Calibrated thin $\boldsymbol {\Pi }_{\mathbf 1}^{\mathbf 1}$ $\sigma$-ideals are $\boldsymbol G_\delta$

Abstract:

Let $E$ be a compact metric space, and let $I \subset \mathcal {K} (E)$ be a calibrated thin $\boldsymbol \Pi _{\mathbf {1}}^{\mathbf {1}}$ $\sigma$-ideal. Then $I$ is $\boldsymbol G_{\delta }$. This solves an open problem, which was posed by Kechris, Louveau and Woodin. Using our result we obtain a new proof of Kaufman’s theorem concerning $U$-sets and $U_{0}$-sets.