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Real Analysis Comprehensive Examination–Math 921/922 Friday, May 29, 2015, 12:00-6:00 p.m. • Work 6 out of 8 problems. • Each problem is worth 20 points, the parts of each problem don’t necessarily carry the same weight. • Write on one side of the paper only and hand your work in order. • Throughout the exam m denotes the Lebesgue measure, L stands for the Lebesgue σ-algebra on R, BR denotes the Borel σ-algebra on R and (X, M, µ) indicates a general measure space unless otherwise indicated. (1) Let µ∗ be an outer measure induced by a finite pre-measure µ0 on an algebra A ⊂ 2X and let M∗ be the associated Caratheodory σ-algebra. (a) Prove that µ∗ M∗ is complete. (b) Prove that for any set E there is B ⊂ Aσδ such that µ∗ (E) = µ∗ (B). (2) The parts of this problem are not directly related. (a) Suppose f : R → R is monotone increasing. Prove that f is Borel-measurable and deduce that any function g : R → R of bounded variation is also Borel-measurable. (b) Let (X, M, µ) be a measure space with positive measure µ. Prove that the set {χE : E ∈ M, µ(E) < ∞} is closed in L1 (X, µ). (3) Suppose f : (0, 1] → R is Riemann integrable on [a, 1] for every 0 < a < 1 and the improper Riemann integral R1 R 0 f (x)dx is convergent (“R” indicates “Riemann”). (a) Prove that if f is bounded on (0, 1), then f ∈ L1 ((0, 1)). Z 1 (b) Give an example of an R-valued function f on (0, 1] (it need not be continuous) such that R f (x)dx is 0 convergent, but f is not in L1 ((0, 1)). (c) Suppose f is not necessarily bounded, but changes sign finitely many times in (0, 1]. Prove f ∈ L1 ((0, 1)). (4) Let (X, M, µ) be a measure space with positive finite measure µ. Suppose fn : X → [0, ∞), n ∈ N, are Mmeasurable and converge point-wise µ-a.e.R to a function f ∈ L1 (X, µ). Assume for every ε > 0 there exists δ > 0 such that E ∈ M with µ(E) < δ implies E fn dµ < ε for n ≥ N , for some N ∈ N. Prove Z Z fn dµ = f dµ . lim n→∞ (5) For i = 1, 2, let µi , νi be finite measures on measurable spaces (Xi , Mi ) such that νi µi . (Remark: in this problem the finiteness assumption is not essential and can be replaced with σ-finiteness, but you need not consider this general case.) Prove the following (you may use without a proof the fact that ν1 × ν2 µ1 × µ2 ): d(ν1 × ν2 ) dν1 dν2 (x1 , x2 ) = (x1 ) · (x2 ) d(µ1 × µ2 ) dµ1 dµ2 µ1 × µ2 -a.e. (6) Let G : R → R be continuous strictly increasing and µG denote the associated Lebesgue-Stieltjes measure such that µG ((x, y]) = G(y) − G(x). Let G(a) = c, G(b) = d. In this problem you may not cite any change of variable theorems. (a) Assume f : R → R is bounded and Borel-measurable. Prove that Z Z f dm = f (G(x))dµG (x) [c,d] [a,b] (b) Extend the result to the case when f is Borel-measurable and in L1 ([c, d]) (instead of being bounded). (7) Let X = [0, 2015]. (a) Prove that there is C > 0 such that for every absolutely continuous function f : X → R with f (0) = 0 kf kL1 (X,m) ≤ Ckf 0 kL1 (X,m) (b) Show that the statement (a) is false if the condition “absolutely continuous” is replaced by “m-a.e. differentiable on X with f 0 ∈ L1 (X, m)”. (8) Let (X, M, µ) be a measure space where µ is positive and σ-finite. (a) Suppose 1 ≤ p < ∞. Prove that weak limits in Lp (X, µ) are unique. In other words, show that if a sequence converges weakly to two functions f, g ∈ Lp (X, µ), then these functions must be equal µ-a.e. (b) Assume p, q > 1 and p1 + 1q = 1. Suppose fk ∈ Lp (X, µ) for k ∈ N and fk → f in Lp (X, µ) as k → ∞. Next, assume functions {gk }k∈N form a bounded subset of Lq (X, µ) and gk → g weakly in Lq (X, µ) as k → ∞. Prove that fk gk → f g weakly in L1 (X, µ).