  Consider an integer $n \geq 2$ and real numbers $\tau>n-1$ and $l>2(\tau+1)$.
Using ideas of Moser, Salamon proved that individual Diophantine tori persist
for Hamiltonian systems which are of class $C^l$. Under the stronger assumption
that the system is a $C^{l+\tau}$ perturbation of an analytic integrable
system, P{\"o}schel proved the persistence of a set of positive measure of
Diophantine tori. We improve the last result by showing it is sufficient for
the perturbation to be of class $C^{l}$ and the integrable part to be of class
$C^{l+2}$.
