Lebesgue Measure on Euclidean Spaces

Prove that there is no Lebesgue measurable set $A \subset \mathbb {R}$ with the property that for any interval $I$,

Let $E$ be a Lebesgue measurable set such that $0<m\left(E\right)<\infty$ and let $\left\{ x_{n}\right\}$ be a sequence in $[0,1]$. Prove that there exists $y \in \mathbb {R}$ and a subsequence $\left\{ x_{n_{k}}\right\}$ such that

Show that if $A \subset [0,1]$ and $m\left(A\right)>0$, then there are $x$ and $y$ in $A$ such that $\left|x-y\right|$ is an irrational number.

Let $f:[0,1] \rightarrow \mathbb {R}$ be a Lipschitz function with constant $K$, and let $E \subset [0,1]$. Prove that

Let $E \subset \mathbb {R}^{n}$, let $\alpha \in \mathbb {R} \backslash \left\{ 0\right\}$, let $\beta \in \mathbb {R}^{n}$, and let $\alpha E+\beta =\left\{ \alpha x+\beta : x \in E\right\}$. Prove that:

Prove that Definition 5.3.3 supplies the same value of $\mu_{*}(A)$ regardless of the choice of a rectangle $R$ in the bounded case, and rectangles $\left\{ R_{k}\right\}$ in the unbounded case.

Definition 5.3.3: Here $\mu$ is the measure on $\mathbb {R}^{n}$ obtained by the Caratheodory extension from the ring of elementary sets, and $\mu^{*}$ is the corresponding outer measure. Let $A$ be a bounded subset of $\mathbb {R}^{n}$ and let $R$ be an open rectangle that contains $A$. The inner measure of $A$ is defined by $\mu_{*}(A)=\mu (R)-\mu^{*}(R \setminus A)$. If $A$ is not bounded, and $\left\{ R_{k}\right\}$ is a sequence of disjoint rectangles with union $\mathbb {R}^{n}$, then $\mu_{*}(A)=\sum_{k=1}^{\infty } \mu_{*}(A \cap R_{k})$.

Let $\left\{ R_{i}\right\}_{i=1}^{\infty }$ be a collection of rectangles in $\mathbb {R}^{n}$, and let $\varepsilon >0$. Prove that there exist open rectangles $\left\{ R_{i}^{\prime }\right\}$ such that $R_{i} \subset R_{i}^{\prime }$, for all $i \in \mathbb {N}$, and

In Exercises 5.3.8-5.3.10, the Lebesgue $\sigma$-algebra $\mathcal{M}_{n}$ and the $n$-dimensional Lebesgue measure $m_{n}$ were obtained using Caratheodory construction, starting with the ring $\mathcal{E}$ of elementary sets and defining the outer measure $m_{n}^{*}$ as in Example 5.2.5.

Example 5.2.5: Let $\mathfrak {E}$ be a ring on $X$ and for $A \subset X$ let $\mu^{*}(A) := \inf \left\{ \sum_{k=1}^{\infty } \mu (E_{k}) : \left\{ E_{k} : k \in \mathbb {N}\right\} \subset \mathfrak {E} \text{ and } A \subset \cup_{k=1}^{\infty } E_{k}\right\}$.

Prove Theorem 5.3.7.

Theorem 5.3.7: All open and all closed subsets of $\mathbb {R}^{n}$ are Lebesgue measurable.

In Exercises 5.3.8-5.3.10, the Lebesgue $\sigma$-algebra $\mathcal{M}_{n}$ and the $n$-dimensional Lebesgue measure $m_{n}$ were obtained using Caratheodory construction, starting with the ring $\mathcal{E}$ of elementary sets and defining the outer measure $m_{n}^{*}$ as in Example 5.2.5.

Example 5.2.5: Let $\mathfrak {E}$ be a ring on $X$ and for $A \subset X$ let $\mu^{*}(A) := \inf \left\{ \sum_{k=1}^{\infty } \mu (E_{k}) : \left\{ E_{k} : k \in \mathbb {N}\right\} \subset \mathfrak {E} \text{ and } A \subset \cup_{k=1}^{\infty } E_{k}\right\}$.

Prove Theorem 5.3.8.

