Comprehensive Exam Real Analysis June 1999

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

1. Let (X, A) be a measurable space.

(a) Define the notion of a σ-finite measure on (X, A).

(b) Let X = ℝ, A be the Borel σ-algebra, and μ be Lebesgue measure. Give an example of a measurable set which is not countable, not bounded and has measure 0.

2. Suppose that f ∈ L_{1}([0, 1], B, m), where m denotes Lebesgue measure and B is the Borel σ-algebra. Show that for every ϵ > 0 there is a function g such that g([0, 1]) is a finite set A, for each a ∈ A, g^{−1}(a) is a finite union of intervals, and ∥f − g∥_{1} < ϵ.

3. Suppose that f, g ∈ L_{1}(ℝ), g > 0 a.e., and that f ≥ 0 a.e. Prove that for every ϵ > 0 there is a δ > 0 such that if A is measurable and

∫_{A} g < δ then ∫_{A} f < ϵ.

4. Suppose that (f_{n})_{n=1}^{∞} is a sequence of functions from [0, 1] to ℝ and there is a finite number M such that |f_{n}(x)| ≤ M and |f′_{n}(x)| ≤ M for all x ∈ [0, 1]. Prove that there is a subsequence of (f_{n})_{n=1}^{∞} which converges uniformly on [0, 1].

5. Let J be the formal identity map i.e., Jf = f. Prove or disprove each of the following. (a) J is well defined and continuous from L_{1}[0, 1] to L_{2}[0, 1].

(b) J is well defined and continuous from L_{2}[0, 1] to L_{1}[0, 1].

(c) If S is a linear subspace of L_{2}[0, 1] and S is closed in the L_{1}[0, 1] norm, then S is closed in L_{2}[0, 1].

6. For y ∈ [0, 1] and f ∈ C[0, 1] show that the formula

T f(y) = ∫_{0}^{1} f(x) sin(xy) dx

defines a bounded linear operator T from C[0, 1] to itself. (Here C[0, 1] has the supremum norm.)