  In this paper we prove a general convergence theorem for almost-additive set
functions on unimodular, amenable groups. These mappings take their values in
some Banach space. By extending the theory of epsilon-quasi tiling techniques,
we set the ground for far-reaching applications in the theory of group
dynamics. In particular, we verify the almost-everywhere convergence of
abstract approximable bounded, additive processes, as well as a Banach space
approximation result for the spectral distribution function (integrated density
of states) for random operators on discrete structures in a metric space.
Further, we include a Banach space valued version of the Lindenstrauss ergodic
theorem for amenable groups.
