  The ergodic decomposition of a family of Hua-Pickrell measures on the space
of infinite Hermitian matrices is studied. Firstly, we show that the ergodic
components of Hua-Pickrell probability measures have no Gaussian factors, this
extends a result of Alexei Borodin and Grigori Olshanski. Secondly, we show
that the sequence of asymptotic eigenvalues of Hua-Pickrell random matrices is
balanced in certain sense and has a "principal value" coincides with the
$\gamma_1$ parameter of ergodic components. This allow us to complete the
program of Borodin and Olshanski on the description of the ergodic
decomposition of Hua-Pickrell probability measures. Finally, we extend the
aforesaid results to the case of infinite Hua-Pickrell measues. By using the
theory of $\sigma$-finite infinite determinantal measures recently introduced
by A. I. Bufetov, we are able to identify the ergodic decomposition of
Hua-Pickrell infinite measures to some explicit $\sigma$-finite determinantal
measures on the space of point configurations in $\mathbb{R}^*$. The paper
resolves a problem of Borodin and Olshanski.
