In this talk we are considering LQG in a symmetry-reduced context. Here symmetries are represented by Lie groups of automorphisms of the underlying bundle. The corresponding invariant connections then serve as a starting point for the construction of a reduced quantum configuration space as, e.g., used in LQC. Alternatively, one might aim at a symmetry reduction directly on the quantum level. This, indeed, can be done by lifting the symmetry on the bundle directly to the quantum space. In this talk we want to compare these two reduction concepts. Moreover, we discuss some measure theoretical aspects of configuration spaces occurring in LQC. If time permits we introduce an algebraic characterization of invariant connections that allows for their explicit calculations in many cases.