Lebesgue Measure on R2

Let $\left\{ R_{k}\right\}_{k=1}^{\infty }$ be rectangles and let $\varepsilon >0$. Prove that there exist open rectangles $\left\{ R_{k}^{\prime }\right\}_{k=1}^{\infty }$ such that $R_{k} \subset R_{k}^{\prime }$, for all $k \in \mathbb {N}$, and such that $\sum_{k=1}^{\infty } \operatorname {vol}\left(R_{k}^{\prime }\right) \leq \sum_{k=1}^{\infty } \operatorname {vol}\left(R_{k}\right)+\varepsilon$.

Let $\left\{ R_{k}\right\}_{k=1}^{\infty }$ be rectangles and let $\varepsilon >0$. Prove that there exist closed rectangles $\left\{ R_{k}^{\prime }\right\}_{k=1}^{\infty }$ such that $R_{k}^{\prime } \subset R_{k}$, for all $k \in \mathbb {N}$, and such that $\sum_{k=1}^{\infty } \operatorname {vol}\left(R_{k}\right) \leq \sum_{k=1}^{\infty } \operatorname {vol}\left(R_{k}^{\prime }\right)+\varepsilon$.

Prove Theorem 4.1.5.

Theorem 4.1.5: Jordan content $c$ has the following properties: (i) Finite additivity: for any disjoint collection $\left\{ E_{k}\right\}_{k=1}^{m} \subset \mathcal{E}$, $c\left(\cup_{k=1}^{m} E_{k}\right) = \sum_{k=1}^{m} c(E_{k})$. (ii) Subadditivity: for any collection $\left\{ E_{k}\right\}_{k=1}^{m} \subset \mathcal{E}$, $c\left(\cup_{k=1}^{m} E_{k}\right) \leq \sum_{k=1}^{m} c(E_{k})$. (iii) Translation invariance: for any $y \in \mathbb {R}^{2}$ and any $E \in \mathcal{E}$, $E+y \in \mathcal{E}$ and $c(E+y)=c(E)$.

Prove that if $A \subset B$ then $c^{*}(A) \leq c^{*}(B)$ and $c_{*}(A) \leq c_{*}(B)$.

Prove that $c^{*}(E)=c^{*}(\bar{E})$ for every set $E$.

Prove that $c_{*}(E)=c_{*}$ (int $E$ ) for every set $E$.

Prove that $c^{*}(E)=\inf \left\{ c^{*}(G): G \text{ is open and } E \subset G\right\}$.

Prove Theorem 4.1.8.

Theorem 4.1.8: Let $A, B \in \mathcal{J}$ (the collection of Jordan measurable sets). (a) $c(\varnothing )=0$. (b) If $A \subset B$, then $c(A) \leq c(B)$ (monotonicity). (c) For any rectangle $R$, $c(R)=\operatorname {vol}(R)$.

Let $A \subset \mathbb {R}^{2}$ and $y \in \mathbb {R}^{2}$. Prove that $c^{*}(A+y)=c^{*}(A)$ and $c_{*}(A+y)= c_{*}(A)$.

Prove that if $\left\{ E_{k}\right\}_{k=1}^{m} \subset \mathcal{J}$ then $c\left(\cup_{k=1}^{m} E_{k}\right) \leq \sum_{k=1}^{m} c\left(E_{k}\right)$.

Prove that the outer content $c^{*}$ is subadditive and that the inner content $c_{*}$ is superadditive.

Prove that if $E=E_{1} \cup E_{2}$ and $d\left(E_{1}, E_{2}\right)>0$, then $c^{*}(E)=c^{*}\left(E_{1}\right)+ c^{*}\left(E_{2}\right)$.

Prove that there are bounded sets $A, B \subset \mathbb {R}^{2}$ such that $c_{*}(B)=0$ and $c_{*}(A \cup B) \neq c_{*}(A)$.

Let $E$ be a set of Jordan content 0 and $A \subset E$. Prove that $A$ is Jordan measurable and $c(A)=0$.

Let $\nu : \mathcal{J} \rightarrow \mathbb {R}$ be a map that is non-negative, finitely additive, and translation invariant. Show that there exists a constant $\alpha \geq 0$ such that $\nu \left(E\right)=\alpha c(E)$ for all Jordan measurable sets $E$.

Let $A$ be a line segment in the $x y$-plane. Prove that $A$ has content 0 .

