  Although algorithmic randomness with respect to various non-uniform
computable measures is well-studied, little attention has been paid to
algorithmic randomness with respect to computable \emph{trivial} measures,
where a measure $\mu$ on $2^\omega$ is trivial if the support of $\mu$ consists
of a countable collection of sequences. In this article, it is shown that there
is much more structure to trivial computable measures than has been previously
suspected.
