The games which have been discussed up to now have been restricted by two conditions mentioned in Chapter 1. These are (1) every move has a finite number of alternatives and (2) every play contains a finite number of moves. It is clear that these restrictions can be relaxed in a wide variety of combinations. Actually, only three general types of infinite games have been studied to any extent: the matrix games with a denumerable number or a continuum of pure strategies  and infinite game trees in which condition (1) above is satisfied . In this chapter, we will only consider matrix games with a continuum of pure strategies. Among the infinite games, they have been studied the most intensively and, indeed, they seem to present the most cogent interpretations for actual situations. To make precise our exact domain of investigation, we formulate the following definition.
DEFINITION 22. A zero-sum two-person game on the unit square is given
by any real valued function A(x, y) defined for 0 ≦ x, y ≦ 1. The values
of x constitute the pure strategies for P1 while the values of y are the pure
strategies for P2.
The interpretation of games on the unit square is exactly analogous to that of finite matrix games. The game is a two-move game in which P2 makes his choice of a value y in ignorance of the choice of a value x by P1. The amount that P2 then pays to P1 is A(x, y).
It is immediately seen that not only are pure strategies insufficient to solve games on the unit square but also finite mixtures of pure strategies will not serve for this purpose. This difficulty is illustrated by the following example.
EXAMPLE. Consider the set of all continuous functions f(x) defined on a closed interval [0, 1]. It is a well-known fact that such a function is uniquely specified by giving its values for the rational values of the argument and hence