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Let $H$ be the space of all Hermitian matrices of infinite order and $U(\infty)$ be the inductive limit of the chain $U(1)\subset U(2)\subset...$ of compact unitary groups. The group $U(\infty)$ operates on the space $H$ by conjugations, and our aim is to classify the ergodic $U(\infty)$-invariant probability measures on $H$ by making use of a general asymptotic approach proposed in Vershik's note \cite{V}. The problem is reduced to studying the limit behavior of orbital integrals of the form $$\int_{B\in\Omega_n}e^{i\op{tr}(AB)}M_n(dB),$$ where $A$ is a fixed $\infty\times\infty$ Hermitian matrix with finitely many nonzero entries, $\Omega_n$ is a $U(n)$-orbit in the space of $n\times n$ Hermitian matrices, $M_n$ is the normalized $U(n)$-invariant measure on the orbit $\Omega_n$, and $n\to\infty$. We also present a detailed proof of an ergodic theorem for inductive limits of compact groups that has been announced in \cite{V}.

Ergodic unitarily invariant measures on the space of infinite Hermitian matrices