The joint continuity of the expected payoff functions

The joint continuity of the expected payoff functions is not obvious at all. A general proof was presented by Glycopantis and Muir [D. Glycopantis, A. Muir, 2000. Continuity of the payoff functions, Economic Theory 16, 239-244]. An alternative proof was obtained by Zarichnyi [M. Zarichnyi, 2004. Continuity of the payoff function revisited, Economics Bulletin 3, 1-4]. A proof can also be obtained by using Billingsley [P. Billingsley, 1968. Convergence of Probability Measures, Wiley, New York, Theorem 3.2, p. 21] or by invoking a result of Balder [E.J. Balder, 1988. Generalized equilibrium results for games with incomplete information, Mathematics of Operations Research 13, 265-276, Theorem 3.1, p. 273]. In this note an alternative proof of the joint continuity of the expected payoff functions is given that is based on measure theoretic techniques.