Let $f:[a, b] \rightarrow \mathbb {R}$ be a continuous function. Show that the graph $\left\{ (x, f(x)): x \in [a, b]\right\}$ is Jordan measurable in $\mathbb {R}^{2}$ with Jordan content zero.

Let $f:[a, b] \rightarrow \mathbb {R}$ be a continuous function. Show that the set $\left\{ (x, t): x \in [a, b], 0 \leq t \leq f(x)\right\}$ is Jordan measurable in $\mathbb {R}^{2}$.

Let $A, B, C$ be three points in $\mathbb {R}^{2}$. Show that the solid triangle with vertices $A, B, C$ is Jordan measurable.

Show that every compact polygon in $\mathbb {R}^{2}$ is Jordan measurable.

Prove that $\partial (A \cup B) \subset \partial A \cup \partial B$.

Prove that $\partial S=\bar{S} \cap \overline{S^{C}}$.

Use Exercise 4.1.22 to prove that $\partial (A \backslash B) \subset \partial A \cup \partial B$.

Let $A$ and $B$ be sets such that their closures $\bar{A}, \bar{B}$ are Jordan sets. Prove that $\overline{A \cap B}$ is a Jordan set.

Give an example of a bounded open set in $\mathbb {R}^{2}$ that is not a Jordan set.

Prove or disprove: Every closed and bounded set in $\mathbb {R}^{2}$ is a Jordan set.

Prove or disprove: a Jordan set in $\mathbb {R}^{2}$ has content 0 if and only if it is nowhere dense.

Let $f$ be a bounded function on a Jordan set $D$, let $A \subset D$ be the set of all the points on which $f$ fails to be continuous, and suppose that $A$ has content 0 . Prove that $f$ is integrable on $D$.

Prove that the set $\mathcal{E}$ is a ring.

Prove that every ring contains the empty set and that it is closed under intersection.

Give an example to show that a collection of sets that is closed under union and intersection need not be a ring.

Let $\Re$ be a ring of subsets of $X$ and let $\mathfrak {S}$ be a ring of subsets of $Y$. Determine whether the set

is a ring.

Let $f: X \rightarrow Y$ and let $\Re$ be a ring of subsets of $X$. Determine whether the set $\left\{ f(A): A \in \Re \right\}$ is a ring.

Let $f: X \rightarrow Y$ and let $\mathfrak {S}$ be a ring of subsets of $Y$. Determine whether the set $\left\{ f^{-1}(B): B \in \mathfrak {S}\right\}$ is a ring.

Prove that the measure of an open set is independent of its representation as a disjoint union of rectangles.

Prove Proposition 4.2.3.

Proposition 4.2.3: If $G$ is an open set in $\mathcal{J}$ then $m(G)=c(G)$.

Prove Proposition 4.2.4.

Proposition 4.2.4: If $G_{1}, G_{2}$ are open sets in $\mathbb {R}^{2}$ such that $G_{1} \subset G_{2}$ then $m(G_{1}) \leq m(G_{2})$.

Prove Proposition 4.2.5.

Proposition 4.2.5: If $\left\{ G_{i}\right\}_{i \in I}$ is a finite or a countable collection of open sets in $\mathbb {R}^{2}$ and $G=\cup_{i \in I} G_{i}$, then $m(G) \leq \sum_{i \in I} m(G_{i})$. If, in addition, the sets $\left\{ G_{i}\right\}_{i \in I}$ are disjoint then $m(G)=\sum_{i \in I} m(G_{i})$.

Prove that the measure of a closed set is independent of the choice of an open rectangle that contains it.

Let $F$ be a closed Jordan set. Using only Definition 4.2.6, prove that $m\left(F\right)=c(F)$.

Definition 4.2.6: Let $F$ be a closed and bounded set in $\mathbb {R}^{2}$ and $R$ an open rectangle such that $F \subset R$. The measure of the closed set $F$ is defined by $m(F) = m(R) - m(R \setminus F)$.

Prove Proposition 4.2.7.

Proposition 4.2.7: If $\left\{ F_{k}\right\}_{k=1}^{n}$ is a finite collection of closed rectangles without common interior points and $F=\cup_{k=1}^{n} F_{k}$, then $m(F)=\sum_{k=1}^{n} m(F_{k})$.

Prove Proposition 4.2.8.

Proposition 4.2.8: If $F$ is a closed set and $G$ an open bounded set such that $F \subset G$, then $m(F) \leq m(G)$.

