140 1. Measure theory Now we return to the monotone case. Inspired by the Lipschitz case, one may hope to recover equality in Proposition 1.6.37 for such functions F . However, there is an important obstruction to this, which is that all the variation of F may be concentrated in a set of measure zero, and thus unde- tectable by the Lebesgue integral of F . This is most obvious in the case of a discontinuous monotone function, such as the (appropriately named) Heav- iside function F := 1[0,+∞) it is clear that F vanishes almost everywhere, but F (b) F (a) is not equal to [a,b] F (x) dx if b and a lie on opposite sides of the discontinuity at 0. In fact, the same problem arises for all jump functions: Exercise 1.6.47. Show that if F is a jump function, then F vanishes al- most everywhere. (Hint: Use the density argument, starting from piecewise constant jump functions and using Proposition 1.6.37 as the quantitative estimate.) One may hope that jump functions—in which all the fluctuation is con- centrated in a countable set—are the only obstruction to the second fun- damental theorem of calculus holding for monotone functions, and that as long as one restricts attention to continuous monotone functions, that one can recover the second fundamental theorem. However, this is still not true, because it is possible for all the fluctuation to now be concentrated, not in a countable collection of jump discontinuities, but instead in an uncount- able set of zero measure, such as the middle thirds Cantor set (Exercise 1.2.9). This can be illustrated by the key counterexample of the Cantor function, also known as the Devil’s staircase function. The construction of this function is detailed in the exercise below. Exercise 1.6.48 (Cantor function). Define the functions F0,F1,F2,... : [0, 1] R recursively as follows: 1. Set F0(x) := x for all x [0, 1]. 2. For each n = 1, 2,... in turn, define Fn(x) := 1 2 Fn−1(3x) if x [0, 1/3], 1 2 if x (1/3, 2/3), 1 2 + 1 2 Fn−1(3x 2) if x [2/3, 1]. (i) Graph F0, F1, F2, and F3 (preferably on a single graph). (ii) Show that for each n = 0, 1,..., Fn is a continuous monotone non- decreasing function with Fn(0) = 0 and Fn(1) = 1. (Hint: Induct on n.)
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