  We consider the one-dimensional nonlinear Schr\"odinger equation with a
nonlinearity of degree $p>1$. We exhibit measures on the space of initial data
for which we describe the non trivial evolution by the linear Schr\"odinger
flow and we show that their nonlinear evolution is absolutely continuous with
respect to this linear evolution. We deduce from this precise description the
global well-posedness of the equation for $p>1$ and scattering for $p>3$. To
the best of our knowledge, it is the first occurence where the description of
quasi-invariant measures allows to get quantitative asymptotics (here
scattering properties) for the nonlinear evolution.
