Ph.D. COMPREHENSIVE EXAM REAL ANALYSIS January 1997

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

1. Let A and B be σ-algebras of subsets of topological spaces X and Y, respectively.

(a) Suppose that f is a function from X into Y. Define the phrases f is measurable and f is Borel measurable.

(b) Suppose that f is a function from X into the real numbers. Define the phrase f is measurable.

(c) Using only the definitions prove that if f and g are measurable real-valued functions on X then f + g is measurable.

2. Let λ denote Lebesgue measure on ℝ. Prove or disprove: (a) L^{2}(λ) ⊂ L^{∞}(λ).

(b) C_{0}(ℝ) is dense (in the norm topology) in L^{∞}(λ).

3. (a) Define absolute continuity for measures and state the Radon-

Nikodym Theorem.

(b) For each measurable subset A of ℝ let

μ(A) = ∫_{A} exp(−x^{2}) dλ(x),

where λ is Lebesgue measure. Show that λ << μ and determine \frac{dλ}{dμ}.

4. (a) Define the weak topology of a Banach space X.

(b) Suppose that x and (x_{n})_{n=1}^{∞} are elements of ℓ^{2}, where ℓ^{2} = L^{2}(μ) and μ is counting measure on the natural numbers ℕ. Prove that if (x_{n})_{n=1}^{∞} converges to x weakly then lim_{n} x_{n}(k) = x(k) for all k ∈ ℕ.

(c) Prove that if in addition to the assumptions in the previous part lim_{n} ∥x_{n}∥ = ∥x∥, then lim_{n} ∥x_{n} − x∥ = 0.

5. Suppose that f is integrable on [0, ∞) (with respect to Lebesgue measure). Define for x ∈ ℝ,

g(x) = ∫_{0}^{∞} f(t) exp(−|x|t) dt.

(a) Prove that g is measurable.

(b) If f is zero on [0, ϵ] for some ϵ > 0, show that g is Lebesgue integrable.