A density approach to Hamilton-Jacobi equations with t-measurable Hamiltonians

In 1985 H. Ishii [Is85] proposed a generalization of the notion of (continuous) viscosity solution for an Hamilton-Jacobi equation with a t-measurable Hamiltonian—that is, a Hamiltonian which is measurable in time and continuous in the other variables. This notion turned out to agree with natural applications, like Control and Differential Games Theory. Since then, several improvements have been achieved for the standard situation when the Hamiltonian is continuous. It is someway an accepted general idea that parallel improvements are likely for t-measurable Hamiltonians as well, though such a job might appear a bit tedious because of the necessarily involved technicalities.