Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices

TL;DR

This paper proves that ergodic, unitarily invariant measures on spaces of infinite matrices are necessarily finite, extending the understanding of measure finiteness in infinite-dimensional matrix spaces and their ergodic components.

Contribution

It establishes the finiteness of ergodic unitarily invariant measures on infinite matrix spaces, a result previously unknown, using a novel approach based on orbital measures and weak precompactness.

Findings

Infinite Hermitian measures invariant under the infinite unitary group are finite.

Ergodic components of infinite Hua-Pickrell measures are finite.

The proof links orbital measure precompactness to measure finiteness.

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 unitarily 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 corresponing sequence of orbital measures is indeed weakly precompact.