130 1. Measure theory The one-sided Hardy-Littlewood maximal inequality has an analogue in this setting: Lemma 1.6.26 (One-sided Hardy-Littlewood inequality). Let F : [a, b] → R be a continuous monotone non-decreasing function, and let λ 0. Then we have m({x ∈ [a, b] : D+F (x) ≥ λ}) ≤ F (b) − F (a) λ and similarly, for the other three Dini derivatives of F . If F is not assumed to be continuous, then we have the weaker inequality m({x ∈ [a, b] : D+F (x) ≥ λ}) ≤ C F (b) − F (a) λ for some absolute constant C 0. Remark 1.6.27. Note that if one naively applies the fundamental theo- rems of calculus, one can formally see that the first part of Lemma 1.6.26 is equivalent to Lemma 1.6.16. We cannot, however, use this argument rigor- ously because we have not established the necessary fundamental theorems of calculus to do this. Nevertheless, we can borrow the proof of Lemma 1.6.16 without diﬃculty to use here, and this is exactly what we will do. Proof. We just prove the continuous case and leave the discontinuous case as an exercise. It suﬃces to prove the claim for D+F by reflection (replacing F (x) with −F (−x), and [a, b] with [−b, −a]), the same argument works for D−F , and then this trivially implies the same inequalities for D+F and D−F . By modifying λ by an epsilon, and dropping the endpoints from [a, b] as they have measure zero, it suﬃces to show that m({x ∈ (a, b) : D+F (x) λ}) ≤ F (b) − F (a) λ . We may apply the rising sun lemma (Lemma 1.6.17) to the continuous function G(x) := F (x) − λx. This gives us an at most countable family of intervals In = (an,bn) in (a, b), such that G(bn) ≥ G(an) for each n, and such that G(y) ≤ G(x) whenever a ≤ x ≤ y ≤ b and x lies outside of all of the In. Observe that if x ∈ (a, b), and G(y) ≤ G(x) for all x ≤ y ≤ b, then D+F (x) ≤ λ. Thus we see that the set {x ∈ (a, b) : D+F (x) λ} is contained in the union of the In, and so by countable additivity, m({x ∈ (a, b) : D+F (x) λ}) ≤ n bn − an. But we can rearrange the inequality G(bn) ≤ G(an) as bn −an ≤ F (bn)−F (an) λ . From telescoping series and the monotone nature of F we have ∑ n F (bn) −

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