1.2. Lebesgue measure 31 for every ε 0, while from countable subadditivity we have m( ∞ n=1 En) ≤ ∞ n=1 m(En). The claim follows. Finally, we handle the case when the En are not assumed to be bounded or closed. Here, the basic idea is to decompose each En as a countable disjoint union of bounded Lebesgue measurable sets. First, decompose Rd as the countable disjoint union Rd = ∞ m=1 Am of bounded measurable sets Am for instance, one could take the annuli Am := {x ∈ Rd : m − 1 ≤ |x| m}. Then each En is the countable disjoint union of the bounded measurable sets En ∩ Am for m = 1, 2,..., and thus m(En) = ∞ m=1 m(En ∩ Am) by the previous arguments. In a similar vein, ∞ n=1 En is the countable disjoint union of the bounded measurable sets En ∩ Am for n, m = 1, 2,..., and thus m( ∞ n=1 En) = ∞ n=1 ∞ m=1 m(En ∩ Am), and the claim follows. From Lemma 1.2.15 one of course can conclude finite additivity, m(E1 ∪ . . . ∪ Ek) = m(E1) + . . . + m(Ek), whenever E1,...,Ek ⊂ Rd are Lebesgue measurable sets. We also have another important result: Exercise 1.2.11 (Monotone convergence theorem for measurable sets). (i) (Upward monotone convergence) Let E1 ⊂ E2 ⊂ . . . ⊂ Rn be a countable non-decreasing sequence of Lebesgue measurable sets. Show that m( ∞ n=1 En) = limn→∞ m(En). (Hint: Express ∞ n=1 En as the countable union of the lacunae En\ n−1 n =1 En .) (ii) (Downward monotone convergence) Let Rd ⊃ E1 ⊃ E2 ⊃ . . . be a countable non-increasing sequence of Lebesgue measurable sets. If at least one of the m(En) is finite, show that m( ∞ n=1 En) = limn→∞ m(En). (iii) Give a counterexample to show that in the hypothesis that at least one of the m(En) is finite in the downward monotone convergence theorem cannot be dropped.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.