The main result of this paper, 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.