  In a paper by Ashtekar and Campiglia, invariance under volume preserving
residual diffeomorphisms has been used to single out the standard
representation of the reduced holonomy-flux algebra in homogeneous loop quantum
cosmology (LQC). In this paper, we use invariance under all residual
diffeomorphisms to single out the standard kinematical Hilbert space of
homogeneous isotropic LQC for both the standard configuration space
$\mathbb{R}_{\mathrm{Bohr}}$, as well as for the Fleischhack one $\mathbb{R}
\sqcup \mathbb{R}_{\mathrm{Bohr}}$. We first determine the scale invariant
Radon measures on these spaces, and then show that the Haar measure on
$\mathbb{R}_{\mathrm{Bohr}}$ is the only such measure for which the momentum
operator is hermitian w.r.t. the corresponding inner product. In particular,
the measure is forced to be identically zero on $\mathbb{R}$ in the Fleischhack
case, so that for both approaches, the standard kinematical LQC-Hilbert space
is singled out.
