$I$-regularity, determinacy, and $\infty$-Borel sets of reals

Daisuke Ikegami

We show under $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$ that every set of reals is $I$-regular for any $\sigma$-ideal $I$ on the Baire space ${\omega }^{\omega }$ such that ${\mathbb{P}}_I$ is proper. T…