PhD COMPREHENSIVE EXAM—REAL ANALYSIS—August 1995

1. Indicate whether each of the following statements is true or false, and briefly explain your answer.

a) Every countable subset of R is Lebesgue measurable, with Lebesgue measure 0.

b) Every subset of R of Lebesgue measure zero is countable.

c) There exists a closed subset of [0,1] with empty interior and Lebesgue measure one.

2. Let (X, M, μ) be a measure space, and suppose {E_{j}}_{j=1}^{∞} ⊂ M. Show that if ∑_{j=1}^{∞} μ(E_{j}) < ∞, then μ(F) = 0, where F = ∩_{k=1}^{∞}{∪_{j=k}^{∞}E_{j}}. Interpretation: F = {x: x belongs to infinitely E_{j}}, so μ(F) = 0 says that almost every x ∈ X belongs to at most finitely many E_{j}.

3. a) State carefully the Monotone Convergence Theorem, Fatou’s Lemma, and the Dominated Convergence Theorem.

b) Prove one of the three theorems in part a); you may assume either of the other two in your proof.

c) Evaluate with explanation

lim_{n→∞} ∫_{0}^{∞} \frac{n sin(x/n)}{x(1 + x^{2})} dx .

4. Suppose that μ and ν are measures on a measurable space (X, M).

a) Define what it means for μ to be absolutely continuous with respect to ν and what it means for μ to be singular with respect to ν.

b) Prove using only the definitions that if μ is both absolutely continuous and singular with respect to ν then μ is the zero measure.

5. If f is a continuous real-valued function on [0,1], the moments of f are the numbers ∫_{0}^{1} f(x)x^{n} dx for n = 0, 1, … Use the Stone-Weierstrass Theorem to show that two functions are identical if they have the same sequence of moments.

6. Recall: If X is a Banach space, {x_{n}} is a sequence in X, and x ∈ X, then x_{n} → x weakly means that ϕ(x_{n}) → ϕ(x) for all ϕ in the dual of X.

a) Fix p so that 1 < p < ∞, and suppose that f_{n} → f weakly in ℓ^{p}(N). Show that f_{n} → f pointwise, that is, for all m ∈ N, f_{n}(m) → f(m) as n → ∞.

b) Find a sequence {f_{n}} in ℓ^{1}(N) so that ∥f_{n}∥_{1} = 1 for all n and f_{n} → 0 pointwise, but f_{n} ↛ 0 weakly in ℓ^{1}(N).