K-trivials Are Never Continuously Random

In [RS07, RS08], Reimann and Slaman raise the question “For which infinite binary sequences X do there exist continuous probability measures μ such that X is effectively random relative to μ?”. They defined the collection NCR1 of binary sequences for which such measures do not exist (we give formal definitions below), and showed, for example, that NCR1 is countable, indeed that every sequence in NCR1 is hyperarithmetic. In this paper we contribute toward the understanding of NCR1 by showing that it contains all sets which are Turing reducible to an incomplete, recursively enumerable set. In particular, NCR1 contains all K-trivial sets.