Theorem 5.3.8: Let $A \subset \mathbb {R}^{n}$ and $\varepsilon >0$. The following are equivalent:

1. $A \in \mathcal{M}_{n}$.

2. There is an open set $G$ such that $A \subset G$ and $m_{n}^{*}(G \setminus A) < \varepsilon$.

3. There is a $G_{\delta }$ set $G$ such that $A \subset G$ and $m_{n}^{*}(G \setminus A) = 0$.

4. There is a closed set $F$ such that $F \subset A$ and $m_{n}^{*}(A \setminus F) < \varepsilon$.

5. There is an $F_{\sigma }$ set $F$ such that $F \subset A$ and $m_{n}^{*}(A \setminus F) = 0$.

In Exercises 5.3.8-5.3.10, the Lebesgue $\sigma$-algebra $\mathcal{M}_{n}$ and the $n$-dimensional Lebesgue measure $m_{n}$ were obtained using Caratheodory construction, starting with the ring $\mathcal{E}$ of elementary sets and defining the outer measure $m_{n}^{*}$ as in Example 5.2.5.

Example 5.2.5: Let $\mathfrak {E}$ be a ring on $X$ and for $A \subset X$ let $\mu^{*}(A) := \inf \left\{ \sum_{k=1}^{\infty } \mu (E_{k}) : \left\{ E_{k} : k \in \mathbb {N}\right\} \subset \mathfrak {E} \text{ and } A \subset \cup_{k=1}^{\infty } E_{k}\right\}$.

Prove Theorem 5.3.9.

Theorem 5.3.9: Every countable set in $\mathbb {R}^{n}$ is measurable and has measure $0$.

Let $E \in \mathcal{M}$ and $0<m\left(E\right)<\infty$. Prove that for every $\alpha \in (0,1)$ there exists an interval $I$ for which $m\left(E \cap I\right)>\alpha m\left(I\right)$.

Prove or disprove: If the boundary of a set $A \subset \mathbb {R}^{2}$ has outer measure zero, then $A$ is measurable.

Prove that the collection of all elementary subsets of $\mathbb {R}^{n}$ is a ring.

Let $A, B \subset \mathbb {R}$ such that $\inf \left\{ \left|x-y\right|: x \in A, y \in B\right\} >0$. Prove or disprove: $m^{*}(A \cup B)=m^{*}(A)+m^{*}(B)$.

Let $E \subset \mathbb {R}$. Prove that $m^{*}(E)=\sum_{n=-\infty }^{\infty } m^{*}(E \cap [n, n+1])$.

Let $E \subset \mathbb {R}^{n}$ and let $\left\{ G_{n}\right\}$ be open subsets of $\mathbb {R}^{n}$ such that $\cup_{n=1}^{\infty } G_{n}= \mathbb {R}^{n}$. Prove that $m^{*}(E)=\lim_{n \rightarrow \infty } m^{*}\left(E \cap \left(\cup_{k=1}^{n} G_{k}\right)\right)$.

Let $A, B \subset \mathbb {R}^{n}$, let $m^{*}(A)=m^{*}(B)=1$ and $m^{*}(A \cup B)=2$. Prove that $m^{*}(A \cap B)=0$.

Prove that the outer measure of the set of irrational numbers in $[0,1]$ is 1 .

Let $f:[0,1] \rightarrow \mathbb {R}$ be a Lipschitz function with constant $K$, and let $E$ be a Lebesgue measurable subset of $[0,1]$. Prove that $f(E)$ is a measurable set.

Construct a closed, uncountable subset of $[0,1]$ that contains no rational numbers.

Prove that a set $A \subset \mathbb {R}^{n}$ is measurable if and only if

for each rectangle $R$ in $\mathbb {R}^{n}$.

Let $\mathcal{A}$ be a $\sigma$-algebra contained in $\mathcal{M}_{n}$. Prove or disprove: the restriction $m_{n} \mid \mathcal{A}$ is $\sigma$-finite.

Let $E \subset \mathbb {R}$ have a finite outer measure and suppose that it is not measurable. Prove that there exists an open set $G \supset E$ that has finite outer measure and for which $m^{*}(G \backslash E)>m^{*}(G)-m^{*}(E)$.

