Multivariable Advanced Calculus

202 THE ABSTRACT LEBESGUE INTEGRALChoose n 2 > n 1 such thatn 3 > n 2 such thatµ(x : |f(x) − f n2 (x)| ≥ 1/2) < 1/2 2 ,µ(x : |f(x) − f n3 (x)| ≥ 1/3) < 1/2 3 ,etc. Now consider what it means for f nk (x) to fail to converge to f(x). Then useProblem 8.10. Suppose (Ω, µ) is a finite measure space (µ (Ω) < ∞) and S ⊆ L 1 (Ω). Then S issaid to be uniformly integrable if for every ε > 0 there exists δ > 0 such that if Eis a measurable set satisfying µ (E) < δ, then∫|f| dµ < εEfor all f ∈ S. Show S is uniformly integrable and bounded in L 1 (Ω) if thereexists an increasing function h which satisfies{∫}h (t)lim = ∞, sup h (|f|) dµ : f ∈ S < ∞.t→∞ tΩS is bounded if there is some number, M such that∫|f| dµ ≤ Mfor all f ∈ S.11. Let (Ω, F, µ) be a measure space and suppose f, g : Ω → (−∞, ∞] are measurable.Prove the sets{ω : f(ω) < g(ω)} and {ω : f(ω) = g(ω)}are measurable. Hint: The easy way to do this is to write{ω : f(ω) < g(ω)} = ∪ r∈Q [f < r] ∩ [g > r] .Note that l (x, y) = x−y is not continuous on (−∞, ∞] so the obvious idea doesn’twork.12. Let {f n } be a sequence of real or complex valued measurable functions. LetS = {ω : {f n (ω)} converges}.Show S is measurable. Hint: You might try to exhibit the set where f n convergesin terms of countable unions and intersections using the definition of a Cauchysequence.13. Suppose u n (t) is a differentiable function for t ∈ (a, b) and suppose that for t ∈(a, b),|u n (t)|, |u ′ n(t)| < K nwhere ∑ ∞n=1 K n < ∞. Show∞∑∞∑( u n (t)) ′ = u ′ n(t).n=1Hint: This is an exercise in the use of the dominated convergence theorem andthe mean value theorem.n=1