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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 thepreviously 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 $\mathsf{RCA}_0$.

Completely determined Borel sets and measurability