Gaussian path integrals play an important role for free quantum field theories, and for the perturbative treatment of interacting quantum field theories. We give a general definition of a Gaussian measure over (Abelian) connections, and study their properties. Then we investigate the absolute continuity of Gaussian measures with respect to representations of the holonomy-flux Weyl algebra for $U(1)$. This shows that for a large subclass, the lengthlike Gaussian measures, there are no representations, where the fluxes can be implemented unitarily.