Math Problems Vault

Problem 4.3.1

Let $\mu : \mathscr {B}^{k} \rightarrow \overline{\mathbb {R}}$ denote a measure that assigns finite measure to bounded $\mathscr {B}^{k}$ -sets. Furthermore, suppose that $\mu$ is translation invariant: $\mu (B \oplus \boldsymbol {x})=\mu (B)$ for all $\boldsymbol {x} \in \mathbb {R}^{k}$ . Then there is $c>0$ such that $\mu (B)=c \lambda_{k}(B)$ for all $B \in \mathscr {B}^{k}$ .

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Problem 4.3.2

Let $A \in \mathscr {L}$ be such that $\bar{\lambda }(A)>0$ , and let $c \in [0,1)$ . There exists an open interval $U$ such that $\bar{\lambda }(A \cap U) \geqslant c \lambda (U) .^{1}$

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Problem 4.3.3

Let $A \in \mathscr {L}$ contain an open interval. Then there exists $a>0$ such that $(-a, a)$ is contained in $D(A)=\left\{ x-y: x, y \in A\right\}$ .

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Problem 4.3.4

Let $A \in \mathscr {L}$ be such that $\bar{\lambda }(A)>0$ . Then there exists $a>0$ such that $(-a, a)$ is contained in $D(A)=\left\{ x-y: x, y \in A\right\}$ .

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Problem 4.3.5

Let $A$ be a dense subset of $\mathbb {R}$ . Then $c A=\left\{ c a: a \in A\right\}$ is dense for any $c \neq 0$ .

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Problem 4.3.6

(a)(b)(c)

Let $\xi$ be an irrational number.

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(a)

Let $A=\left\{ n+m \xi : n, m \in \mathbb {Z}\right\}$ . Then $A$ is a dense subset of $\mathbb {R}$ .

(b)

Let $B=\left\{ n+m \xi : n, m \in \mathbb {Z}, n$ even $\right\}$ . Then $B$ is a sense subset of $\mathbb {R}$ .

(c)

Let $C=\left\{ n+m \xi : n, m \in \mathbb {Z}, n$ odd $\right\}$ . Then $C$ is a sense subset of $\mathbb {R}$ .

Problem 4.3.7

For $x, y \in \mathbb {R}$ write $x \sim y$ iff $x-y \in A=\left\{ n+m \xi : n, m \in \mathbb {Z}\right\}$ , where $\xi$ is a fixed irrational number as in the previous exercise. Then $\sim$ is an equivalence relation, and hence $\mathbb {R}$ may be partitioned into disjoint equivalence classes.

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Problem 4.3.8

We now invoke ( $A C$ ) to form a set $E_{0}$ consisting of exactly one element from each of the equivalence classes in the previous exercise. We will now show that $E_{0} \notin \mathscr {L}$ .

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(a)

There exist Borel subsets of $E_{0}$ .

(b)

Let $F \subseteq E_{0}$ be a Borel set. Then $D(F)$ cannot contain any nonzero elements of $A$ , where $A$ is the set in Exercise 4.3.6.

(c)

By (b), there cannot exist an open interval containing the origin that is contained in $D(F)$ , hence $\lambda (F)=0$ .

(d)

From (c), we have $\lambda_{*}\left(E_{0}\right)=0$ .

(e)

If $a_{1}$ and $a_{2}$ are distinct elements of $A=\left\{ n+m \xi : m, n \in \mathbb {Z}\right\}$ , then $E_{0} \oplus a_{1}$ and $E_{0} \oplus a_{2}$ are disjoint.

(f)

$\mathbb {R}=\bigcup \left\{ E_{0} \oplus a: a \in A\right\}$ , the countable union being disjoint.

(g)

If $E_{0} \in \mathscr {L}$ , then $\bar{\lambda }\left(E_{0} \oplus a\right)=0$ for each $a \in A$ , hence $\bar{\lambda }(\mathbb {R})=0$ . Therefore, since the assumption that $E_{0} \in \mathscr {L}$ leads to an absurdity, it must be the case that $E_{0} \notin \mathscr {L}$ , and hence there exists a subset of $\mathbb {R}$ that fails to be a Lebesgue set.

Problem 4.3.9

In conjunction with the previous exercises, it is not possible to have Lebesgue measure defined for every subset of $\mathbb {R}$ . That is, a measure extension $\mu$ of $\Delta_{F}$ [where $F \in \mathfrak {S}$ is the identity function] from the semiring $\mathcal{A}$ consisting of $\varnothing$ and the bounded rsc intervals $(a, b]$ to $2^{\mathbb {R}}$ fails to exist.

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Remark.

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Argue by contradiction for a quick answer: if such a measure extension $\mu$ to $2^{\mathbb {R}}$ exists then the previous work would force $\mu (\mathbb {R})=0$ , contradicting what we expect of length.

Problem 4.3.10

Let $B \in \mathscr {L}$ be such that $\bar{\lambda }(B)>0$ . We will show that there is $E_{1} \subseteq B$ with $E_{1} \notin \mathscr {L}$ by following the outline below. The set $A$ appearing below is the set $A$ from Exercise 4.3.7, consisting of the numbers $n+m \xi$ , where $n, m \in \mathbb {Z}$ and $\xi$ is a fixed irrational number.

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(a)

$\lambda_{*}$ is monotone and translation invariant.

(b)

$B=\bigcup \left\{ (E_{0} \oplus a) \cap B: a \in A\right\}$ , the union being disjoint.

(c)

There exists $a^{*} \in A$ such that $\lambda^{*}\left((E_{0} \oplus a^{*}) \cap B\right)>0$ .

(d)

Let $E_{1}=(E_{0} \oplus a^{*}) \cap B$ . Then $E_{1} \subseteq B$ and $E_{1} \notin \mathscr {L}$ .

(e)

We may actually generalize further and state that any subset $B \subseteq \mathbb {R}$ with $\lambda^{*}(B)>0$ [ $B$ not necessarily in $\mathscr {L}$ ] contains a subset outside of $\mathscr {L}$ .

Remark.

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By (b), we have $0<\bar{\lambda }(B)=\lambda^{*}(B) \leqslant \sum \left\{ \lambda^{*}((E_{0} \oplus a) \cap B): a \in A\right\}$ ; and by (a) and part (d) of Exercise 4.3.8, $\lambda_{*}(E_{1})=\lambda_{*}((E_{0} \oplus a^{*}) \cap B) \leqslant \lambda_{*}(E_{0} \oplus a^{*})=\lambda_{*}(E_{0})=0$ , while at the same time $\lambda^{*}(E_{1})>0$ ; invoke the relevant part of the Approximation Theorem.

Problem 4.3.11

Let $A \in \mathscr {L}$ , let $B$ be a dense subset of $\mathbb {R}$ , and suppose that $\bar{\lambda }(A \triangle (A \oplus x))=0$ for every $x \in B$ . Then either $\bar{\lambda }(A)=0$ or $\bar{\lambda }(A^{c})=0$ . This exercise is independent of the previous exercises.

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