Prove Theorem 4.2.10.

Theorem 4.2.10: The measure of a closed bounded set $F \subset \mathbb {R}^{2}$ is the infimum of the measures of all open sets that contain $F$.

Prove Proposition 4.2.12.

Proposition 4.2.12: Let $E \subset \mathbb {R}^{2}$. Then $c_{*}(E) \leq m_{*}(E) \leq m^{*}(E) \leq c^{*}(E)$.

Prove that the empty set is measurable and that $m\left(\varnothing \right)=0$.

Let $E$ be an elementary set in $\mathbb {R}^{2}$. Prove or disprove: $m_{*}\left(E\right)=c(E)$. What if $E$ is a Jordan measurable set?

Let $E \subset \mathbb {R}^{2}$ be compact. Prove that $m\left(E\right)=c^{*}(E)$.

Prove Proposition 4.2.16.

Proposition 4.2.16: Every countable set is measurable and it has measure $0$.

Give an example of an unbounded set with Lebesgue outer measure 0 .

Prove Proposition 4.2.17.

Proposition 4.2.17: Let $A, F, G, E \subset \mathbb {R}^{2}$. (a) If $G$ is open then $m_{*}(G) = m^{*}(G) = m(G)$. (b) If $F$ is closed then $m_{*}(F) = m^{*}(F) = m(F)$. (c) Monotonicity: If $A \subset E$ then $m_{*}(A) \leq m_{*}(E)$ and $m^{*}(A) \leq m^{*}(E)$.

Prove Theorem 4.2.19.

Theorem 4.2.19 (Superadditivity of inner measure): If a set $E \subset \mathbb {R}^{2}$ is a finite or a countable disjoint union of the sets $\left\{ E_{i}\right\}_{i \in I}$, $E = \cup_{i \in I} E_{i}$, then $m_{*}(E) \geq \sum_{i \in I} m_{*}(E_{i})$.

An interval $I$ is unbounded if it is of the form $(-\infty , b),(-\infty , b]$, $(a,+\infty ),[a,+\infty )$, or $(-\infty , \infty )$. A rectangle $R=I_{1} \times I_{2}$ is unbounded if at least one of the intervals $I_{1}, I_{2}$ is unbounded. Prove that, if $R$ is an unbounded rectangle, then $m^{*}(R)=\operatorname {vol}(R)=\infty$.

Prove that if $m^{*}(A)=0$, then $m^{*}(A \cup B)=m^{*}(B)$.

Prove that $\left|m^{*}(A)-m^{*}(B)\right| \leq m^{*}(A \triangle B)$ for any $A, B \subset \mathbb {R}^{2}$, provided that at least one of the numbers $m^{*}(A), m^{*}(B)$ is finite.

Let $A \sim B$ if and only if $m^{*}(A \triangle B)=0$. Prove that:

Nikodym (1929). Prove that the metric space in Exercise 4.2.21 is complete.

Let $A$ be the set of points in $[0,1] \times [0,1]$ with both coordinates irrational. Prove that $m^{*}(A)=1$.

Let $A$ and $B$ be bounded subsets of $\mathbb {R}^{2}$ for which there is $\alpha >0$ such that $\left\| a-b\right\| \geq \alpha$ for all $a \in A, b \in B$. Prove that $m^{*}(A \cup B)=m^{*}(A)+m^{*}(B)$.

Let $E \subset \mathbb {R}^{2}$ be a countable union of rectangles $\left\{ R_{k}\right\}_{k=1}^{\infty }$ with disjoint interiors. Prove that $m\left(E\right)=\sum_{k=1}^{\infty } m\left(R_{k}\right)$.

Prove Corollary 4.2.20.

Corollary 4.2.20: If a set $E \subset \mathbb {R}^{2}$ is a finite or a countable disjoint union of measurable sets $\left\{ E_{i}\right\}_{i \in I}$, $E = \cup_{i \in I} E_{i}$, then $E$ is measurable and $m(E) = \sum_{i \in I} m(E_{i})$.

Let $E \subset \mathbb {R}^{2}$ and let $\left\{ R_{k}\right\}_{k=1}^{n}$ be disjoint open rectangles. Prove that:

Give examples to show that each of the inequalities $c_{*}(A) \leq m_{*}\left(A\right)$ and $m^{*}(A) \leq c^{*}(A)$ can be strict.

Prove Corollary 4.2.23.

