  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 $\mathsf{RCA}_0$.
