  In loop quantum gravity in the connection representation, the quantum
configuration space $\bar{\mathcal{A}/\mathcal{G}}$, which is a compact space,
is much larger than the classical configuration space $\mathcal{A}/%
\mathcal{G}$ of connections modulo gauge transformations. One finds that $%
\bar{\mathcal{A}/\mathcal{G}}$ is homeomorphic to the space $Hom(%
\mathcal{L}_{\ast},G))/Ad$. We give a new, natural proof of this result,
suggesting the extension of the hoop group $\mathcal{L}_{\ast}$ to a larger,
compact group $\mathcal{M}(\mathcal{L}_{\ast})$ that contains $%
\mathcal{L}_{\ast}$ as a dense subset. This construction is based on almost
periodic functions. We introduce the Hilbert algebra $L_{2}(\mathcal{M}(%
\mathcal{L}_{\ast}))$ of $\mathcal{M}(\mathcal{L}_{\ast})$ with respect to the
Haar measure $\xi $ on $\mathcal{M}(\mathcal{L}_{\ast})$. The measure $% \xi $
is shown to be invariant under 3-diffeomorphisms. This is the first step in a
proof that $L_{2}(\mathcal{M}(\mathcal{L}_{\ast}))$ is the appropriate Hilbert
space for loop quantum gravity in the loop representation. In a subsequent
paper, we will reinforce this claim by defining an extended loop transform and
its inverse.
