This paper studies existence problems in L¹ for the linear, space-inhomogeneous Boltzmann equation with periodic or (perfectly) absorbing boundary conditions under realistic assumptions on the cross-sections. By an iteration technique, solutions are first constructed to an integral equation variant of the transport equation in the case of bounded impact parameters and an L¹ type of cross-sections. They are then used to study the existence of solutions of a measure form of the transport equation in the case of unbounded impact parameters. These solutions conserve mass. Estimates of their higher moments are also given. In particular the results hold for inverse kth-power forces with 3 < k ≤ 5.