Summary: It is shown that given any nonnegative, continuous function \(g\) on the lateral boundary of a cylinder, a Radon measure \(\mu\) satisfying (1.4) on the bottom and a Radon measure \(\lambda\) at the corner of the bottom, there is a unique continuous very weak solution to the porous medium equation in the slow diffusion case which is continuous up to the boundary for positive time. Moreover, it is equal to \(g\) along the lateral boundary, and takes \((\mu,\lambda)\) as its initial trace.