Corollary 4.2.23: A set $E$ is measurable if and only if its complement is measurable.

Let $E \subset \mathbb {R}^{2}$ and let $A \in \mathcal{E}$. Prove that $c(A)=m_{*}\left(A \cap E\right)+m^{*}\left(A \cap E^{C}\right)$.

Let $E \subset [0,1] \times [0,1]$ with the property that if $F$ is a closed measurable set of positive measure then $E \cap F \neq \varnothing$. Prove that $m^{*}(E)=1$.

Let $\mathcal{F}$ be the collection of all closed subsets of $\mathbb {R}^{2}$, and let $\mathcal{F}^{\prime } \subset \mathcal{F}$ consists of sets of positive measure. Prove that $\mathcal{F}^{\prime }$ has the same cardinality as $\mathbb {R}$.

Let $E$ be a set in $\mathbb {R}^{2}$ for which $m^{*}(E)>0$ and let $0<\alpha <m^{*}(E)$. Prove that there is a compact set $K \subset E$ such that $m\left(K\right)=\alpha$.

Let $E$ be a subset of $\mathbb {R}^{2}$. Prove or disprove: $\operatorname {Int}(E)=\varnothing$ if and only if $m^{*}(E)=0$.

Let $A \subset \mathbb {R}^{2}$ intersect every compact set in $\mathbb {R}^{2}$. Prove that $m^{*}(A)=\infty$.

Let $A \subset \mathbb {R}^{2}$ be a bounded set with the property that

Prove or disprove: $A$ is measurable.

Find the Lebesgue measure of a subset of $[0,1] \times [0,1]$ consisting of points $(x, y)$ such that $\left|\sin x\right|<1 / 2$ and $\cos (x+y) \in \mathbb {Q}$.

Let $A, B \subset \mathbb {R}^{2}$, let $A \in \mathcal{M}$, and suppose that $m^{*}(A \triangle B)=0$. Prove that $B \in \mathcal{M}$ and $m\left(B\right)=m\left(A\right)$.

Prove that a set $B \subset \mathbb {R}^{2}$ is nonmeasurable if and only if there exists $\varepsilon >0$ such that for every measurable set $A \subset B, m^{*}(B \backslash A) \geq \varepsilon$.

Prove Corollary 4.3.9.

Corollary 4.3.9: Let $E \subset \mathbb {R}^{2}$. There exists a $G_{\delta }$ set $G \subset \mathbb {R}^{2}$ such that $E \subset G$ and $m(G) = m^{*}(E)$.

Let $A, B \subset \mathbb {R}^{2}$ be sets of finite outer measure such that $A \cup B$ is measurable and $m\left(A \cup B\right)=m^{*}(A)+m^{*}(B)$. Prove that $A, B$ are both measurable sets.

Prove that the Lebesgue outer measure on $\mathbb {R}^{2}$ is translation invariant.

Let $A \subset \mathbb {R}^{2}$ have a finite outer measure. Prove that $A^{C}-A^{C}=\mathbb {R}^{2}$.

The purpose of this exercise is to establish the following result: If $A \subset \mathbb {R}^{2}$ is a Lebesgue measurable set of positive measures, then the set $A-A:=\left\{ x-y: x \in A, y \in A\right\}$ contains a neighborhood of the origin.

Let $\mathfrak {M}$ be a $\sigma$-algebra of subsets of $X$ and let $f: X \rightarrow Y$. Prove or disprove: $f(\mathfrak {M})$ is a $\sigma$-algebra.

Let $\mathfrak {M}$ be a $\sigma$-algebra of subsets of $X$ and let $f: X \rightarrow Y$. Prove or disprove: $\mathfrak {N}:=\left\{ E: f^{-1}(E) \in \mathfrak {M}\right\}$ is a $\sigma$-algebra.

Let $f: X \rightarrow Y$ and let $\mathfrak {N}$ be a $\sigma$-algebra of subsets of $Y$. Prove or disprove: $\mathfrak {M}:=\left\{ f^{-1}(E): E \in \mathfrak {N}\right\}$ is a $\sigma$-algebra.

Let $X=\cup_{n=1}^{\infty } X_{n}$ and let $\mathfrak {M}_{n}$ be a $\sigma$-algebra on $X_{n}$, for each $n \in \mathbb {N}$. Prove that the set

is a $\sigma$-algebra on $X$.

