1.2. Lebesgue measure 35 (i) (Empty set) m(∅) = 0. (ii) (Countable additivity) If E1,E2,... ⊂ Rd is a countable sequence of disjoint Lebesgue measurable sets, then m( ∞ n=1 En)= ∑∞ n=1 m(En). (iii) (Translation invariance) If E is Lebesgue measurable and x ∈ Rd, then m(E + x) = m(E). (iv) (Normalisation) m([0, 1]d) = 1. Hint: First show that m must match elementary measure on elementary sets, then show that m is bounded by outer measure. Exercise 1.2.24 (Lebesgue measure as the completion of elementary mea- sure). The purpose of the following exercise is to indicate how Lebesgue measure can be viewed as a metric completion of elementary measure in some sense. To avoid some technicalities we will not work in all of Rd, but in some fixed elementary set A (e.g. A = [0, 1]d). (i) Let 2A := {E : E ⊂ A} be the power set of A. We say that two sets E, F ∈ 2A are equivalent if EΔF is a null set. Show that this is a equivalence relation. (ii) Let 2A/ ∼ be the set of equivalence classes [E] := {F ∈ 2A : E ∼ F } of 2A with respect to the above equivalence relation. Define a distance d: 2A/ ∼ ×2A/ ∼→ R+ between two equivalence classes [E], [E ] by defining d([E], [E ]) := m∗(EΔE ). Show that this distance is well defined (in the sense that m(EΔE ) = m(F ΔF ) whenever [E] = [F ] and [E ] = [F ]) and gives 2A/ ∼ the structure of a complete metric space. (iii) Let E ⊂ 2A be the elementary subsets of A, and let L ⊂ 2A be the Lebesgue measurable subsets of A. Show that L/ ∼ is the closure of E/ ∼ with respect to the metric defined above. In particular, L/ ∼ is a complete metric space that contains E/ ∼ as a dense subset in other words, L/ ∼ is a metric completion of E/ ∼. (iv) Show that Lebesgue measure m: L → R+ descends to a continuous function m: L/ ∼→ R+, which by abuse of notation we shall still call m. Show that m: L/ ∼→ R+ is the unique continuous exten- sion of the analogous elementary measure function m: E/ ∼→ R+ to L/ ∼. For a further discussion of how measures can be viewed as completions of elementary measures, see §2.1 of An epsilon of room, Vol. I. Exercise 1.2.25. Define a continuously differentiable curve in Rd to be a set of the form {γ(t) : a ≤ t ≤ b} where [a, b] is a closed interval and γ : [a, b] → Rd is a continuously differentiable function.

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