  We show that the space of generalized fluxes - momentum space - for loop
quantum gravity cannot be constructed by Fourier transforming the projective
limit construction of the space of generalized connections - position space -
due to the non-abelianess of the gauge group SU(2). From the abelianization of
SU(2), U(1)^3, we learn that the space of generalized fluxes turns out to be an
inductive limit, and we determine the consistency conditions the fluxes should
satisfy under coarse-graining of the underlying graphs. We comment on the
applications to loop quantum cosmology, in particular, how the characterization
of the Bohr compactification of the real line as a projective limit opens the
way for a similar analysis for LQC.
