706 CHAPTER 21. THE BOCHNER INTEGRAL

Thus, unlike the classical definition of the derivative, when a function and its deriva-tive are both in L1, it has a representative f̂ which equals the function a.e. such thatf̂ can be recovered from its derivative. Recall the well known example of this notworking out which is based on the Cantor function which you should see in a realanalysis course. This function had zero derivative a.e. and yet it climbed from 0 to1 on the unit interval. Thus one could not recover it from integrating its classicalderivative.