1.1. Prologue: The problem of measure 9 Jordan measurability also inherits many of the properties of elementary measure: Exercise 1.1.6. Let E, F ⊂ Rd be Jordan measurable sets. (1) (Boolean closure) Show that E ∪ F , E ∩ F , E\F , and EΔF are Jordan measurable. (2) (Non-negativity) m(E) ≥ 0. (3) (Finite additivity) If E, F are disjoint, then m(E ∪ F ) = m(E) + m(F ). (4) (Monotonicity) If E ⊂ F , then m(E) ≤ m(F ). (5) (Finite subadditivity) m(E ∪ F ) ≤ m(E) + m(F ). (6) (Translation invariance) For any x ∈ Rd, E + x is Jordan measur- able, and m(E + x) = m(E). Now we give some examples of Jordan measurable sets: Exercise 1.1.7 (Regions under graphs are Jordan measurable). Let B be a closed box in Rd, and let f : B → R be a continuous function. (1) Show that the graph {(x, f(x)) : x ∈ B} ⊂ Rd+1 is Jordan mea- surable in Rd+1 with Jordan measure zero. (Hint: On a compact metric space, continuous functions are uniformly continuous.) (2) Show that the set {(x, t) : x ∈ B 0 ≤ t ≤ f(x)} ⊂ Rd+1 is Jordan measurable. Exercise 1.1.8. Let A, B, C be three points in R2. (1) Show that the solid triangle with vertices A, B, C is Jordan mea- surable. (2) Show that the Jordan measure of the solid triangle is equal to 1 2 |(B − A) ∧ (C − A)|, where |(a, b) ∧ (c, d)| := |ad − bc|. (Hint: It may help to first do the case when one of the edges, say AB, is horizontal.) Exercise 1.1.9. Show that every compact convex polytope6 in Rd is Jordan measurable. Exercise 1.1.10. (1) Show that all open and closed Euclidean balls B(x, r) := {y ∈ Rd : |y − x| r}, B(x, r) := {y ∈ Rd : |y − x| ≤ r} in Rd are Jordan measurable, with Jordan measure cdrd for some constant cd 0 depending only on d. 6A closed convex polytope is a subset of Rd formed by intersecting together finitely many closed half-spaces of the form {x ∈ Rd : x · v ≤ c}, where v ∈ Rd, c ∈ R, and · denotes the usual dot product on Rd. A compact convex polytope is a closed convex polytope which is also bounded.

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