Multivariable Advanced Calculus

The Abstract LebesgueIntegralThe general Lebesgue integral requires a measure space, (Ω, F, µ) and, to begin with,a nonnegative measurable function. I will use Lemma 2.3.3 about interchanging twosupremums frequently. Also, I will use the observation that if {a n } is an increasingsequence of points of [0, ∞] , then sup n a n = lim n→∞ a n which is obvious from thedefinition of sup.8.1 Definition For Nonnegative Measurable Functions8.1.1 Riemann Integrals For Decreasing FunctionsFirst of all, the notationis short for[g < f]{ω ∈ Ω : g (ω) < f (ω)}with other variants of this notation being similar. Also, the convention, 0 · ∞ = 0 willbe used to simplify the presentation whenever it is convenient to do so.Definition 8.1.1 For f a nonnegative decreasing function defined on a finiteinterval [a, b] , define∫ baf (λ) dλ ≡limM→∞∫ baM ∧ f (λ) dλ = supM∫ baM ∧ f (λ) dλwhere a ∧ b means the minimum of a and b. Note that for f bounded,supM∫ baM ∧ f (λ) dλ =∫ baf (λ) dλwhere the integral on the right is the usual Riemann integral because eventually M > f.For f a nonnegative decreasing function defined on [0, ∞),∫ ∞0fdλ ≡ limR→∞∫ R0fdλ = supR>1∫ R0fdλ = supRsupM>0∫ R0f ∧ MdλSince decreasing bounded functions are Riemann integrable, the above definition iswell defined. Now here are some obvious properties.181