  We prove that the ergodic Ces\' aro averages generated by a positive
Dunford-Schwartz operator in a noncommutative space $L^p(\mathcal M,\tau)$,
$1<p<\infty$, converge almost uniformly (in Egorov's sense). This problem goes
back to the original paper of Yeadon \cite{ye}, where bilaterally almost
uniform convergence of these averages was established for $p=1$.
