denote the Urysohn sphere and let $\mathrm{M}\mathrm{o}\mathrm{d}(\mathcal{L},\mathbb{U})$ be the space of metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any $\mathrm{I}\mathrm{s}\mathrm{o}(\mathbb{U})$-invariant Borel function $f\colon $\mathrm{M}\mathrm{o}\mathrm{d}$($\mathcal{L}$, $\mathbb{U}$)\rightarrow \lbrack 0,1]$, there exists a sentence $\varphi$ of ${\mathcal{L}}_{{\omega }_1\omega }$ such that for all $M\in $\mathrm{M}\mathrm{o}\mathrm{d}$($\mathcal{L}$,$\mathbb{U}$)$wehave$f(M)=\phi ^{M}$. At the same time we introduce a variant ${\mathcal{L}}_{{\omega }_1\omega }^{\ast }$ of ${\mathcal{L}}_{{\omega }_1\omega }$ in which the usual quantifiers are replaced with category quantifiers, and establish the analogous theorem for ${\mathcal{L}}_{{\omega }_1\omega }^{\ast }$. This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group action is Borel isomorphic to the isomorphism relation on the set of models of a given ${\mathcal{L}}_{{\omega }_1\omega }$-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.