Uniqueness of Measures in Loop Quantum Cosmology

In a paper of Ashtekar and Campiglia, residual diffeomorphisms have been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). We show that, in the homogeneous isotropic case, unitarity of the translations w.r.t. the extended $\mathbb{R}$-action (exponentiated reduced fluxes in the standard approach) singles out the Bohr measure on both the standard quantum configuration space $\mathbb{R}_\mathrm{Bohr}$ as well as on the Fleischhack one. Thus, in both situation, the same condition singles out the standard kinematical Hilbert space of LQC.