References for lesson 3: Existence of Nash equilibria
 

Border, Kim [1985]: Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge Univ. Press, Cambridge.
Good reference for fixed points and their applications in game theory and economics

Dunford, Nelson, and Jacob T. Schwartz [1958]:Linear operators, Interscience, New York (NY).
A classical reference for functional analysis and measure theory

Glicksberg, Irving L. [1952]: A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points, Proc. Amer. Math. Soc., 3, 170-174.
Extension of Nash's theorem to the case of infinite sets of pure strategies (spaces with compact sets of pure strategies and continuous payoffs)

Kakutani, Shizuo [1941]: A Generalization of Brouwer's Fixed Point Theorem, Duke Math. J., 8, 457-458.
The fixed point theorem used for the proof of existence for Nash equilibria (for mixed extensions of finite games)

Klein, Erwin and Anthony C. Thompson [1984]: Theory of correspondences, Wiley, New York.
A good reference for correspondences and topologies on ``hyperspaces'' (sets of subsets of a topological space). With applications to game theory and mathematical economics

Osborne, Martin and Ariel Rubinstein [1994]: A course in Game Theory, MIT Press, Cambridge (MA).
A very good general reference.