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. In this paper we show that Ishii's definition of viscosity solution coincides with the one which would arise by extending by density the standard definition. Namely, we regard a t-measurable Hamiltonian H as an element of the closure (for suitable topologies) of a class of continuous Hamiltonians. On the other hand, we show that the set of Ishii's (sub-, super-) solutions for H is nothing but the limit set of the (sub-, super-) solutions corresponding to continuous Hamiltonians approaching H. This put us in the condition of establishing comparison, existence, and regularity results by deriving them from the analogous results for the case of continuous Hamiltonians.