  For a singular measure $\mu$, Ahlfors regular of order $\alpha>0,$ with
compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator
$\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator
$\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$ Here $P$ is the signed
measure, $P=V\mu$ with density $V$ belonging to the Orlicz class $L^{\Psi,\mu}$
with $\Psi(t)=(t+1)\log(t+1)-t.$ Using eigenvalue estimates for such operators,
obtained in \texttt{arXiv:2011.14877}, we establish eigenvalue asymptotics of
$\mathbf{T}(P,\mathbf{A})$ for a class of measures, including the ones
supported on uniformly rectifiable sets. These results lead to the
measurability in the sense of A.Connes of operators $\mathbf{T}(P,\mathbf{A})$
and a formula for the singular trace of these operators, producing a
noncommutative version of integral with respect to singular measure.
