The Ergodic Decomposition of Infinite Pickrell Measures III. The infinite Bessel process as the limit of radial parts of finite-dimensional projections of infinite Pickrell measures

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.