  By applying Rohlin's result on the classification of homomorphisms of
Lebesgue space, the random inertial manifold of a stochastic damped nonlinear
wave equations with singular perturbation is proved to be approximated almost
surely by that of a stochastic nonlinear heat equation which is driven by a new
Wiener process depending on the singular perturbation parameter. This
approximation can be seen as the Smolukowski--Kramers approximation as time
goes to infinity. However, as time goes infinity, the approximation changes
with the small parameter, which is different from the approximation on a finite
interval.
