  Projective structures have successfully been used for the construction of
measures in the framework of loop quantum gravity. In the present paper, we
establish such structures for the configuration space $\mathbb{R}\sqcup
\mathbb{R}_{\mathbb{Bohr}}$, recently introduced in the context of homogeneous
isotropic loop quantum cosmology. In contrast to the traditional space
$\mathbb{R}_{\mathbb{Bohr}}$, the first one is canonically embedded into the
quantum configuration space of the full theory. In particular, for the
embedding of states into a corresponding symmetric sector of loop quantum
gravity, this is advantageous. However, in contrast to the traditional space,
there is no Haar measure on $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$
defining a canonical kinematical $L^2$-Hilbert space on which operators can be
represented. The introduced projective structures allow to construct a family
of natural measures on $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$ whose
corresponding $L^2$-Hilbert spaces we finally investigate.
