# Convergence of empirical measure in case of proliferation

I am currently working on the theory of mean field limits of interacting particles. Here are two slides of a talk from an Italian researcher:

I don't understand why he calls $u(t,x)$ a time dependent probability measure (Pic 2) because first $S_t^N$ is not a probability measure as there is no reason why it should be less or equal than 1 as the particles proliferate and second the total time derivative of $u(t,x)$ then should be 0 which is obviously not the case if it fulfills the given PDE. Can somebody help?