Comprehensive Examination Real Analysis June 1998

A passing mark will be given if 5 problems are completely solved.

1. Let X = [0, 1] = Y, A = {A : A ⊂ X} and B = {B : B ⊂ X, B is countable or X \ B is countable}.

(a) Define σ-algebra and show that B is a σ-algebra.

(b) Show that there is a non-measurable map from (X, B) to (Y, A) and that every map

from (Y, A) to (X, B) is measurable.

2. Suppose that f is a monotone non-decreasing real-valued function on [0, 1]. (a) Show that f has at most countably many points of discontinuity.

(b) Prove or disprove: There is a sequence of absolutely continuous functions (f_{n}) such

that lim_{n→∞} f_{n}(t) = f(t) for all t ∈ [0, 1].

3. Suppose that f : X → R is measurable with respect to the measure space (X, M, μ)

and, for any x ∈ X, 0 ≤ f(x) ≤ 1. Let E = f^{−1}(1). Also, suppose that g ∈ L^{1}(μ). Prove that

lim_{n→∞} ∫ f^{n} g dμ = ∫_{E} g dμ.

4. Let

f(x) = \{ \begin{matrix}x & if 0 ≤ x < .5 \\ 2x & if .5 ≤ x ≤ 1 \end{matrix}

(a) What is the Lebesgue-Stieltjes measure ν on [0,1] induced by f? (What is an explicit formula for calculating ν(A) for all Borel sets A?)

(b) State the Lebesgue Decomposition Theorem.

(c) Determine the Lebesgue Decomposition of ν with respect to Lebesgue measure on

[0, 1].

5. Suppose that 1 ≤ r < p < ∞.

(a) Show that L_{p}[0, 1] ⊂ L_{r}[0, 1].

(b) Prove or disprove: L_{p}[0, ∞) ⊂ L_{r}[0, ∞).

6. Suppose that (f_{n})_{n=1}^{∞} is a sequence of differentiable functions in C([0, 1]) such that for each rational q ∈ [0, 1], lim_{n→∞} f_{n}(q) exists and there exists a constant M < ∞ such that |f′_{n}(t)| ≤ M for all n ∈ N and t ∈ [0, 1]. Show that there is a continuous function f such that (f_{n})_{n=1}^{∞} converges uniformly to f.