  In this work we investigate the question, under what conditions Hilbert
spaces that are induced by measures on the space of generalized connections
carry a representation of certain non-Abelian analogues of the electric flux.
We give the problem a precise mathematical formulation and start its
investigation. For the technically simple case of U(1) as gauge group, we
establish a number of "no-go theorems" asserting that for certain classes of
measures, the flux operators can not be represented on the corresponding
Hilbert spaces.
  The flux-observables we consider play an important role in loop quantum
gravity since they can be defined without recourse to a background geometry,
and they might also be of interest in the general context of quantization of
non-Abelian gauge theories.
