Abstract We consider the radial nonlinear Schrödinger equation i ∂ t u + Δ u = | u | p −1 u in dimension d ⩾ 2 for p ∈ 1 , 1 + 4 d and construct a natural Gaussian measure μ 0 which support is almost L rad 2 and such that μ 0 —almost every initial data gives rise to a unique global solution. Furthermore, for p > 1 + 2 d and d ⩽ 10, the solutions constructed scatter in a space which is almost L 2 . This paper can be viewed as a higher dimensional counterpart of the work of Burq and Thomann (2020 arXiv: 2012.13571 ), in the radial case.