Time-dependent measurable Hamilton-Jacobi equations

We study the equation u(t) + H(x, Du) = 0 in R-N x (0, + infinity) with Hamiltonian H(x, p) measurable with respect to the state variable, and convex and coercive in p. We introduce a notion of solution based on viscosity test functions, appropriate averages in measure-theoretic sense of the Hamiltonian, and t-partial-sup-convolutions. We get existence results, comparison principles, and stability properties. We show that our solutions are uniform limits of viscosity solutions (in the sense of Crandall and Lions) of approximated continuous equations.