PROBLEM SET 4

1. Compute the integral of the Cantor-Lebesgue function ∫_{0}^{1} F (x)dm from Chapter 2.

2. If f_{n}(x) = sin^{2}(nx) find both ∫_{0}^{2π} lim inf_{n→∞} f_{n}(x)dm and lim inf_{n→∞} ∫_{0}^{2π} f_{n}(x)dm.

3. What is lim_{n→∞} ∫_{−∞}^{∞} x^{n}e^{−n|x|}dm? Find the limit and prove it is correct.

4. Suppose {f_{n}} is a sequence of functions that converges almost everywhere to a function f and define F_{n} = sup_{k=1,...,n} |f_{n}|. Show that if the integrals of F_{n} remain bounded as n → ∞ then lim_{n} ∫ f_{n}dm = ∫ fdm.

5. Given a measureable function f define its “maximal function” as

Mf(x) = sup_{t>0} \frac{1}{2t} ∫_{x−t}^{x+t} |f(y)|dm(y).

If f is integrable on the reals, does Mf have to be integrable?

6. Show that ∑_{n=1}^{∞} cos^{n}(2^{n}x) converges for a.e. x, but diverges on a dense set of x’s.