Let $X$ be a proper subspace of $\mathbb {R}^{n}$. Prove that $m\left(X\right)=0$, where $m$ is the $n$-dimensional Lebesgue measure.

Let $\mu$ be a translation invariant measure on $\mathcal{M}_{n}$. Prove that there exists a constant $\alpha \geq 0$ such that $\mu \left(E\right)=\alpha m_{n}\left(E\right)$ for all Lebesgue measurable sets $E \subset \mathbb {R}^{n}$.

Prove that Proposition 5.3.2 holds in every Euclidean space $\mathbb {R}^{n}$. That is, prove that $\mu^{*}(A)=m^{*}(A)$, where $m^{*}(A)$ is as in Definition 4.2.11 and $\mu^{*}(A)$ as in Example 5.2.5, with $\mathfrak {E}$ the ring of elementary subsets of $\mathbb {R}^{n}$.

Definition 4.2.11: Let $A$ be a bounded subset of $\mathbb {R}^{2}$. The (Lebesgue) outer measure of $A$ is $m^{*}(A) := \inf \left\{ m(G) : G \text{ is an open set and } A \subset G\right\}$. The (Lebesgue) inner measure of $A$ is $m_{*}(A) := \sup \left\{ m(F) : F \text{ is a compact set and } F \subset A\right\}$. The set $A$ is (Lebesgue) measurable if $m^{*}(A) = m_{*}(A)$, in which case this number is denoted by $m(A)$ and called the (Lebesgue) measure of $A$. If $A$ is unbounded it is measurable if $A \cap R$ is measurable for every rectangle $R$.

Example 5.2.5: Let $\mathfrak {E}$ be a ring on $X$, $\mu$ a measure on $\mathfrak {E}$, and for $A \subset X$ let $\mu^{*}(A) := \inf \left\{ \sum_{k=1}^{\infty } \mu (E_{k}) : \left\{ E_{k} : k \in \mathbb {N}\right\} \subset \mathfrak {E} \text{ and } A \subset \cup_{k=1}^{\infty } E_{k}\right\}$. Then $\mu^{*}$ is an outer measure, and $\mu^{*}(A)=\mu (A)$ if $A \in \mathfrak {E}$.

Prove that Theorem 4.2.22 holds in every Euclidean space $\mathbb {R}^{n}$. That is, prove that if $E$ is a bounded set in $\mathbb {R}^{n}$ and $R$ an open rectangle containing $E$ then,

where $m^{*}$ and $m_{*}$ are defined analogously to Definition 4.2.11.

Definition 4.2.11: Let $A$ be a bounded subset of $\mathbb {R}^{2}$. The (Lebesgue) outer measure of $A$ is $m^{*}(A) := \inf \left\{ m(G) : G \text{ is an open set and } A \subset G\right\}$. The (Lebesgue) inner measure of $A$ is $m_{*}(A) := \sup \left\{ m(F) : F \text{ is a compact set and } F \subset A\right\}$. The set $A$ is (Lebesgue) measurable if $m^{*}(A) = m_{*}(A)$, in which case this number is denoted by $m(A)$ and called the (Lebesgue) measure of $A$. If $A$ is unbounded it is measurable if $A \cap R$ is measurable for every rectangle $R$.

Prove that Proposition 5.3.4 holds in every Euclidean space $\mathbb {R}^{n}$. That is, prove that $\mu_{*}(A)=m_{*}\left(A\right)$, where $\mu_{*}(A)$ is as in Definition 5.3.3 and $m^{*}(A)$ as in Definition 4.2.11.

Definition 5.3.3: Here $\mu$ is the measure on $\mathbb {R}^{n}$ obtained by the Caratheodory extension from the ring of elementary sets, and $\mu^{*}$ is the corresponding outer measure. Let $A$ be a bounded subset of $\mathbb {R}^{n}$ and let $R$ be an open rectangle that contains $A$. The inner measure of $A$ is defined by $\mu_{*}(A)=\mu (R)-\mu^{*}(R \setminus A)$. If $A$ is not bounded, and $\left\{ R_{k}\right\}$ is a sequence of disjoint rectangles with union $\mathbb {R}^{n}$, then $\mu_{*}(A)=\sum_{k=1}^{\infty } \mu_{*}(A \cap R_{k})$.

