Completely Determined Borel Sets and Measurability

We consider the reverse math strength of the statement CD-M: " Every completely determined Borel set is measurable.” Over WWKL, we obtain the following results analogous to the previously studied category case: (1) CD-M lies strictly between ATR₀ and L_{ω1, ω}-CA. (2) Whenever M ⊆ 2^ω is the second-order part of an ω-model of CD-M, then for every Z ∈ M, there is a R ∈ M such that R is Δ¹₁-random relative to Z. On the other hand, without WWKL, all sets have measure zero (as measured according to CD-M), and it follows vacuously that WWKL implies CD-M over RCA₀.