Comprehensive Exam Real Analysis August 1999

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

1. Let f : [0, 1] → ℝ be a bounded Lebegue measurable function. Prove that for every ϵ > 0 there is a simple function g on [0, 1] such that ∥f − g∥_{∞} < ϵ.

2. Suppose that (X, A, μ) is a finite measure space and that (f_{n})_{n=1}^{∞} is a sequence of realvalued measurable functions on X. For each of the following types of convergence of a sequence (f_{n})_{n=1}^{∞} to a function f U: uniform convergence A: convergence almost everywhere M: convergence in measure indicate which of the others are implied (but do not prove) and in each case for which the implication is false give a counterexample.

3. Let (A_{n})_{n=1}^{∞} be a sequence of disjoint intervals such that m(A_{n}) = 2^{−n} for each n, where m is Lebesgue measure, and ∪_{n=1}^{∞}A_{n} = [0, 1]. Define f : [0, 1] → ℝ by f(x) = n for all x ∈ A_{n}. If g is a measurable function, consider the expression

T_{f}(g) = ∫_{0}^{1} f(x)g(x) dx.

For each of the following Banach spaces of (equivalence classes of) functions X determine whether T_{f} defines a bounded linear functional on X and justify your answer.

(a) X = L_{1}[0, 1] (b) X = L_{2}[0, 1] (c) X = L_{∞}[0, 1] (d) X = C[0, 1]

4. Suppose that μ is a finite measure on some measurable space (X, A), f, g ∈ L_{1}(μ) and define

ν(A) = ∫_{A} f dμ  and  γ(A) = ∫_{A} g dμ for all A ∈ A.

(a) Suppose ν ⊥ γ. What is the relationship between f and g?

(b) Suppose ν ≪ γ. What is the relationship between f and g?

5. Suppose that f : [0, ∞) → ℝ is continuous and bounded. Prove that if

∫_{0}^{∞} f(x)e^{−nx} dx = 0 for n = 1, 2, …,

then f = 0.

6. Let K(x, y) ∈ L_{∞}([0, 1] × [0, 1]) and define

T f(x) = ∫_{0}^{1} K(x, y)f(y) dy.

(a) Prove or disprove: T is a well defined bounded linear operator on L_{2}[0, 1].

(b) Let (A_{i})_{i=1}^{k} and (B_{i})_{i=1}^{k} be two finite sequences of measurable subsets of [0, 1]. If K(x, y) = ∑_{i=1}^{k} 1_{A_{i}}(x)1_{B_{i}}(y), for all x, y, prove that the range of T, T (L_{2}[0, 1]), is finite dimensional and deduce that \overline{T (\{f : ∥f∥_{2} ≤ 1\})} is compact.