Completely determined Borel sets and measurability

Abstract

We consider the reverse math strength of the statement $\mathsf{C\text-DM}$:"Every completely determined Borel set is measurable." Over $\mathsf{WWKL}_0$, we obtain the following results analogous to the previously studied category case. $\mathsf{C\text-DM}$ 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 $\mathsf{C\text-DM}$, 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 $\mathsf{C\text-DM}$), and it follows vacuously that $\neg \mathsf{WWKL}_0$ implies $\mathsf{C\text-DM}$ over $\mathsf{RCA}_0$.