  The third part of the paper concludes the proof of the main result --- the
description of the ergodic decomposition of infinite Pickrell measures. First
it is shown that the scaling limit of radial parts of finite-dimensional
infinite Pickrell measures is precisely the infinite Bessel point process. It
is then established that the "gaussian parameter" almost surely vanishes for
our ergodic components, and the convergence to the scaling limit is then
established in the space of finite measures on the space of finite measures.
Finally, singularity is established for Pickrell measures corresponding to
different values of the parameter.
