Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures

This paper is the first in a series of three. We give an explicit description of the ergodic decomposition of infinite Pickrell measures on the space of infinite complex matrices. A key role is played by the construction of $\sigma$-finite analogues of determinantal measures on spaces of configurations, including the infinite Bessel process, a scaling limit of the $\sigma$-finite analogues of the Jacobi orthogonal polynomial ensembles. Our main result identifies the infinite Bessel process with the decomposing measure of an infinite Pickrell measure.