Approachability in infinite dimensional spaces

The approachability theorem of Blackwell (1956b) is extended to infinite dimensional spaces. Two players play a sequential game whose payoffs are random variables. A set C of random variables is said to be approachable by player 1 if he has a strategy that ensures that the difference between the average payoff and its closest point in C, almost surely converges to zero. Necessary conditions for a set to be approachable are presented. Received February 2002/Final version July 2002  I acknowledge Eilon Solan for his helpful comments. The author acknowledges the support of the Israel Science Foundation, grant no. 178/99.