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The third part of the paper concludes the proof of the main result --- thedescription of the ergodic decomposition of infinite Pickrell measures. Firstit is shown that the scaling limit of radial parts of finite-dimensionalinfinite Pickrell measures is precisely the infinite Bessel point process. Itis then established that the "gaussian parameter" almost surely vanishes forour ergodic components, and the convergence to the scaling limit is thenestablished in the space of finite measures on the space of finite measures.Finally, singularity is established for Pickrell measures corresponding todifferent values of the parameter.

izvestiya. mathematics

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