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In the canonical approach to Lorentzian Quantum General Relativity in fourspacetime dimensions an important step forward has been made by Ashtekar, Ishamand Lewandowski some eight years ago through the introduction of an appropriateHilbert space structure.  This Hilbert space, together with its generalization due to Baez and Sawin,is appropriate for semi-classical quantum general relativity if the spacetimeis spatially compact. In the spatially non-compact case, however, an extensionof the Hilbert space is needed in order to approximate metrics that aremacroscopically nowhere degenerate.  For this purpose, in this paper we apply von Neumann's theory of the InfiniteTensor Product (ITP) of Hilbert Spaces to Quantum General Relativity. Thecardinality of the number of tensor product factors can take the value of anypossible Cantor aleph as is needed for our problem, where a Hilbert space isattached to each edge of an arbitrarily complicated, generally infinite graph.  The new framework opens a pandora's box full of techniques, appropriate topose fascinating physical questions such as quantum topology change,semi-classical quantum gravity, effective low energy physics etc. from theuniversal point of view of the ITP. In particular, the study of photons andgravitons propagating on fluctuating quantum spacetimes is now in reach, thetopic of the next paper in this series.Comment: 60 pages, LATEX, no figure

Gauge field theory coherent states (GCS): IV. Infinite tensor product and thermodynamical limit