  We study a summability method called almost convergence for bounded
measurable functions defined on a locally compact abelian group. We define
almost convergence using topologically invariant means and exhibit two
different kinds of necessary and sufficient conditions, one is analytic and the
other is functional analytic, for a given function to be almost convergent. As
an application, we show complex Tauberian theorems for almost convergence on
the integers and the real numbers. In particular, the latter one can be viewed
as an analogue of the Wiener-Ikehara theorem.
