Completely determined Borel sets and measurability

Abstract

We consider the reverse math strength of the statement $\mathsf{C}\text{-}\mathsf{DM}$:"Every completely determined Borel set is measurable." Over ${\mathsf{WWKL}}_0$, we obtain the following results analogous to the previously studied category case. $\mathsf{C}\text{-}\mathsf{DM}$ lies strictly between ${\mathsf{ATR}}_0$ and ${\mathsf{L}}_{{\omega }_1,\omega }\text{-}\mathsf{CA}$. Whenever $M\subseteq 2^{\omega }$ is the second-order part of an $\omega$-model of $\mathsf{C}\text{-}\mathsf{DM}$, then for every $Z\in M$, there is a $R\in M$ such that $R$ is ${\Delta }_1^1$-random relative to $Z$. On the other hand, without ${\mathsf{WWKL}}_0$, all sets have measure zero (as measured according to $\mathsf{C}\text{-}\mathsf{DM}$), and it follows vacuously that $\neg {\mathsf{WWKL}}_0$ implies $\mathsf{C}\text{-}\mathsf{DM}$ over ${\mathsf{RCA}}_0$.

arXiv comment19 pages, minor revisions

References in corpus (1)

• Algorithmic randomness, reverse mathematics, and the dominated convergence theorem