Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment XIII.

1. Problem 1. a) Compute the Hardy-Littlewood maximal function M1_{[0,1]}(x), of the characteristic function of segment [0, 1].

b) Let f(x) = x^{−1}(log x)^{−2} if x ∈ (0, 1/2), and zero on the rest of R. Prove that f is integrable. On the other hand, show that Mf(x) ≥ |2x log(2x)|^{−1}, if x ∈ (0, 1/4), so

that ∫_{0}^{1} Mf(x)dx = ∞.

c) Assume that both f and Mf are integrable on R^{n}. Prove that f(x) = 0 almost everywhere. Hint. Show that there exists a constant c = c(f) such that Mf(x) ≥ c(f)|x|^{−n} whenever x is sufficiently large.

2. Problem 2. Let f(x) = 1 if x ∈ [0, 1], and zero otherwise. Define h_{c}(x) := sup_{n∈N} n^{c}f(nx). Prove that

a) h_{c} is integrable on R if c ∈ (0, 1).

b) h_{1} is in weak-type L but not integrable on R.

c) h_{c} is not in weak-type L if c > 1.

Hint. For a.e. x ∈ R, h_{c}(x) = ∑_{n=1}^{∞} 1_{(1/(n+1),1/n]}(x)n^{c}.

3. Problem 3. For any set E ⊂ R^{2}, the boundary ∂E of E is, by definition, the closure of E minus the interior of E.

a) Show that E is Lebesgue measurable whenever m(∂E) = 0.

b) Suppose that E is the union of a (possibly uncountable) collection of closed discs in R^{2} whose radii are at least 1 and at most 2. Use a) to show that E is Lebesgue measurable. Hint. Observe that the density of the boundary points is always < 1. On the other hand, if m(∂E) > 0, then the Lebesgue theorem gives 1 for almost all x ∈ ∂E.

c) Show that the conclusion of b) is true even when the radii are unrestricted.

d) Show that some unions of closed discs of radius 1 are not Borel sets. Hint. What if all the discs touch the straight line?

4. Problem 4. Let f be periodic with the period 1 and Lebesgue integrable with respect to the Lebesgue measure on [0, 1]. Define the Riemann sum

f_{n}(x) := \frac{1}{n} ∑_{k=1}^{n} f(x + \frac{k}{n}), n ∈ N.

ℵ) Prove that a sequence (f_{2^{n}}(x))_{n=1}^{∞} converges almost everywhere to A = ∫_{0}^{1} f(x)dx by completing the following steps.

a) Assume that f ≥ 0. Show that s_{N}f(x) := sup_{0≤n≤N} f_{2^{n}}(x) is in weak type. More precisely, prove that

∀λ > 0 λm(E_{λ}) ≤ ∫_{E_{λ}} f(x)dx, E_{λ} := {x ∈ [0, 1] : s_{N}f(x) > λ}.

Hint. Define E_{n} := {x ∈ [0, 1] : f_{2^{n}}(x) > λ}, 0 ≤ n ≤ N, N ∈ N, and observe that

E_{λ} = ∪_{n=1}^{N}E_{n} = ∪_{n=1}^{N}A_{n},  A_{N} := E_{N},  A_{n} := E_{n} \ (∪_{i=n+1}^{N}E_{i}), n = 0, ..., N − 1,

where A_{n} are disjoint. Prove that all A_{n} are periodic with period 2^{−n}, and conclude that

∫_{A_{n}} f(x)dx = ∫_{A_{n}} f_{2^{n}}(x)dx ≥ λm(A_{n}).

This implies ℵ).

b) Letting N → ∞ prove that

λm({x ∈ [0, 1] : sup_{n∈{0}∪N} f_{2^{n}}(x) > λ}) ≤ ∫_{0}^{1} f(x)dx.

c) Define Φ(x) := \overline{lim}_{n→∞} f_{2^{n}}(x). Prove that Φ has arbitrary small periods, hence Φ(x) = const almost everywhere. Hint. Use Problem 1, d) of Assignment XI.

d) Prove that Φ ≤ A by taking any λ < Φ. Conclude that Φ = A. Hint. Replace f(x) by −f(x). ℶ) Show that in fact any sequence (f_{m_{n}}(x))_{n=1}^{∞} converges to A almost everywhere, provided m_{n−1} divides m_{n}. ℷ) Find an integrable periodic function f such that a sequence of Riemann sums (f_{n}(x))_{n=1}^{∞} diverges almost everywhere. Hint. Assume (do not prove) that for every irrational x there are infinitely many integers n such that |x − m/n| < 1/n^{2} for some integer m. Fix 0 < α < 1/2. Take f(x) = |x|^{−1+α} for |x| ≤ 1/2, and define f(x) for all other x by periodicity. Then f(x − m/n) > f(1/n^{2}) = n^{2−2α}, so f_{n}(x) > n^{1−2α}.