  (This is a report for the Proceedings of ``Journees Relativistes 1993''
written in September 1993. Containes a short description of the results
published elsewhere in the joint paper with A. Ashtekar) Integral calculus on
the space of gauge equivalent connections is developed. By carring out a
non-linear generalization of the theory of cylindrical measures on topological
vector spaces, a faithfull, diffeomorphism invariant measure is introduced on a
suitable completion of the quotient space. The strip (i.e. momentum) operators
are densely-defined in the resulting Hilbert space and interact with the
measure correctly, to become essentially self adjoint operators.

General Relativity and Quantum Cosmology