Constructive Approach to the Global Markov Property in Euclidean Quantum Field Theory I. Construction of transition kernels by

The trace properties of the sample paths of su ciently regular generalized random elds are studied. In particular, nice localisation properties are shown in the case of hyperplanes. Using techniques of Euclidean quantum eld theory a constructive description of the conditional expectation values with respect to some Gibbs measures describing Euclidean quantum eld theory models and the -algebras localised in halfspaces is given. In particular the Global Markov property with respect to hyperplanes follows from these constructions in an explicit way.