  In the canonical approach to Lorentzian Quantum General Relativity in four
spacetime dimensions an important step forward has been made by Ashtekar, Isham
and Lewandowski some eight years ago through the introduction of an appropriate
Hilbert space structure.
  This Hilbert space, together with its generalization due to Baez and Sawin,
is appropriate for semi-classical quantum general relativity if the spacetime
is spatially compact. In the spatially non-compact case, however, an extension
of the Hilbert space is needed in order to approximate metrics that are
macroscopically nowhere degenerate.
  For this purpose, in this paper we apply von Neumann's theory of the Infinite
Tensor Product (ITP) of Hilbert Spaces to Quantum General Relativity. The
cardinality of the number of tensor product factors can take the value of any
possible Cantor aleph as is needed for our problem, where a Hilbert space is
attached to each edge of an arbitrarily complicated, generally infinite graph.
  The new framework opens a pandora's box full of techniques, appropriate to
pose fascinating physical questions such as quantum topology change,
semi-classical quantum gravity, effective low energy physics etc. from the
universal point of view of the ITP. In particular, the study of photons and
gravitons propagating on fluctuating quantum spacetimes is now in reach, the
topic of the next paper in this series.
