1.3. The Lebesgue integral 53 One can also define the upper unsigned Lebesgue integral Rd f(x) dx := inf h≥f h simple Simp Rd h(x) dx, but we will use this integral much more rarely. Note that both integrals take values in [0, +∞], and that the upper Lebesgue integral is always at least as large as the lower Lebesgue integral. In the definition of the lower unsigned Lebesgue integral, g is required to be bounded by f pointwise everywhere, but it is easy to see that one could also require g to just be bounded by f pointwise almost everywhere without affecting the value of the integral, since the simple integral is not affected by modifications on sets of measure zero. The following properties of the lower Lebesgue integral are easy to es- tablish: Exercise 1.3.10 (Basic properties of the lower Lebesgue integral). Let f, g : Rd [0, +∞] be unsigned functions (not necessarily measurable). (i) (Compatibility with the simple integral) If f is simple, then we have Rd f(x) dx = Rd f(x) dx = Simp Rd f(x) dx. (ii) (Monotonicity) If f g pointwise almost everywhere, then we have Rd f(x) dx Rd g(x) dx and Rd f(x) dx Rd g(x) dx. (iii) (Homogeneity) If c [0, +∞), then Rd cf(x) dx = c Rd f(x) dx. (The claim unfortunately fails for c = +∞, but this is somewhat tricky to show.) (iv) (Equivalence) If f, g agree almost everywhere, then Rd f(x) dx = Rd g(x) dx and Rd f(x) dx = Rd g(x) dx. (v) (Superadditivity) Rd f(x) + g(x) dx Rd f(x) dx + Rd g(x) dx. (vi) (Subadditivity of upper integral) Rd f(x)+g(x) dx Rd f(x) dx+ Rd g(x) dx. (vii) (Divisibility) For any measurable set E, one has Rd f(x) dx = Rd f(x)1E(x) dx + Rd f(x)1Rd\E(x) dx. (viii) (Horizontal truncation) As n ∞, Rd min(f(x),n) dx converges to Rd f(x) dx. (ix) (Vertical truncation) As n ∞, Rd f(x)1|x|≤n dx converges to Rd f(x) dx. Hint: From Exercise 1.2.11 one has m(E {x : |x| n}) m(E) for any measurable set E.
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