  In this paper we prove the almost sure existence of global weak solution to
the 3D incompressible Navier-Stokes Equation for a set of large data in
$\dot{H}^{-\alpha}(\mathbb{R}^{3})$ or $\dot{H}^{-\alpha}(\mathbb{T}^{3})$ with
$0<\alpha\leq 1/2$. This is achieved by randomizing the initial data and
showing that the energy of the solution modulus the linear part keeps finite
for all $t\geq0$. Moreover, the energy of the solutions is also finite for all
$t>0$. This improves the recent result of Nahmod, Pavlovi\'{c} and Staffilani
on (SIMA, [1])in which $\alpha$ is restricted to $0<\alpha<\frac{1}{4}$.
