Convolution semigroups on hypergroups

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.