AN UPCROSSING INEQUALITY WITH APPLICATIONS

The theorem of Lebesgue that a function of bounded variation has a derivative almost everywhere, the ergodic theorem of Birkhoff, and the martingale theorem have the common property of being nonconstructive; they are not true in intuitionist mathematics. (See Brouwer [1] for an example of a function of bounded variation that does not have a derivative almost everywhere.) This paper developed from an attempt to save the phenomena…