Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices

Abstract

The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of corners of finite size, must be finite. A similar result, Theorem 1, is also established for uni-tarily invariant measures on the space of all infinite complex matrices. These results, combined with the ergodic decomposition theorem of [3], imply that the infinite Hua-Pickrell measures of Borodin and Olshanski [2] have finite ergodic components. The proof is based on the approach of Olshanski and Vershik [6]. First, it is shown that if the sequence of orbital measures assigned to almost every point is weakly precompact, then our ergodic measure must indeed be finite. The second step, which completes the proof, shows that if a unitarily-invariant measure admits well-defined projections onto the quotient space of finite corners, then for almost every point the corre-sponing sequence of orbital measures is indeed weakly precompact.