  Sofic groups were defined implicitly by Gromov in [Gr99] and explicitly by
Weiss in [We00]. All residually finite groups (and hence every linear group) is
sofic. The purpose of this paper is to introduce, for every countable sofic
group $G$, a family of measure-conjugacy invariants for measure-preserving
$G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai
entropy for actions of amenable groups. They are computed exactly for Bernoulli
shifts over $G$, leading to a complete classification of Bernoulli systems up
to measure-conjugacy for many groups including all countable linear groups.
Recent rigidity results of Y. Kida and S. Popa are utilized to classify
Bernoulli shifts over mapping class groups and property T groups up to orbit
equivalence and von Neumann equivalence respectively.
