  We prove existence and uniqueness of solutions to a class of porous media
equations driven by the fractional Laplacian when the initial data are positive
finite Radon measures on the Euclidean space. For given solutions without a
prescribed initial condition, the problem of existence and uniqueness of the
initial trace is also addressed. By the same methods we can also treat weighted
fractional porous media equations, with a weight that can be singular at the
origin, and must have a sufficiently slow decay at infinity (power-like). In
particular, we show that the Barenblatt-type solutions exist and are unique.
Such a result has a crucial role in [24], where the asymptotic behavior of
solutions is investigated. Our uniqueness result solves a problem left open,
even in the non-weighted case, in [42]
