  We prove that Poisson measures are invariant under (random) intensity
preserving transformations whose finite difference gradient satisfies a cyclic
vanishing condition. The proof relies on moment identities of independent
interest for adapted and anticipating Poisson stochastic integrals, and is
inspired by the method applied in [22] on the Wiener space, although the
corresponding algebra is more complex than in the Wiener case. The examples of
application include transformations conditioned by random sets such as the
convex hull of a Poisson random measure.
