The Full Decomposability Conjecture

Let ${\mathcal A}$ be an analytic subset of a Polish space and let ${\mathcal Y}$ be separable metrizable. For a function $f:{\mathcal A}\to{\mathcal Y}$, interpret $f^{-1}\boldsymbol{\Sigma}^0_{1+\eta}\subseteq\boldsymbol{\Sigma}^0_{1+\xi}$ as the corresponding preimage condition, and let a $\boldsymbol{\Delta}^0_{1+\xi}$-cover mean a countable cover by sets in that pointclass. The Full Decomposability Conjecture. For any countable ordinals $\eta\leq\xi<\omega_1$, the following are equivalent: (1) $f^{-1}\boldsymbol{\Sigma}^0_{1+\eta}\subseteq\boldsymbol{\Sigma}^0_{1+\xi}$; (2) this inclusion holds continuous-uniformly; and (3) there is a $\boldsymbol{\Delta}^0_{1+\xi}$-cover $({\mathcal A}_i)_{i\in\omega}$ of ${\mathcal A}$ such that, for every $i$, $f\upharpoonright{{\mathcal A}_i}$ is $\boldsymbol{\Sigma}^0_{1+\theta_i}$-measurable for some ordinal $\theta_i$ satisfying $\theta_i+\eta\leq\xi$. This conjecture extends finite-level decomposability to transfinite Borel levels and corrects the ordinal calculation in the previously mentioned formulation; its resolution status is not specified in the supplied text.