Ph.D. COMPREHENSIVE EXAM REAL ANALYSIS May 1991

Justify your answers. Six complete solutions will constitute a passing score.

1. Suppose that (Ω, B) is a measurable space and that μ is a finitely additive nonnegative real-valued function on B. If μ(Ω) < ∞, prove that μ is countably additive if and only if for all (A_{n}) ⊂ B, A_{n} ↓ ∅, μ(A_{n}) → 0. If μ(Ω) = ∞, is this still true? (A_{n} ↓ ∅ means that A_{n} ⊃ A_{n+1} for all n and ∩_{n}A_{n} = ∅.)

2. Consider the following two statements about a Lebesgue measurable function f on [0, 1]. a. f is continuous almost everywhere. b. There is a continuous function g such that f = g almost everywhere. What is the relationship between a. and b.? (In other words prove or disprove

each of the statements a. ⇒ b. and b. ⇒ a.)

3. Suppose that f and g are functions from ℝ to ℝ such that f is Borel measurable and g is continuous. Prove or disprove: f ∘ g is Borel measurable.

4. Suppose that p, q, and r ∈ [1, ∞) and \frac{1}{p} + \frac{1}{q} = \frac{1}{r}. Show that fg ∈ L_{r} for all g ∈ L_{q} if and only if f ∈ L_{p}. You may assume that the measure space is [0, 1] with Lebesgue measure.

5. Is the cardinality of the set of Borel subsets of [0, 1] the same as the cardinality of the set of Lebesgue measurable subsets of [0, 1] ?

6. Suppose that g is a continuous periodic odd function on ℝ and that f ∈ L_{1}([0, 1]). Compute lim_{t→∞} ∫_{ℝ} f(x)g(tx) dλ(x) where λ denotes Lebesgue measure.

7. Let P = {h(x, y) : h(x, y) = f(x)g(y) for some f, g ∈ C[0, 1]}. Is the “sup” norm closure of P equal to C([0, 1] × [0, 1])?

8. Consider the following two conditions on a subset F of L_{1}^{loc}(ℝ). (Below λ denotes Lebesgue measure.) a. For every ϵ > 0 there is a δ > 0 such that for all f ∈ F, if A is a measurable

set with λ(A) < δ then ∫_{A} |f|dλ < ϵ. b. For every ϵ > 0 there is an M ∈ ℝ such that for all f ∈ F, ∫_{{x:|f(x)|≥M}} |f|dλ <

ϵ. Prove or disprove: a. ⟺ b.

9.

a. Carefully state the Radon-Nikodym Theorem. b. Suppose (Ω, B, μ) is a measure space with μ(Ω) = 1 and that A is a sub-σ-

algebra of B. Show that for every g ∈ L_{1}(Ω, B, μ) there is an A-measurable function f on Ω such that for every A ∈ A, ∫_{A} fdμ = ∫_{A} gdμ.

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