On the existence of equitable cake divisions

Let the cake be represented by the unit interval and let each player have a valuation expressed by a nonatomic probability measure. A cake division is said to be equitable if the value of the piece assigned to a player by his measure is the same for all players. We show that for any number n of players in any order an equitable division exists, giving each player a contiguous cake piece. Moreover, there is at least one order in which the common value is not smaller than 1/n.