Comprehensive Examination Real Analysis January 1998

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

1. Suppose that (X, A) is a measurable space. Show that if f and g are measurable real-valued functions on X, then f + g is measurable.

2. Let (X, A, μ) be a measure space and let (f_{n})_{n=1}^{∞} be a sequence of measurable real-valued functions on X.

(a) Define (f_{n})_{n=1}^{∞} converges in measure.

(b) Describe an example of a sequence which converges in measure

but does not converge almost everywhere.

3. Let m be Lebesgue measure on R and fix a measurable set A_{0} with m(A_{0}) < ∞. Define the measure ν on the Lebesgue measurable sets by ν(E) = m(A_{0} ∩ E). Find the Radon–Nikodym derivative of ν with respect to m. Justify your answer.

4. Let f be a function from [0, 1] into R.

(a) Define f is absolutely continuous.

(b) Show that if f is a Lipschitz function on [0,1], it is absolutely continuous. (f is Lipschitz if there exists a constant K such that |f(x) − f(y)| ≤ K|x − y| for all x, y.)

(c) Is every absolutely continuous function on [0,1] Lipschitz?

Justify your answer.

5. Suppose that μ is a finite regular Borel measure, i.e., a Radon measure, on [−1, 1] such that for n = 0, 1, 2, …, ∫ x^{n} dμ = 0. Show that μ = 0.

6. Let C([0, 1]) be the space of continuous functions on [0, 1] with ∥f∥ = sup{|f(x)| : x ∈ [0, 1]}.

(a) State the Riesz Representation Theorem for C([0, 1])^{∗}?

(b) Suppose that (f_{n}) is a sequence in C([0, 1]) with ∥f_{n}∥ ≤ 1 for all n such that lim_{n→∞} f_{n}(t) = 0 for all t, show that (f_{n}) converges to 0 weakly.