PhD COMPREHENSIVE EXAM—REAL ANALYSIS—June 1993

1. Indicate whether each of the following statements is true or false, and briefly explain your answer. In a) and b), E denotes a Lebesgue measurable subset of R, and m denotes Lebesgue measure on R.

a) If E contains a nonempty open set then m(E) > 0.

b) If m(E) > 0 then E contains a nonempty open set.

c) There exists a continuous increasing function on [0, 1] which is differentiable at no point.

d) Every continuous increasing function on [0, 1] is absolutely continuous.

2. Let m denote Lebesgue measure on R, and assume f ∈ L^{1}(m) with f ≥ 0. Show that

(∗)  ∫ f(x + t) dm(x) = ∫ f(x) dm(x)

if t ∈ R. You may assume that m(E + t) = m(E) if E is a measurable set, but you may not assume the major theorem of which (∗) is a special case.

3. Suppose that μ is a measure, f ∈ L^{1}(μ), and f ≥ 0. Define E_{n} = {x: f(x) ≥ n}. a) Show that lim_{n→∞} ∫_{E_{n}} f dμ = 0.

b) Use a) to show that lim_{n→∞} nμ(E_{n}) = 0.

4. Let X be a nonempty set and μ^{∗} an outer measure on X.

a) If E ⊂ X and μ^{∗}(E) = 0, show that E is μ^{∗}-measurable. (Recall that E is said to be μ^{∗}-measurable if

μ^{∗}(A) = μ^{∗}(A ∩ E) + μ^{∗}(A \ E)

for all A ⊂ X.)

b) Let M denote the collection of μ^{∗}-measurable sets and μ the restriction of μ^{∗} to M. Assuming that μ is a measure, show that it is complete.

5. a) Suppose f is a real-valued continuous function on [0, 1] and ϵ > 0. Show that there exist real numbers a_{0}, a_{1}, …, a_{n} so that, for all x ∈ [0, 1],

|a_{0} + a_{1}x^{2} + a_{2}x^{4} + ⋯ + a_{n}x^{2n} − f(x)| < ϵ.

b) Is the result of a) true if [0, 1] is replaced by [−1, 1]? Explain your answer.

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).