32 1. Measure theory Exercise 1.2.12. Show that any map E → m(E) from Lebesgue mea- surable sets to elements of [0, +∞] that obeys the above empty set and countable additivity axioms will also obey the monotonicity and countable subadditivity axioms from Exercise 1.2.3, when restricted to Lebesgue mea- surable sets of course. Exercise 1.2.13. We say that a sequence En of sets in Rd converges point- wise to another set E in Rd if the indicator functions 1En converge pointwise to 1E. (i) Show that if the En are all Lebesgue measurable, and converge pointwise to E, then E is Lebesgue measurable also. (Hint: Use the identity 1E(x) = lim infn→∞ 1En (x) or 1E(x) = lim supn→∞ 1En (x) to write E in terms of countable unions and intersections of the En.) (ii) (Dominated convergence theorem) Suppose that the En are all con- tained in another Lebesgue measurable set F of finite measure. Show that m(En) converges to m(E). (Hint: Use the upward and downward monotone convergence theorems, Exercise 1.2.11.) (iii) Give a counterexample to show that the dominated convergence theorem fails if the En are not contained in a set of finite measure, even if we assume that the m(En) are all uniformly bounded. In later sections we will generalise the monotone and dominated con- vergence theorems to measurable functions instead of measurable sets see Theorem 1.4.43 and Theorem 1.4.48. Exercise 1.2.14. Let E ⊂ Rd. Show that E is contained in a Lebesgue measurable set of measure exactly equal to m∗(E). Exercise 1.2.15 (Inner regularity). Let E ⊂ Rd be Lebesgue measurable. Show that m(E) = sup K⊂E,K compact m(K). Remark 1.2.16. The inner and outer regularity properties of measure can be used to define the concept of a Radon measure (see §1.10 of An epsilon of room, Vol. I.). Exercise 1.2.16 (Criteria for finite measure). Let E ⊂ Rd. Show that the following are equivalent: (i) E is Lebesgue measurable with finite measure. (ii) (Outer approximation by open) For every ε 0, one can contain E in an open set U of finite measure with m∗(U\E) ≤ ε.

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