10 1. Measure theory (2) Establish the crude bounds 2 √ d d ≤ cd ≤ 2d. (An exact formula for cd is cd = 1 d ωd, where ωd := 2πd/2 Γ(d/2) is the volume of the unit sphere Sd−1 ⊂ Rd and Γ is the Gamma function, but we will not derive this formula here.) Exercise 1.1.11. This exercise assumes familiarity with linear algebra. Let L: Rd → Rd be a linear transformation. (1) Show that there exists a non-negative real number D such that m(L(E)) = Dm(E) for every elementary set E (note from previous exercises that L(E) is Jordan measurable). (Hint: Apply Exercise 1.1.3 to the map E → m(L(E)).) (2) Show that if E is Jordan measurable, then L(E) is also, and m(L(E)) = Dm(E). (3) Show that D = | det L|. (Hint: Work first with the case when L is an elementary transformation, using Gaussian elimination. Alter- natively, work with the cases when L is a diagonal transformation or an orthogonal transformation, using the unit ball in the latter case, and use the polar decomposition.) Exercise 1.1.12. Define a Jordan null set to be a Jordan measurable set of Jordan measure zero. Show that any subset of a Jordan null set is a Jordan null set. Exercise 1.1.13. Show that (1.1) holds for all Jordan measurable E ⊂ Rd. Exercise 1.1.14 (Metric entropy formulation of Jordan measurability). Define a dyadic cube to be a half-open box of the form i1 2n , i1 + 1 2n × . . . × id 2n , id + 1 2n for some integers n, i1,...,id. Let E ⊂ Rd be a bounded set. For each integer n, let E∗(E, 2−n) denote the number of dyadic cubes of sidelength 2−n that are contained in E, and let E∗(E, 2−n) be the number of dyadic cubes7 of sidelength 2−n that intersect E. Show that E is Jordan measurable if and only if lim n→∞ 2−dn(E∗(E, 2−n) − E∗(E, 2−n)) = 0, in which case one has m(E) = lim n→∞ 2−dnE∗(E, 2−n) = lim n→∞ 2−dnE∗(E, 2−n). 7This quantity could be called the (dyadic) metric entropy of E at scale 2−n.

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