Tt-functionals and martin-löf randomness for bernoulli measures

For r ∈ [0, 1], the Bernoulli measure μ_r on the Cantor space {0, 1}^ℕ assigns measure r to the set of sequences with 1 at a fixed position. In [5] it is shown that for r, s ∈ [0, 1], μ_s is continuously reducible to μ_r if and only if r and s satisfy certain purely number theoretic conditions (binomial reducibility). We bring these results into the context of computability theory and Martin-Löf randomness and show that the continuous maps arising in [5] are truthtable functionals (tt-functionals) on {0, 1}^ℕ. This allows us to extend the characterization of continuous reductions between Bernoulli measures to include tt-functionals. It then follows from the conservation of randomness under tt-functionals that if s is binomially reducible to r, then there is a tt-functional that maps every Martin-Löf random sequence for μ_s to a Martin-Löf random sequences for μ_r. We are also able to show using results in [2] that the converse of this statement is not true.