For discrete measured groupoids preserving a probability measure we introduce a notion of sofic dimension that measures the asymptotic growth of the number of sofic approximations on larger and larger finite sets. In the case of groups we give a form
ula for free products with amalgamation over an amenable subgroup. We also prove a free product formula for measure-preserving actions.
We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes t
he computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have a generating countable partition with finite entropy if the base has infinite entropy, which in the amenable case is well known and in the case that the acting group contains the free group on two generators was established by Bowen using a different argument.
