1.3. The Lebesgue integral 51 Remark 1.3.10. Lemma 1.3.9 tells us that if f : Rd → [0, +∞] is measur- able, then f −1(E) is Lebesgue measurable for many classes of sets E. How- ever, we caution that it is not necessarily the case that f −1(E) is Lebesgue measurable if E is Lebesgue measurable. To see this, we let C be the Cantor set C := { ∞ j=1 aj3−j : aj ∈ {0, 2} for all j} and let f : R → [0, +∞] be the function defined by setting f(x) := ∞ j=1 2bj3−j whenever x ∈ [0, 1] is not a terminating binary decimal, and so has a unique binary expansion x = ∑ ∞ j=1 bj2−j for some bj ∈ {0, 1}, and f(x) := 0 otherwise. We thus see that f takes values in C, and is bijective on the set A of non-terminating decimals in [0, 1]. Using Lemma 1.3.9, it is not diﬃcult to show that f is measurable. On the other hand, by modifying the construction from the previous notes, we can find a subset F of A which is non-measurable. If we set E := f(F ), then E is a subset of the null set C and is thus itself a null set but f −1(E) = F is non-measurable, and so the inverse image of a Lebesgue measurable set by a measurable function need not remain Lebesgue measurable. However, we will later see that it is still true that f −1(E) is Lebesgue measurable if E has a slightly stronger measurability property than Lebesgue measurability, namely Borel measurability see Exercise 1.4.29(iii). Now we can define the concept of a complex-valued measurable function. As discussed earlier, it will be convenient to allow for such functions to only be defined almost everywhere, rather than everywhere, to allow for the possibility that the function becomes singular or otherwise undefined on a null set. Definition 1.3.11 (Complex measurability). An almost everywhere defined complex-valued function f : Rd → C is Lebesgue measurable, or measurable for short, if it is the pointwise almost everywhere limit of complex-valued simple functions. As before, there are several equivalent definitions: Exercise 1.3.7. Let f : Rd → C be an almost everywhere defined complex- valued function. Then the following are equivalent: (i) f is measurable. (ii) f is the pointwise almost everywhere limit of complex-valued simple functions.

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