  We consider the $3d$ energy critical nonlinear Schr\" odinger equation with
data distributed according to the Gaussian measure with covariance operator
$(1-\Delta)^{-s}$, where $\Delta$ is the Laplace operator and $s$ is
sufficiently large. We prove that the flow sends full measure sets to full
measure sets. We also discuss some simple applications. This extends a previous
result by Planchon-Visciglia and the second author from $1d$ to higher
dimensions.
