  We consider the radial nonlinear Schr\"odinger equation $i\partial_tu +\Delta
u = |u|^{p-1}u$ in dimension $d\geqslant 2$ for $p\in
\left(1,1+\frac{4}{d}\right]$ and construct a natural Gaussian measure $\mu_0$
which support is almost $L^2_{\text{rad}}$ and such that $\mu_0$ - almost every
initial data gives rise to a unique global solution. Furthermore, for
$p>1+\frac{2}{d}$ and $d\in\{2, \dots, 10\}$ the solutions constructed scatters
in a space which is almost $L^2$. This paper can be viewed as the higher
dimensional counterpart of the work of Burq and Thomann, in the radial case.
