1.3. The Lebesgue integral 45 (iii) (Vanishing) We have Simp Rd f(x) dx = 0 if and only if f is zero almost everywhere. (iv) (Equivalence) If f and g agree almost everywhere, then we have Simp Rd f(x) dx = Simp Rd g(x) dx. (v) (Monotonicity) If f(x) ≤ g(x) for almost every x ∈ Rd, then Simp Rd f(x) dx ≤ Simp Rd g(x) dx. (vi) (Compatibility with Lebesgue measure) For any Lebesgue measur- able E, one has Simp Rd 1E(x) dx = m(E). Furthermore, show that the simple unsigned integral f → Simp Rd f(x) dx is the only map from the space Simp+(Rd) of unsigned simple functions to [0, +∞] that obeys all of the above properties. We can now define an absolutely convergent counterpart to the simple unsigned integral. This integral will soon be superceded by the absolutely Lebesgue integral, but we give it here as motivation for the more general notion of integration. Definition 1.3.6 (Absolutely convergent simple integral). A complex-valued simple function f : Rd → C is said to be absolutely integrable of Simp Rd |f(x)| dx ∞. If f is absolutely integrable, the integral Simp Rd f(x) dx is defined for real signed f by the formula Simp Rd f(x) dx := Simp Rd f+(x) dx − Simp Rd f−(x) dx where f+(x) := max(f(x), 0) and f−(x) := max(−f(x), 0) (note that these are unsigned simple functions that are pointwise dominated by |f| and thus have finite integral), and for complex-valued f by the formula13 Simp Rd f(x) dx := Simp Rd Re f(x) dx + i Simp Rd Im f(x) dx. Note from the preceding exercise that a complex-valued simple func- tion f is absolutely integrable if and only if it has finite measure support (since finiteness almost everywhere is automatic). In particular, the space 13Strictly speaking, this is an abuse of notation as we have now defined the simple integral Simp Rd three different times, for unsigned, real signed, and complex-valued simple functions, but one easily verifies that these three definitions agree with each other on their common domains of definition, so it is safe to use a single notation for all three.

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