Given a set A in the unit interval and the associated Lebesgue measure λ, it is a natural question whether we may (in some sense) compute the measure [Formula: see text] in terms of the set A. Under the moniker measure theoretic uniformity, Tanaka and Sacks have (independently) provided a positive answer for the well-known class of hyperarithmetical sets of reals, and provided a basis theorem for such sets of positive measure. The hyperarithmetical sets are exactly the sets computable in terms of the functional [Formula: see text], in the sense of Kleene’s S1–S9. In turn, Kleene’s [Formula: see text] essentially corresponds to arithmetical comprehension as in [Formula: see text]. In this paper, we generalise the aforementioned results to the ‘next level’, namely [Formula: see text], in the form of the Suslin functional, or the equivalent hyperjump. We also generalise the Tanaka-Sacks basis theorem to sets of positive measure that are semi-computability relative to the Suslin functional. Finally, we discuss similar generalisations for infinite time Turing machines.