1.6. Differentiation theorems 131 F (an) ≤ F (b) − F (a) (this is easiest to prove by first working with a finite subcollection of the intervals (an,bn), and then taking suprema), and the claim follows. The discontinuous case is left as an exercise. Exercise 1.6.31. Prove Lemma 1.6.26 in the discontinuous case. (Hint: The rising sun lemma is no longer available, but one can use either the Vitali-type covering lemma (which will give C = 3) or the Besicovitch lemma (which will give C = 2), by modifying the proof of Theorem 1.6.20. Exercise 1.6.32. Let μ be a finite Borel measure on R. Show that |{x ∈ R : sup r0 1 2r μ([x − r, x + r]) ≥ λ}| ≤ C λ μ(R) for any λ 0 and some absolute constant C 0. Sending λ → ∞ in the above lemma (cf. Exercise 1.3.18), and then send- ing [a, b] to R, we conclude as a corollary that all the four Dini derivatives of a continuous monotone non-decreasing function are finite almost every- where. So to prove Theorem 1.6.25 for continuous monotone non-decreasing functions, it suﬃces to show that (1.29) holds for almost every x. In view of the trivial inequalities, it suﬃces to show that D+F (x) ≤ D−F (x) and D−F (x) ≤ D+F (x) for almost every x. We will just show the first inequal- ity, as the second follows by replacing F with its reflection x → −F (−x). It will suﬃce to show that for every pair 0 r R of real numbers, the set E = Er,R := {x ∈ R : D+F (x) R r D−F (x)} is a null set, since by letting R, r range over rationals with R r 0 and taking countable unions, we would conclude that the set {x ∈ R : D+F (x) D−F (x)} is a null set (recall that the Dini derivatives are all non-negative when F is non-decreasing), and the claim follows. Clearly E is a measurable set. To prove that it is null, we will establish the following estimate: Lemma 1.6.28 (E has density less than one). For any interval [a, b] and any 0 r R, one has m(Er,R ∩ [a, b]) ≤ r R |b − a|. Indeed, this lemma implies that E has no points of density, which by Exercise 1.6.24 forces E to be a null set. Proof. We begin by applying the rising sun lemma to the function G(x) := rx + F (−x) on [−b, −a] the large number of negative signs present here is needed in order to properly deal with the lower left Dini derivative D−F . This gives an at most countable family of disjoint intervals −In = (−bn, −an) in (−b, −a), such that G(−an) ≥ G(−bn) for all n, and such that G(−x) ≤

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.