The purpose of this paper is to
establish a unified treatment of many disparate theorems of Levy-Hinčin type. The
appropriate framework to do this is the theory of commutative hypergroups. In this
way we not only generalize the results mentioned above but also settle some
asymmetries indicated above. Roughly speaking a hypergroup K is a space in which
the product of two elements is a probability measure on this space satisfying certain
conditions. If K is commutative and if the space K of characters is a hypergroup
under pointwise operations a Levy-Hinčin formula for convolution semigroups is
obtained. Before setting up some notation we show how the examples fit
in.
