  Let P -> M be a principal G-bundle. Using techniques from the loop
representation of gauge theory, we construct well-defined substitutes for
``Lebesgue measure'' on the space A of connections on P and for ``Haar
measure'' on the group Ga of gauge transformations. More precisely, we define
algebras of ``cylinder functions'' on the spaces A, Ga, and A/Ga, and define
generalized measures on these spaces as continuous linear functionals on the
corresponding algebras. Borrowing some ideas from lattice gauge theory, we
characterize generalized measures on A, Ga, and A/Ga in terms of graphs
embedded in M. We use this characterization to construct generalized measures
on A and Ga, respectively. The ``uniform'' generalized measure on A is
invariant under the group of automorphisms of P. It projects down to the
generalized measure on A/Ga considered by Ashtekar and Lewandowski in the case
G = SU(n). The ``generalized Haar measure'' on Ga is right- and left-invariant
as well as Aut(P)-invariant. We show that averaging any generalized measure on
A against generalized Haar measure gives a gauge-invariant generalized measure
on A.
