We consider the incompressible Navier-Stokes equations on the whole space $R^n$, $n\geq 3$. Especially, we consider the Cauchy problem in the frame work of the weak Lebesgue space $L^{n,\infty}(R^n)$ with scale invariant forces in $BC([0,T);L^{n/3, \infty}(R^n))$. It is well known that $C^{\infty}_0(R^n)$ is not dense in $L^{n,\infty}(R^n)$ and that the Stokes semigroup is not strongly continuous at $t=0$. For this difficulty, the existence of local mild solutions and its uniqueness are not yet completely clarified in $L^{n,\infty}(R^n)$. Firstly we consider the local existence of weak mild solutions of (N-S) with restriction of the singularity of initial data and external forces. Due to the local solution, we may apply it to the uniqueness theorem within the weak mild solutions in the class $BC([0,T);L^{n,\infty}(R^n))$. As an application of the uniqueness theorem, we are able to investigate the regularity of weak mild solutions by the singularity of initial data.