Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment XI.

1. Problem 1. a) Let E := ∪_{k=1}^{∞}E_{k} ⊂ R, E_{k} := [2^{−2k−1}, 2^{−2k}] ∪ [−2^{−2k}, −2^{−2k−1}]. Show that at 0, E has upper density 2/3 and lower density 1/3. In other words, show that

\overline{lim}_{δ→0} \frac{m(E ∩ (−δ, δ))}{2δ} = \frac{2}{3},  \underline{lim}_{δ→0} \frac{m(E ∩ (−δ, δ))}{2δ} = \frac{1}{3}.

b) Construct a set E ⊂ R which has density 1/2 at the origin.

c) Suppose that E is a measurable set of real numbers with arbitrary small periods, i.e., there exists a sequence (p_{i})_{i=1}^{∞}, p_{i} > 0, p_{i} → 0 as i → ∞, so that E + p_{i} = E ∀i. Prove that either m(E) = 0 or m(R \ E) = 0. Hint: Pick α ∈ R, and put f(x) := m(E ∩ [α, x]) for x > α. Show that

f(x + p_{i}) − f(x − p_{i}) = f(y + p_{i}) − f(y − p_{i}), α + p_{i} < x < y.

What does this imply about f′(x) if m(E) > 0?

d) Suppose that f is Lebesgue measurable on R with periods s, t, s/t ∈ R\Q (∀x ∈ R, f(x + t) = f(x), f(x + s) = f(x)). Prove that f(x) = const almost everywhere, but f need not to be a constant. Hint: Apply c) to the set {x ∈ R : f(x) > λ}, where λ > 0.

2. Problem 2. In this exercise f is a continuous function on [a, b], F, E ⊂ [a, b] are measurable.

a) Let F be closed. Prove that f(F) is closed.

b) Let F be F_{σ}, then f(F) is F_{σ}. Definition (Lusin). We say that f has N-property, if m(f(E)) = 0 ∀E : m(E) = 0.

c) Prove that absolutely continuous function has N-property.

d) Show that f(E) is measurable ∀E measurable iff f has N-property. Hint. To prove part if, observe that f does not have N-property, implies the existence of E, such that m(E) = 0, and μ^{∗}(f(E)) > 0. Then there exists B ⊂ f(E) which is not measurable (why?). Consider A := f^{−1}(B) ⊂ E. Show that A is measurable, and get a contradiction.

e) Prove that an absolutely continuous function maps measurable sets onto measurable sets.

f) Construct a continuous one-to-one mapping of [0, 1] onto [0, 2] which does not have N-property. Hint: Consider f(x) = c(x) + x, where c(x) is the Cantor ladder, defined in Problem 3, Assignment 7. What is the measure of the image of the Cantor set?

3. Problem 3. This exercise helps to understand “the change of variables” in the Lebesgue integral. Let (a, b) ⊂ R be bounded.

a) Choose intervals W_{n} ⊂ (a, b) in such away that ∪_{n}W_{n} is dense in (a, b), and the set K := (a, b) \ ∪_{n}W_{n} has positive measure.

b) Choose continuous functions ϕ_{n} on W_{n} so that ϕ_{n}(x) = 0 outside W_{n}, and 0 < ϕ_{n}(x) < 2^{−n} in W_{n}. Put

ϕ(x) := ∑_{n=1}^{∞} ϕ_{n}(x),  T (x) := ∫_{a}^{x} ϕ(t)dt,  x ∈ (a, b).

c) Prove that T is one-to-one, and differentiable at every point.

d) Prove that T′ is continuous, T′(x) = 0, ∀x ∈ K, m(T (K)) = 0.

e) Prove that if E is a nonmeasurable subset of K (why it exists?), then 1_{T (E)}(x) is measurable, but 1_{T (E)}(T x) is not. Observe that ϕ_{n} can be chosen such that T is infinitely differentiable homeomorphism of (a, b) onto some segment in R.

4. Problem 4. This exercise is a part of the proof of “the change of variables” in the Lebesgue integral. Let ϕ be increasing and absolutely continuous function on [p, q], x = ϕ(t), a = ϕ(p), b = ϕ(q).

a) Let E_{t} me a measurable subset of [p, q], E_{x} := ϕ(E_{t}). Prove that

m(E_{x}) = ∫_{E_{t}} ϕ′(t)dt

Hint. Prove this at first for E_{t} = [α, β]. Then for (α, β), open and closed E_{t}. In the general case, find closed F_{x}, and open G_{x} such that ∀ϵ > 0,

F_{x} ⊂ E_{x} ⊂ G_{x} ⊂ (a, b),  m(F_{x}) > m(E_{x}) − ϵ,  m(G_{x}) < m(E_{x}) + ϵ.

Consider F_{t} := ϕ^{−1}(F_{x}), G_{t} := ϕ^{−1}(G_{x}), and observe that

m(F_{x}) ≤ ∫_{E_{t}} ϕ′(t)dt ≤ m(G_{x}).

b) Let all notation be as in a), and let e_{x} ⊂ [a, b], m(e_{x}) = 0, e_{t} = ϕ^{−1}(e_{x}). Prove that

m(e_{t}^{∗}) = 0, e_{t}^{∗} := {t ∈ e_{t} : ϕ′(t) = 0 is false}.

Hint: Observe that

e_{x} ⊂ G_{x}^{n} ⊂ G_{x}^{n−1} ⊂ ... ⊂ G_{x}^{2} ⊂ G_{x}^{1} ⊂ (a, b), m(G_{x}^{n}) → 0.

Define

H_{x} := ∩_{n=1}^{∞}G_{x}^{n},  H_{t} := ϕ^{−1}(H_{x}),  G_{t}^{n} := ϕ^{−1}(G_{x}^{n}),

and prove that H_{t} is measurable, and m(H_{x}) = 0. Show that ∫_{H_{t}} ϕ′(t)dt = 0.