  This paper is the first in a series of three. The main result, Theorem 1.11,
gives an explicit description of the ergodic decomposition for infinite
Pickrell measures on spaces of infinite complex matrices. The main construction
is that of sigma-finite analogues of determinantal measures on spaces of
configurations. An example is the infinite Bessel point process, the scaling
limit of sigma-finite analogues of Jacobi orthogonal polynomial ensembles. The
statement of Theorem 1.11 is that the infinite Bessel point process (subject to
an appropriate change of variables) is precisely the ergodic decomposition
measure for infinite Pickrell measures.
