  Motivated by applications to transition semigroups, we introduce the notion
of a norming dual pair and study a Pettis-type integral on such pairs. In
particular, we establish a sufficient condition for integrability. We also
introduce and study a class of semigroups on such dual pairs which are an
abstract version of transition semigroups. Using our results, we give
conditions ensuring that a semigroup consisting of kernel operators has a
Laplace transform which also consists of kernel operators. We also provide
conditions under which a semigroup is uniquely determined by its Laplace
transform.