Definition 4.2.11: Let $A$ be a bounded subset of $\mathbb {R}^{2}$. The (Lebesgue) outer measure of $A$ is $m^{*}(A) := \inf \left\{ m(G) : G \text{ is an open set and } A \subset G\right\}$. The (Lebesgue) inner measure of $A$ is $m_{*}(A) := \sup \left\{ m(F) : F \text{ is a compact set and } F \subset A\right\}$. The set $A$ is (Lebesgue) measurable if $m^{*}(A) = m_{*}(A)$, in which case this number is denoted by $m(A)$ and called the (Lebesgue) measure of $A$. If $A$ is unbounded it is measurable if $A \cap R$ is measurable for every rectangle $R$.

Prove the unbounded case in Proposition 5.3.4.

Proposition 5.3.4: If $A \subset \mathbb {R}^{2}$ then $\mu^{*}(A) = m^{*}(A)$. (The bounded case is proved in the text using Proposition 5.3.2 and Theorem 4.2.22; only the unbounded case remains.)

Prove or disprove: If $A$ is a subset of $[0,1]$ such that $m\left(\operatorname {Int}(A)\right)= m\left(\bar{A}\right)$, then $A$ is measurable.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be continuous. Determine the measure of its graph.

Show that if a set $E \subset \mathbb {R}^{n}$ has a positive outer measure, then there is a bounded subset of $E$ that also has a positive outer measure.

Let $E$ be a subset of $\mathbb {R}$ such that $m^{*}(E)<\infty$ and let $\varepsilon >0$. Prove that $E$ is a disjoint union of a finite number of sets, each of which has an outer measure at most $\varepsilon$. Does the assertion remain true in $\mathbb {R}^{n}$ ? In any measurable space?

Find the Lebesgue measure of a subset of $[0,1]$ whose decimal expansion contains a digit 2 before a digit 3 .

Prove that Cantor set $F$ is measurable and that $m\left(F\right)=0$.

Let $F$ denote the Fat Cantor set obtained by deleting at step $n$ from each of the $2^{n-1}$ intervals the middle interval of length $\alpha^{n} I$, where $0<\alpha \leq 1 / 3$. (In the classical Cantor set $\alpha =1 / 3$.) Prove that $F$ is measurable and find its measure.

Let $0<\lambda <1$ and construct a set $K$ as follows: Remove from $[0,1]$ its middle part of length $\lambda$; we are left with two intervals $\left(a_{1}, b_{1}\right)$ and $\left(a_{2}, b_{2}\right)$. Remove from each of them a middle part of length $\lambda \left(b_{i}-a_{i}\right)$, etc. Prove that $m\left(K\right)=0$.

Give an example of a subset of $[0,1]$ that is of the first category and of Lebesgue measure 1.

Prove that there exists an open subset of $\mathbb {R}$ such that its boundary has a positive measure.

Prove or disprove: There exists a dense uncountable subset of $\mathbb {R}$ with Lebesgue measure zero.

Prove or disprove: If $A$ is an open subset of $[0,1]$, then $m\left(A\right)=m\left(\bar{A}\right)$.

Prove or disprove: There exists an open dense subset $A$ of $[0,1] \times [0,1]$ such that its complement $([0,1] \times [0,1]) \backslash A$ has positive Lebesgue measure.

Given a nonempty set $A \subset \mathbb {R}$ construct a function $f: \mathbb {R} \rightarrow \mathbb {R}$ which is constant a.e. and $f(\mathbb {R})=A$.

Prove that the Lebesgue $\sigma$-algebra $\mathcal{M}$ has the same cardinality as $\mathbb {P}(\mathbb {R})$.

Let $\left\{ r_{n}\right\}$ a be a sequence that is dense in $\mathbb {R}$, and let

Prove that $A$ is an uncountable set of measure 0 .

Let $A \subset [0,1]$ be a measurable set of positive measures. Show that there exist two distinct points $x, y \in A$ such that $x-y$ is a rational number.

Prove that there exists a measurable set $A \subset [0,1]$ such that $0< m\left(A \cap I\right)<m\left(I\right)$ for every interval $I \subset [0,1]$.