  We give a short and simple proof that the Lorentzian 10j symbol, which forms
a key part of the Barrett-Crane model of Lorentzian quantum gravity, is finite.
The argument is very general, and applies to other integrals. For example, we
show that the Lorentzian and Riemannian causal 10j symbols are finite, despite
their singularities. Moreover, we show that integrals that arise in
Cherrington's work are finite. Cherrington has shown that this implies that the
Lorentzian partition function for a single triangulation is finite, even for
degenerate triangulations. Finally, we also show how to use these methods to
prove finiteness of integrals based on other graphs and other homogeneous
domains.
