46 1. Measure theory Simpabs(Rd) of absolutely integrable simple functions is closed under addi- tion and scalar multiplication by complex numbers, and is thus a complex vector space. The properties of the unsigned simple integral then can be used to deduce analogous properties for the complex-valued integral: Exercise 1.3.2 (Basic properties of the complex-valued simple integral). Let f, g : Rd → C be absolutely integrable simple functions. (i) (*-linearity) We have Simp Rd f(x) + g(x) dx = Simp Rd f(x) dx + Simp Rd g(x) dx and (1.11) Simp Rd cf(x) dx = c × Simp Rd f(x) dx for all c ∈ C. Also, we have Simp Rd f(x) dx = Simp Rd f(x) dx. (ii) (Equivalence) If f and g agree almost everywhere, then we have Simp Rd f(x) dx = Simp Rd g(x) dx. (iii) (Compatibility with Lebesgue measure) For any Lebesgue measur- able E, one has Simp Rd 1E(x) dx = m(E). (Hints: Work out the real-valued counterpart of the linearity property first. To establish (1.11), treat the cases c 0,c = 0,c = −1 separately. To deal with the additivity for real functions f, g, start with the identity f + g = (f + g)+ − (f + g)− = (f+ − f−) + (g+ − g−) and then rearrange the second inequality so that no subtraction appears.) Furthermore, show that the complex-valued simple integral f → Simp Rd f(x) dx is the only map from the space Simpabs(Rd) of absolutely integrable simple functions to C that obeys all of the above properties. We now comment further on the fact that (simple) functions that agree almost everywhere, have the same integral. We can view this as an asser- tion that integration is a noise-tolerant operation: One can have “noise” or “errors” in a function f(x) on a null set, and this will not affect the final value of the integral. Indeed, once one has this noise tolerance, one can even

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