Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment III.

1. Problem 1.

Definition 1. Let E := [0, 1] × [0, 1], and let A ⊂ E. The inner measure μ_{∗}(A) (of A) is a number defined as

μ_{∗}(A) := 1 − μ^{∗}(E \ A).

Prove that μ_{∗}(A) ≤ μ^{∗}(A). Hint. μ^{∗}(E \ A) + μ^{∗}(A) ≥ μ^{∗}(E).

2. Problem 2. Prove that A ⊆ E is Lebesgue measurable if and only if μ_{∗}(A) = μ^{∗}(A). Hint. Let A ⊆ E be Lebesgue measurable. Then ∀ϵ > 0 there exists an elementary set B such that μ^{∗}(A△B) < ϵ. Prove that

μ^{∗}(B) − ϵ ≤ μ_{∗}(A) ≤ μ^{∗}(A) ≤ μ^{∗}(B) + ϵ.

Conversely, assume that μ_{∗}(A) = μ^{∗}(A). Then prove the following chain of statements leading to the result.

a) ∀ϵ > 0 there exist elementary sets P_{n}, Q_{n}, n = 1, 2, ..., such that

A ⊂ ∪_{n=1}^{∞}P_{n}, μ^{∗}(A) ≥ ∑_{n=1}^{∞} m′(P_{n}) − ϵ,

and

(E \ A) ⊂ ∪_{n=1}^{∞}Q_{n}, μ^{∗}(E \ A) ≥ ∑_{n=1}^{∞} m′(Q_{n}) − ϵ.

Conclude that

∑_{n=1}^{∞}(m′(P_{n}) + m′(Q_{n})) ≤ 1 + 2ϵ,

and that there exists N such that

∑_{n=N+1}^{∞}(m′(P_{n}) + m′(Q_{n})) < ϵ.

b) Denote

∪_{n=1}^{∞}P_{n} = P,  ∪_{n=1}^{∞}Q_{n} = Q,  ∪_{n=1}^{N}P_{n} = P_{N}  ∪_{n=1}^{N} Q_{n} = Q_{N}.

Observe that

μ^{∗}(P_{N}△A) ≤ μ^{∗}(P_{N} \ A) + μ^{∗}(A \ P_{N}).

c) Prove that

μ^{∗}(A \ P_{N}) ≤ ∑_{n=N+1}^{∞} m′(P_{n}).

d) Observe that

P_{N} \ A ⊂ (P_{N} ∩ Q_{N}) ∪ (P_{N} ∩ (Q \ Q_{n})) ⊂ (P_{N} ∩ Q_{N}) ∪ (Q \ Q_{n}),

and conclude

μ^{∗}(P_{N} \ A) ≤ μ^{∗}(P_{N} ∩ Q_{N}) + ∑_{n=N+1}^{∞} m′(Q_{n}).

e) Observe that E ⊆ (P ∪ Q), and show that

1 = μ^{∗}(E) ≤ μ^{∗}(P_{N} ∪ Q_{N}) + μ^{∗}(P \ P_{N}) + μ^{∗}(Q \ Q_{N}).

Use the fact that for elementary sets C, D,

m′(C ∪ D) = m′(C) + m′(D) − m′(C ∩ D),

and a) to conclude that

1 ≤ ∑_{n=1}^{∞}(m′(P_{n}) + m′(Q_{n})) − μ^{∗}(P_{N} ∩ Q_{N}) ≤ 1 + 3ϵ − μ^{∗}(P_{N} ∩ Q_{N}),

and μ^{∗}(P_{N} ∩ Q_{N}) ≤ 3ϵ.

f) Use d) to show that

μ^{∗}(P_{N} \ A) ≤ 2ϵ + ∑_{n=N+1}^{∞} m′(Q_{n}).

g) ”Glue” pieces b), c), f) to obtain the desired result.

3. Problem 3. Prove that the capacity of all Lebesgue measurable sets is greater that continuum.

Hint. Consider P (C) (the set of all subsets of the Cantor set).

4. Problem 4^{∗}. Let A ⊂ [0, 1] be such that in the decimal decomposition of every x ∈ A you meet 2 before you meet 3. Find the Lebesgue measure of A.

Hint. This is “Cantor-like” exercise. Think first about numbers you would not like to have (go for the complement). At first take out of [0, 1] the set [0.3, 0.4]. Then take out eight (why not nine?) sets of the type [0.n_{1}3, 0.n_{1}4], where n_{1} = 0, 1, 4, 5, 6, 7, 8, 9, and so on...

5. Problem 5. Let E = [0, 1] × [0, 1] be a unit square in the plane, and let

A := \{(x, y) ∈ E : | sin x| < \frac{1}{2}, cos(x + y) ∈ R \ Q\}.

Find the Lebesgue measure of A. Hint. What is the complement of A?

6. Problem 6*. A set A ⊆ E is called Caratheodory measurable if ∀Z ⊆ E we have

μ^{∗}(Z) = μ^{∗}(Z ∩ A) + μ^{∗}(Z \ A).

Prove that A is Caratheodory measurable if and only if it is Lebesgue measurable. Hint. Let A be Caratheodory measurable. Then by taking Z = E, we have

μ^{∗}(E) = μ^{∗}(A) + μ^{∗}(E \ A) = μ^{∗}(A) + μ_{∗}(A).

This gives (see Problem 2) the Lebesgue measurability of A. Conversely, let A be Lebesgue measurable. Then ∀Z ⊆ E, we have

μ^{∗}(Z) ≤ μ^{∗}(Z ∩ A) + μ^{∗}(Z \ A),

and it remains to prove

μ^{∗}(Z) ≥ μ^{∗}(Z ∩ A) + μ^{∗}(Z \ A).

To this end, prove the following (important) Lemma. ∀Z ⊆ E there exists a Lebesgue measurable set Z_{1} such that Z ⊆ Z_{1} ⊆ E, and μ^{∗}(Z) = μ^{∗}(Z_{1}). Then conclude that

μ^{∗}(Z) = μ^{∗}(Z_{1}) = μ^{∗}(Z_{1} ∩ A) + μ^{∗}(Z \ A) ≥ μ^{∗}(Z ∩ A) + μ^{∗}(Z \ A).