We study proton-(anti)proton collisions at the LHC or Tevatron in the presence of experimental restrictions on the hadronic final state and for generic parton momentum fractions. At the scale of the hard interaction, factorization does not yield standard parton distribution functions (PDFs) for the initial state. The measurement restricting the hadronic final state introduces a new scale and probes the proton prior to the hard collision. This corresponds to evaluating the PDFs at the scale . After the proton is probed, the incoming hard parton is contained in an initial-state jet, and the hard collision occurs between partons inside these jets rather than inside protons. The proper description of such initial-state jets requires “beam functions”. At the scale , the beam function factorizes into a convolution of calculable Wilson coefficients and PDFs. Below , the initial-state evolution is described by the usual PDF evolution which changes , while above it is governed by a different renormalization group evolution that sums double logarithms of and leaves fixed. As an example, we prove a factorization theorem for “isolated Drell-Yan”, where is restricted to have no central jets. We comment on the extension to cases where the hadronic final state contains a certain number of isolated central jets.