TT-Functionals and Martin-Löf Randomness for Bernoulli Measures

Abstract

For $r \in [0,1]$, the Bernoulli measure $\mu_r$ on the Cantor space $\{0,1\}^{\mathbb{N}}$ assigns measure $r$ to the set of sequences with $1$ at a fixed position. In [5] it is shown that for $r,s \in [0,1]$, $\mu_s$ is continuously reducible to $\mu_r$ if any only if $r$ and $s$ satisfy certain purely number theoretic conditions (binomial reduciblility). We bring these results into the context of computability theory and Martin-Löf randomness and show that the continuous maps arising in [5] are truth-table functionals (tt-functionals) on $\{0,1\}^{\mathbb{N}}$. This allows us 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 $\mu_s$ to a Martin-Löf random sequences for $\mu_r$. We are also able to show using results in [2] that the converse of this statement is not true.