  Let $G$ be a group with a finite subgroup $H$. We define the
$L^2$-multiplicity of an irreducible representation of $H$ in the
$L^2$-homology of a proper $G$-CW-complex. These invariants generalize the
$L^2$-Betti numbers. Our main results are approximation theorems for
$L^2$-multiplicities which extend the approximation theorems for $L^2$-Betti
numbers of L\"uck, Farber and Elek-Szab\'o respectively. The main ingredient is
the theory of characters of infinite groups and a method to induce characters
from finite subgroups. We discuss applications to the cohomology of
(arithmetic) groups.
