126 1. Measure theory it turns out, from the theory of Radon measures, all locally finite Borel mea- sures have this property, but we will not prove this here see §1.10 of An epsilon of room, Vol. I.) Establish the Hardy-Littlewood maximal inequality μ({x ∈ R : sup x∈I 1 μ(I) I |f(y)| dμ(y) ≥ λ}) ≤ 2 λ R |f(y)| dμ(y) for any absolutely integrable function f ∈ L1(μ), where the supremum ranges over all open intervals I that contain x. Note that this essentially generalises Exercise 1.6.11, in which μ is replaced by Lebesgue measure. (Hint: Repeat the proof of the usual Hardy-Littlewood maximal inequality, but use the Besicovich covering lemma in place of the Vitali-type covering lemma. Why do we need the former lemma here instead of the latter?) Exercise 1.6.23 (Cousin’s theorem). Prove Cousin’s theorem: given any function δ : [a, b] → (0, +∞) on a compact interval [a, b] of positive length, there exists a partition a = t0 t1 . . . tk = b with k ≥ 1, together with real numbers tj ∗ ∈ [tj−1,tj] for each 1 ≤ j ≤ k and tj − tj−1 ≤ δ(tj ∗). (Hint: Use the Heine-Borel theorem, which asserts that any open cover of [a, b] has a finite subcover, followed by the Besicovitch covering lemma.) This theo- rem is useful in a variety of applications related to the second fundamental theorem of calculus, as we shall see below. The positive function δ is known as a gauge function. Now we turn to consequences of the Lebesgue differentiation theorem. Given a Lebesgue measurable set E ⊂ Rd, call a point x ∈ Rd a point of density for E if m(E∩B(x,r)) m(B(x,r)) → 1 as r → 0. Thus, for instance, if E = [−1, 1]\{0}, then every point in (−1, 1) (including the boundary point 0) is a point of density for E, but the endpoints −1, 1 (as well as the exterior of E) are not points of density. One can think of a point of density as being an “almost interior” point of E it is not necessarily the case that one can fit a small ball B(x, r) centered at x inside of E, but one can fit most of that small ball inside E. Exercise 1.6.24. If E ⊂ Rd is Lebesgue measurable, show that almost every point in E is a point of density for E, and almost every point in the complement of E is not a point of density for E. Exercise 1.6.25. Let E ⊂ Rd be a measurable set of positive measure, and let ε 0. (i) Using Exercise 1.6.15 and Exercise 1.6.24, show that there exists a cube Q ⊂ Rd of positive sidelength such that m(E ∩ Q) (1 − ε)m(Q).

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