An essential part of high-energy hadronic collisions is the soft hadronic activity that underlies the primary hard interaction. It includes soft radiation from the primary hard partons, secondary multiple parton interactions (MPI), and factorization-violating effects. The invariant mass spectrum of the leading jet in Z+jet and H+jet events is directly sensitive to these effects, and we use a QCD factorization theorem to predict its dependence on the jet radius R, jet p_{T}, jet rapidity, and partonic process for both the perturbative and nonperturbative components of primary soft radiation. We prove that the nonperturbative contributions involve only odd powers of R, and the linear R term is universal for quark and gluon jets. The hadronization model in Pythia8 agrees well with these properties. The perturbative soft initial state radiation (ISR) has a contribution that depends on the jet area in the same way as the underlying event, but this degeneracy is broken by dependence on the jet p_{T}. The size of this soft ISR contribution is proportional to the color state of the initial partons, yielding the same positive contribution for gg→Hg and gq→Zq, but a negative interference contribution for qq[over ¯]→Zg. Hence, measuring these dependencies allows one to separate hadronization, soft ISR, and MPI contributions in the data.