  We consider a randomization of a function on $\mathbb R^d$ that is naturally
associated to the Wiener decomposition and, intrinsically, to the modulation
spaces. Such randomized functions enjoy better integrability, thus allowing us
to improve the Strichartz estimates for the Schr\"odinger equation. As an
example, we also show that the energy-critical cubic nonlinear Schr\"odinger
equation on $\mathbb R^4$ is almost surely locally well-posed with respect to
randomized initial data below the energy space.
