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52 1. Measure theory (iii) The (magnitudes of the) positive and negative parts of Re(f) and Im(f) are unsigned measurable functions. (iv) f −1(U) is Lebesgue measurable for every open set U ⊂ C. (v) f −1(K) is Lebesgue measurable for every closed set K ⊂ C. From the above exercise, we see that the notion of complex-valued mea- surability and unsigned measurability are compatible when applied to a function that takes values in [0, +∞) = [0, +∞] ∩ C everywhere (or almost everywhere). Exercise 1.3.8. (i) Show that every continuous function f : Rd → C is measurable. (ii) Show that a function f : Rd → C is simple if and only if it is measurable and takes on at most finitely many values. (iii) Show that a complex-valued function that is equal almost every- where to an measurable function, is itself measurable. (iv) Show that if a sequence fn of complex-valued measurable functions converges pointwise almost everywhere to an complex-valued limit f, then f is also measurable. (v) If f : Rd → C is measurable and φ: C → C is continuous, show that φ ◦ f : Rd → C is measurable. (vi) If f, g are measurable functions, show that f + g and fg are mea- surable. Exercise 1.3.9. Let f : [a, b] → R be a Riemann integrable function. Show that if one extends f to all of R by defining f(x) = 0 for x ∈ [a, b], then f is measurable. 1.3.3. Unsigned Lebesgue integrals. We are now ready to integrate un- signed measurable functions. We begin with the notion of the lower unsigned Lebesgue integral, which can be defined for arbitrary unsigned functions (not necessarily measurable): Definition 1.3.12 (Lower unsigned Lebesgue integral). Let f : Rd →[0, +∞] be an unsigned function (not necessarily measurable). We define the lower unsigned Lebesgue integral Rd f(x) dx to be the quantity Rd f(x) dx := sup 0≤g≤f g simple Simp Rd g(x) dx where g ranges over all unsigned simple functions g : Rd → [0, +∞] that are pointwise bounded by f.