Completely determined Borel sets and measurability

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