Projective limits in Euclidean quantum field theory, I: Free scalar fields

We present two constructions of projective systems of measures associated to discretizations of free scalar Euclidean quantum fields. The first one is obtained using only purely combinatorial data and applies to free massless scalar fields on polyhedral cell complexes. The second construction utilizes the suitably discretized covariance of the free massive scalar field on a compact Riemannian manifold.