  The stable Ramsey's theorem for pairs has been the subject of numerous
investigations in mathematical logic. We introduce a weaker form of it by
restricting from the class of all stable colorings to subclasses of it that are
non-null in a certain effective measure-theoretic sense. We show that the sets
that can compute infinite homogeneous sets for non-null many computable stable
colorings and the sets that can compute infinite homogeneous sets for all
computable stable colorings agree below $\emp'$ but not in general. We also
answer the analogs of two well known questions about the stable Ramsey's
theorem by showing that our weaker principle does not imply $\mathsf{COH}$ or
$\mathsf{WKL}_0$ in the context of reverse mathematics.
