Measure-theoretic uniformity and the Suslin functional

Given a set A in the unit interval and associated Lebesgue measure ?, it is natural question whether we may (in some sense) compute ? ( ) terms of A. Under moniker theoretic uniformity, Tanaka Sacks have (independently) provided positive answer for well-known class hyperarithmetical sets reals, basis theorem such measure. The are exactly computable functional 2 E, sense Kleene’s S1–S9. In turn, E essentially corresponds to arithmetical comprehension as ACA 0 . this paper, generalise aforementioned results ‘next level’, namely ? 1 -CA , form Suslin functional, or equivalent hyperjump. We also Tanaka-Sacks that semi-computability relative functional. Finally, discuss similar generalisations infinite time Turing machines.