Motivated by the theory of graph limits, we introduce and study the
convergence and limits of linear representations of finite groups over finite
fields. The limit objects are infinite dimensional representations of free
groups in continuous algebras. We show that under a certain integrality
condition, the algebras above are skew fields. Our main result is the extension
of Schramm's characterization of hyperfiniteness to linear representations.Comment: Final version. To appear in the Journal of Algebr