Math Problems Vault

Problem 2D.1

(a)(b)

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(a)

Show that the set consisting of those numbers in $(0,1)$ that have a decimal expansion containing one hundred consecutive 4s is a Borel subset of $\mathbb {R}$ .

(b)

What is the Lebesgue measure of the set in part (a)?

Problem 2D.2

Prove that there exists a bounded set $A \subseteq \mathbb {R}$ such that $\left|F\right| \leq \left|A\right|-1$ for every closed set $F \subseteq A$ .

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Problem 2D.3

Prove that there exists a set $A \subseteq \mathbb {R}$ such that $\left|G \backslash A\right| = \infty$ for every open set $G$ that contains $A$ .

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Problem 2D.4

(a)(b)(c)

The phrase nontrivial interval is used to denote an interval of $\mathbb {R}$ that contains more than one element. Recall that an interval might be open, closed, or neither.

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(a)

Prove that the union of each collection of nontrivial intervals of $\mathbb {R}$ is the union of a countable subset of that collection.

(b)

Prove that the union of each collection of nontrivial intervals of $\mathbb {R}$ is a Borel set.

(c)

Prove that there exists a collection of closed intervals of $\mathbb {R}$ whose union is not a Borel set.

Problem 2D.5

Prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable, then there exists an increasing sequence $F_{1} \subseteq F_{2} \subseteq \cdots$ of closed sets contained in $A$ such that

\left|A \backslash \bigcup _{k = 1}^{\infty } F_{k}\right| = 0

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Problem 2D.6

Suppose $A \subseteq \mathbb {R}$ and $\left|A\right| < \infty$ . Prove that $A$ is Lebesgue measurable if and only if for every $\varepsilon > 0$ there exists a set $G$ that is the union of finitely many disjoint bounded open intervals such that $\left|A \backslash G\right|+\left|G \backslash A\right| < \varepsilon$ .

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Problem 2D.7

Prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable, then there exists a decreasing sequence $G_{1} \supseteq G_{2} \supseteq \cdots$ of open sets containing $A$ such that

\left|\left(\bigcap _{k = 1}^{\infty } G_{k}\right) \backslash A\right| = 0

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Problem 2D.8

Prove that the collection of Lebesgue measurable subsets of $\mathbb {R}$ is translation invariant. More precisely, prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable and $t \in \mathbb {R}$ , then $t+A$ is Lebesgue measurable.

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Problem 2D.9

Prove that the collection of Lebesgue measurable subsets of $\mathbb {R}$ is dilation invariant. More precisely, prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable and $t \in \mathbb {R}$ , then $t A$ (which is defined to be $\left\{ t a: a \in A\right\}$ ) is Lebesgue measurable.

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Problem 2D.10

Prove that if $A$ and $B$ are disjoint subsets of $\mathbb {R}$ and $B$ is Lebesgue measurable, then $\left|A \cup B\right| = \left|A\right|+\left|B\right|$ .

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Problem 2D.11

Prove that if $A \subseteq \mathbb {R}$ and $\left|A\right| > 0$ , then there exists a subset of $A$ that is not Lebesgue measurable.

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Problem 2D.12

Suppose $b < c$ and $A \subseteq (b, c)$ . Prove that $A$ is Lebesgue measurable if and only if $\left|A\right|+\left|(b, c) \backslash A\right| = c-b$ .

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Problem 2D.13

Suppose $A \subseteq \mathbb {R}$ . Prove that $A$ is Lebesgue measurable if and only if

\left|(-n, n) \cap A\right|+\left|(-n, n) \backslash A\right| = 2 n

for every $n \in \mathbb {N}$ .

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Problem 2D.14

Show that $\frac{1}{4}$ and $\frac{9}{13}$ are both in the Cantor set.

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Problem 2D.15

Show that $\frac{13}{17}$ is not in the Cantor set.

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Problem 2D.16

List the eight open intervals whose union is $G_{4}$ in the definition of the Cantor set (2.74).

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Problem 2D.17

Let $C$ denote the Cantor set. Prove that $\left\{ \frac{1}{2} x+\frac{1}{2} y: x, y \in C\right\} = [0,1]$ .

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Problem 2D.18

Prove that every open interval of $\mathbb {R}$ contains either infinitely many or no elements in the Cantor set.

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Problem 2D.19

Evaluate $\int_{0}^{1} \Lambda$ , where $\Lambda$ is the Cantor function.

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Problem 2D.20

(a)(b)

Evaluate each of the following:

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(a)

$\Lambda \left(\frac{9}{13}\right)$ ;

(b)

$\Lambda (0.93)$ .

Problem 2D.21

(a)(b)

Find each of the following sets:

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(a)

$\Lambda^{-1}\left(\left\{ \frac{1}{3}\right\} \right)$ ;

(b)

$\Lambda^{-1}\left(\left\{ \frac{5}{16}\right\} \right)$ .

Problem 2D.22

(a)(b)

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(a)

Suppose $x$ is a rational number in $[0,1]$ . Explain why $\Lambda (x)$ is rational.

(b)

Suppose $x \in C$ is such that $\Lambda (x)$ is rational. Explain why $x$ is rational.

Problem 2D.23

Show that there exists a function $f: \mathbb {R} \rightarrow \mathbb {R}$ such that the image under $f$ of every nonempty open interval is $\mathbb {R}$ .

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Problem 2D.24

(a)(b)

For $A \subseteq \mathbb {R}$ , the quantity

\sup \left\{ \left|F\right|: F \text{ is a closed bounded subset of } \mathbb {R} \text{ and } F \subseteq A\right\}

is called the inner measure of $A$ .

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(a)

Show that if $A$ is a Lebesgue measurable subset of $\mathbb {R}$ , then the inner measure of $A$ equals the outer measure of $A$ .

(b)

Show that inner measure is not a measure on the $\sigma$ -algebra of all subsets of $\mathbb {R}$ .