On the $L^p$-spaces of projective limits of probability measures

The present article describes the precise structure of the $L^p$-spaces of projective limit measures by introducing a category theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive Quantum Field Theory (QFT) is given through the Osterwalder-Schrader axioms.