1.2. Lebesgue measure 29 (Hint: Some of these deductions are either trivial or very easy. To deduce (i) from (vi), use the ε/2n trick to show that E is contained in a Lebesgue mea- surable set Eε with m∗(EεΔE) ≤ ε, and then take countable intersections to show that E differs from a Lebesgue measurable set by a null set.) Exercise 1.2.8. Show that every Jordan measurable set is Lebesgue mea- surable. Exercise 1.2.9 (Middle thirds Cantor set). Let I0 := [0, 1] be the unit interval, let I1 := [0, 1/3]∪[2/3, 1] be I0 with the interior of the middle third interval removed, let I2 := [0, 1/9]∪[2/9, 1/3]∪[2/3, 7/9]∪[8/9, 1] be I1 with the interior of the middle third of each of the two intervals of I1 removed, and so forth. More formally, write In := a1,...,an∈{0,2} [ n i=1 ai 3i , n i=1 ai 3i + 1 3n ]. Let C := ∞ n=1 In be the intersection of all the elementary sets In. Show that C is compact, uncountable, and a null set. Exercise 1.2.10. (This exercise presumes some familiarity with point-set topology.) Show that the half-open interval [0, 1) cannot be expressed as the countable union of disjoint closed intervals. (Hint: It is easy to prevent [0, 1) from being expressed as the finite union of disjoint closed intervals. Next, assume for the sake of contradiction that [0, 1) is the union of infinitely many closed intervals, and conclude that [0, 1) is homeomorphic to the middle thirds Cantor set, which is absurd. It is also possible to proceed using the Baire category theorem (§1.7 of An epsilon of room, Vol. I.) For an additional challenge, show that [0, 1) cannot be expressed as the countable union of disjoint closed sets. Now we look at the Lebesgue measure m(E) of a Lebesgue measurable set E, which is defined to equal its Lebesgue outer measure m∗(E). If E is Jordan measurable, we see from (1.2) that the Lebesgue measure and the Jordan measure of E coincide, thus Lebesgue measure extends Jordan measure. This justifies the use of the notation m(E) to denote both Lebesgue measure of a Lebesgue measurable set, and Jordan measure of a Jordan measurable set (as well as elementary measure of an elementary set). Lebesgue measure obeys significantly better properties than Lebesgue outer measure, when restricted to Lebesgue measurable sets: Lemma 1.2.15 (The measure axioms). (i) (Empty set) m(∅) = 0. (ii) (Countable additivity) If E1,E2,... ⊂ Rd is a countable of disjoint Lebesgue measurable sets, then m( ∞ n=1 En)= ∑sequence ∞ n=1 m(En).

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