  The most common spin foam models of gravity are widely believed to be
discrete path integral quantizations of the Plebanski action. However, their
derivation in present formulations is incomplete and lower dimensional simplex
amplitudes are left open to choice. Since their large-spin behavior determines
the convergence properties of the state-sum, this gap has to be closed before
any reliable conclusion about finiteness can be reached. It is shown that these
amplitudes are directly related to the path integral measure and can in
principle be derived from it, requiring detailed knowledge of the constraint
algebra and gauge fixing. In a related manner, minimal requirements of
background independence provide non trivial restrictions on the form of an
anomaly free measure. Many models in the literature do not satisfy these
requirements. A simple model satisfying the above consistency requirements is
presented which can be thought of as a spin foam quantization of the
Husain--Kuchar model.
