Iseki [11] defined a general notion of ergodicity suitable for functions

where

*J* is an arbitrary abelian
semigroup
and

*X* is a Banach space. In this paper we develop the theory of such
functions, showing in particular that it fits the general framework
established by Eberlein [9] for ergodicity of semigroups of operators
acting on

*X*. Moreover, let

be a translation invariant
closed subspace of the space of all bounded functions from

*J* to

*X*. We
prove that if

contains the constant functions and

is an ergodic function whose
differences lie in

then

.
This result has applications to spaces
of sequences facilitating new proofs of theorems of Gelfand and
Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of
ergodic vectors of a representation

generalizing
results known for the case

.
Finally, when

*J* is a
subsemigroup of a locally compact abelian group

*G*, we compare the Iseki
integrals with the better known Cesàro integrals.