34 1. Measure theory Define a Gδ set to be a countable intersection ∞ n=1 Un of open sets, and an Fσ set to be a countable union ∞ n=1 Fn of closed sets. Exercise 1.2.19. Let E ⊂ Rd. Show that the following are equivalent: (i) E is Lebesgue measurable. (ii) E is a Gδ set with a null set removed. (iii) E is the union of a Fσ set and a null set. Remark 1.2.17. From the above exercises, we see that when describing what it means for a set to be Lebesgue measurable, there is a tradeoff between the type of approximation one is willing to bear, and the type of things one can say about the approximation. If one is only willing to approximate to within a null set, then one can only say that a measurable set is approximated by a Gδ or a Fσ set, which is a fairly weak amount of structure. If one is willing to add on an epsilon of error (as measured in the Lebesgue outer measure), one can make a measurable set open dually, if one is willing to take away an epsilon of error, one can make a measurable set closed. Finally, if one is willing to both add and subtract an epsilon of error, then one can make a measurable set (of finite measure) elementary, or even a finite union of dyadic cubes. Exercise 1.2.20 (Translation invariance). If E ⊂ Rd is Lebesgue measur- able, show that E + x is Lebesgue measurable for any x ∈ Rd, and that m(E + x) = m(E). Exercise 1.2.21 (Change of variables). If E ⊂ Rd is Lebesgue measurable, and T : Rd → Rd is a linear transformation, show that T(E) is Lebesgue measurable, and that m(T(E)) = | det T|m(E). We caution that if T : Rd → Rd is a linear map to a space Rd of strictly smaller dimension than Rd, then T(E) need not be Lebesgue measurable see Exercise 1.2.27. Exercise 1.2.22. Let d, d ≥ 1 be natural numbers. (i) If E ⊂ Rd and F ⊂ Rd , show that (md+d )∗(E × F ) ≤ (md)∗(E)(md )∗(F ), where (md)∗ denotes d-dimensional Lebesgue measure, etc. (ii) Let E ⊂ Rd, F ⊂ Rd be Lebesgue measurable sets. Show that E × F ⊂ Rd+d is Lebesgue measurable, with md+d (E × F ) = md(E) · md (F ). (Note that we allow E or F to have infinite measure, and so one may have to divide into cases or take advantage of the monotone convergence theorem for Lebesgue measure, Exercise 1.2.11.) Exercise 1.2.23 (Uniqueness of Lebesgue measure). Show that Lebesgue measure E → m(E) is the only map from Lebesgue measurable sets to [0, +∞] that obeys the following axioms:

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