  We consider the Wigner equation corresponding to a nonlinear Schroedinger
evolution of the Hartree type in the semiclassical limit $\hbar\to 0$. Under
appropriate assumptions on the initial data and the interaction potential, we
show that the Wigner function is close in $L^2$ to its weak limit, the solution
of the corresponding Vlasov equation. The strong approximation allows the
construction of semiclassical operator-valued observables, approximating their
quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a
pointwise-positivity manipulation, which seems necessary in working with the
$L^2$ norm and the precise form of the nonlinearity. We employ the Husimi
function as a pivot between the classical probability density and the Wigner
function, which -- as it is well known -- is not pointwise positive in general.
