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Ph.D. Comprehensive Exam: Real Analysis August 2010 1. True or False? If True, prove; if False, give a counterexample. a. Suppose that A and B are subsets of R such that the Lebesgue outer measure of A is zero, i.e., m∗ (A) = 0 then m∗ (A ∪ B) = m∗ (B). b. If A is a subset of [0, 1] and m∗ (A) = 1, then A contains an interval of positive length. c. If fn → 0 in Lp ([0, 1]) for some p ∈ [1, ∞) then fn → 0 a.e. d. If m is the Lebesgue measure on R2 and if f : R → R is Borel-measurable then m{(x, f (x)) : x ∈ R} = 0. 2. Evaluate lim Z ∞ −x(3+n−2 sin(x)) e 1 + n−2 x n→∞ 0 dx justifying your answer. 3. Assume that f is non-negative and Lebesgue integrable on Rd . Show that for every > 0 there exists a δ > 0 such that if A is a measurable subset of Rd such that µ(A) ≤ δ then Z f dµ ≤ . A 4. Let µ, ν, λ be σ-finite positive measures on (X, M). a. Show that µ µ + λ. b. Show that if ν µ and λ µ then ν + λ µ and d(ν + λ) dν dλ = + dµ dµ dµ µ − a.e. c. Show that if λ ν µ then λ µ and dλ dλ dν = dµ dν dµ 1 µ − a.e.