  The intention of this thesis is to provide general tools and concepts that
allow to perform a mathematically substantiated symmetry reduction in (quantum)
gauge field theories. Here, the main focus is on the framework of loop quantum
gravity (LQG), where we concentrate on the reduction of the quantum
configuration space, and the construction of a normalized Radon measures on the
reduced one. More precisely, we introduce a new way to symmetry reduce the
LQG-configuration space directly on the quantum level, and then show that this
always leads to a (strictly) larger reduced space than quantizing the classical
configuration space of invariant connections (traditional approach). We prove a
general classification theorem for such invariant connections, which we then
use to calculate the classical configuration space for the homogeneous and the
spherically symmetric case. Here, the backbone of the introduced reduction
concept is a lifting result for group actions on sets to spectra of
$C^*$-subalgebras of the bounded functions thereon; and as a further
application of this, we single out the standard kinematical Hilbert space of
homogeneous isotropic loop quantum cosmology by means of the same invariance
condition for both the standard configuration space
$\mathbb{R}_{\mathrm{Bohr}}$, as well as for the Fleischhack one
$\mathbb{R}\sqcup\mathbb{R}_{\mathrm{Bohr}}$. Along the way, symmetries of
embedded analytic curves under a given analytic Lie group action are
investigated, and a first classification result is proven for the case that the
action is proper or pointwise proper and transitive, and only admits normal
stabilizers.
