  We introduce a notion of measuring scales for quantum abelian gauge systems.
At each measuring scale a finite dimensional affine space stores information
about the evaluation of the curvature on a discrete family of surfaces. Affine
maps from the spaces assigned to finer scales to those assigned to coarser
scales play the role of coarse graining maps. This structure induces a
continuum limit space which contains information regarding curvature evaluation
on all piecewise linear surfaces with boundary. The evaluation of holonomies
along loops is also encoded in the spaces introduced here; thus, our framework
is closely related to loop quantization and it allows us to discuss effective
theories in a sensible way. We develop basic elements of measure theory on the
introduced spaces which are essential for the applicability of the framework to
the construction of quantum abelian gauge theories.

High Energy Physics - Theory;General Relativity and Quantum Cosmology;Differential Geometry