In this chapter our basic tool, the dynamic law of large numbers (LLN, whose descriptive explanation was given in Chapter 1 ), is set on a firm mathematical foundation. We prove several versions of this LLN, with different regularity assumptions on the coefficients, with and without major players, and finally with a distinguished (or tagged) player, the latter version being used later, in Part II. convergence is proved with rather precise estimates of the error terms, which is, of course, crucial for any practical applications. For instance, we show that in the case of smooth coefficients, the convergence rates, measuring the difference between the various bulk characteristics of the dynamics of N players and the limiting evolution (corresponding to an infinite number of players), are of order 1 / N. For example, for N=10 players, this difference is about 10%, showing that the number N does not have to be “very large” for the approximation (1.3 ) of “infinitely many players” to give reasonable predictions.