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In this paper we prove the almost sure existence of global weak solution tothe 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 andshowing that the energy of the solution modulus the linear part keeps finitefor 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 Staffilanion (SIMA, [1])in which $\alpha$ is restricted to $0<\alpha<\frac{1}{4}$.

Almost Sure Existence of Global Weak Solutions to the 3D Incompressible Navier-Stokes Equation