Existence and uniqueness of nonsimple multiple SLE

We prove the existence and uniqueness of multiple SLE$_\kappa$ associated with any given link pattern for $\kappa\in (4,6]$. We also have the uniqueness for $\kappa\in (6,8)$. The multiple SLE$_\kappa$ law is constructed by first inductively constructing a $\sigma$-finite multiple SLE$_\kappa$ measure and then normalizing the measure whenever it is finite. The total mass of the measure satisfies the conformal covariance, asymptotics and PDE for multiple SLE$_\kappa$ partition functions in the literature subject to the assumption that it is smooth.