Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment V.

1. Problem 1. Let E = [0, 1] × [0, 1] ⊂ R^{2}, and let S be a subring of rectangles of type T_{ab} := {a ≤ x < b, 0 ≤ y ≤ 1}. Define m(T_{ab}) := b − a.

a) Describe the Lebesgue continuation of this measure. What sets are going to be measurable?

b) Prove that T̃ := {0 ≤ x ≤ 1, y = 1/2} is not measurable, and find its outer measure.

Hint. The set is Lebesgue measurable if and only if its outer measure is equal to its inner measure.

2. Problem 2.

Definition 1. Let U be a collection of all open subsets of the real line. Then R(U) is called Borel sets (the minimal ring containing U).

Prove that any Lebesgue measurable set on the real line is a union of a Borel set and a set of measure zero.

Hint. Let A ⊂ R be measurable. According to Assignment III, Problem 2, ∀ϵ > 0, there exists a closed set B_{ϵ} ⊂ A such that μ^{∗}(A \ B_{ϵ}) < ϵ. The set you are looking for is ∪_{n=1}^{∞}B_{1/n}.

3. Problem 3.

Definition 2. We say that a measure μ (defined on a corresponding subring S) is invariant under the transformation T : S → S if

∀A ∈ S, μ(T^{−1}(A)) ≡ μ(A).

a) It is known (take it as granted) that a real number x ∈ [0, 1] can be written as a continuous fraction

x = \frac{1}{n_{1} + \frac{1}{n_{2}+...}} , n_{k} ∈ N,

where a rational number can be written as a finite fraction, and an irrational number as an infinite one. Define the transformation T on [0, 1] as T := {1/x}, where {·} stands for the fractional part of a number. Prove that (in terms of sequences (n_{k})_{k=1}^{∞}), T has the form T((n_{k})_{k=1}^{∞}) = (n_{k+1})_{k=1}^{∞}.

b) Let μ be a measure on [0, 1], defined as

μ([α, β)) := log_{2} \frac{1 + β}{1 + α}.

Prove that μ is invariant under T defined in a).

4. Problem 4. Let m be a measure on a subring S, and let μ be its extension to R(S). Prove that the following statements are equivalent for μ, and might be not equivalent for m. ℵ) σ-additivity.

μ( ∪_{k=1}^{∞} A_{k}) = ∑_{k=1}^{∞} μ(A_{k});

ℶ) upper semicontinuity.

A_{1} ⊃ A_{2} ⊃ A_{3} ⊃ ...,  A = ∩_{k=1}^{∞}A_{k},  ⇒  μ(A) = lim_{k→∞} μ(A_{k});

ℷ) lower semicontinuity.

A_{1} ⊂ A_{2} ⊂ A_{3} ⊂ ...,  A = ∪_{k=1}^{∞}A_{k},  ⇒  μ(A) = lim_{k→∞} μ(A_{k});

ℸ) continuity.

μ( lim_{k→∞} A_{k}) = lim_{k→∞} μ(A_{k}).

Hint. Prove that ℵ) ⟺ ℶ), ℵ) ⟺ ℷ), ℸ) ⇒ ℵ). Prove that ℶ), ℷ) imply

μ( \overline{lim}_{n→∞} A_{n}) = μ(∩_{k=1}^{∞} ∪_{n=k}^{∞} A_{n}) = lim_{k→∞} μ(∪_{n=k}^{∞}A_{n}) ≥ \overline{lim}_{k→∞} μ(A_{k}),

μ( \underline{lim}_{n→∞} A_{n}) = μ(∪_{k=1}^{∞} ∩_{n=k}^{∞} A_{n}) = lim_{k→∞} μ(∩_{n=k}^{∞}A_{n}) ≤ \underline{lim}_{n→∞} μ(A_{k}).

This will give you ℵ) ⟺ ℸ). Moreover, consider the following example of a measure, which is not σ-additive. Take a subring S of subsets of [0, 1) ∩ Q, and define

S := {s_{a,b} := [a, b) ∩ [0, 1) ∩ Q}, m(s_{a,b}) = b − a.

Prove that for m, ℶ) and ℷ) are true, but ℵ) and ℸ are not. On the other hand for the extension μ of m, all ℵ), ℶ), ℷ), ℸ) are not true (they are equivalent).

5. Problem 5.

Definition 3. A pair (X, d) is called a metric space, if X is a set, and d is a distance. More precisely, d is a nonnegative real function d(x, y) defined for any x, y ∈ X, and satisfying

1) d(x, y) = 0 ⟺ x = y,

2) d(x, y) = d(y, x) (the axiom of symmetry)

3) d(x, z) ≤ d(x, y) + d(y, z) (the axiom of triangle). Let μ be a σ-additive measure on a subring S ⊂ P (X), and let μ^{∗} be the corresponding outer measure on P (X).

a) We say that A ∼ B if μ^{∗}(A△B) = 0. Prove that this is an equivalence relation.

b) Let X̃ be a set of all classes Ã of equivalence. Prove that (X̃, d) is a metric space, where d(Ã, B̃) := μ^{∗}(A△B). Here Ã, B̃ are classes of equivalence containing A, B.

6. Problem 6. Is it possible to construct a set G ⊂ [0, 1] such that ℵ) G is dense on [0, 1], ℶ) G has measure (length) zero, ℷ) G is not countable? Hint. Consider G := [0, 1] \ E, where E is a union of sets E_{i} constructed in the previous Assignment, Problem 5, b)*.