K-trivials are never continuously random
G. Barmpalias, N. Greenberg, A. Montalban and T. Slaman
Summary
Reimann and Slaman raised the question "For which
infinite binary sequences X do there exist continuous probability measures m such
that X is effectively random relative to m?". They defined the collection
NCR of binary sequences for which such measures do not exist, and showed that NCR is countable;
indeed that every sequence in NCR is hyperarithmetic. In this paper we
contribute toward the understanding of NCR by showing that it contains
all sets which are Turing reducible to an incomplete, recursively enumerable
set. In particular, NCR contains all K-trivial sets.