Let $X$ be an uncountable set and let

Prove that $\mathcal{A}$ is a $\sigma$-algebra.

Let $\mathcal{A}_{1}, \mathcal{A}_{2}$ be $\sigma$-algebras on $X$. Prove or disprove: $\mathcal{A}_{1} \cap \mathcal{A}_{2}$ is a $\sigma$ algebra.

Let $\mathcal{A}_{1}, \mathcal{A}_{2}$ be $\sigma$-algebras on $X$. Prove or disprove: $\mathcal{A}_{1} \cup \mathcal{A}_{2}$ is a $\sigma$ algebra.

Let $\left\{ \mathcal{A}_{n}\right\}_{n=1}^{\infty }$ be an increasing sequence of $\sigma$-algebras. Prove or disprove: $\cup_{n=1}^{\infty } \mathcal{A}_{n}$ is a $\sigma$-algebra.

Let $X \subset \mathbb {R}^{2}$ be a countable set and let

Prove that $\mathcal{A}$ is the smallest $\sigma$-algebra that contains $\left\{ x\right\}$ for each $x \in X$.

Let $\mathcal{S}$ be the collection of distinct subsets of $\mathbb {R}$ that are symmetric about the origin. (A set $A$ is symmetric if $x \in A$ if and only if $-x \in A$.) Prove that $\mathcal{S}$ is a $\sigma$-algebra.

If $\mathcal{S}$ is a collection of subsets of $\mathbb {R}^{2}$, we denote by $\sigma (\mathcal{S})$ the smallest $\sigma$-algebra that contains $\mathcal{S}$.

Let $X$ be a set and $\mathcal{A}$ a subset of $\mathbb {P}(X)$. Prove that there is a unique smallest $\sigma$-algebra of subsets of $X$ that contains $\mathcal{A}$.

Let $X$ be a set and $\mathcal{A}$ a $\sigma$-algebra of subsets of $X$. Suppose that $T: X \rightarrow X$ is a bijection such that $T(A) \in \mathcal{A}$ for all $A \in \mathcal{A}$. Prove that $\left\{ A \in \mathcal{A}: T(A)=A\right\}$ is a $\sigma$-algebra.

Let $X, Y$ be nonempty sets and let $\mathcal{A} \subset \mathbb {P}(X)$. Prove that $\sigma (\mathcal{A}) \times Y$ is a $\sigma$-algebra on $X \times Y$.

Let $X$ be a set and $\mathcal{A}$ a $\sigma$-algebra of subsets of $X$. Suppose that $S \subset X$ is a set that is not in $\mathcal{A}$. Prove that

Let $X$ be a set and $\mathcal{A}$ a nonempty subset of $\mathbb {P}(X)$. Prove that

Let $X$ be a set and $\mathcal{A}$ a countable collection of disjoint subsets of $X$ such that their union covers $X$. Describe $\sigma (\mathcal{A})$.

Let $X \subset \mathbb {R}^{2}$, let $\mathcal{S}$ be a collection of subsets of $X$, and let $A \in \sigma (\mathcal{S})$. Prove that there is a countable subset $\mathcal{C}$ of $\mathcal{S}$ such that $A \in \sigma (\mathcal{C})$.

In Exercises 4.3.26-4.3.28, $Y$ is a nonempty subset of $X$ and, for each $\mathcal{A} \subset \mathbb {P}(X), \mathcal{A}_{Y}=\left\{ A \cap Y: A \in \mathcal{A}\right\}$.

Prove that if $\mathcal{A}$ is a $\sigma$-algebra then $\mathcal{A}_{Y}$ is a $\sigma$-algebra on $Y$.

In Exercises 4.3.26-4.3.28, $Y$ is a nonempty subset of $X$ and, for each $\mathcal{A} \subset \mathbb {P}(X), \mathcal{A}_{Y}=\left\{ A \cap Y: A \in \mathcal{A}\right\}$.

Prove that $\left\{ E \subset X: E \cap Y \in \sigma \left(\mathcal{A}_{Y}\right)\right\}$ is a $\sigma$-algebra.

In Exercises 4.3.26-4.3.28, $Y$ is a nonempty subset of $X$ and, for each $\mathcal{A} \subset \mathbb {P}(X), \mathcal{A}_{Y}=\left\{ A \cap Y: A \in \mathcal{A}\right\}$.

Prove that $\sigma \left(\mathcal{A}_{Y}\right)=\sigma (\mathcal{A})_{Y}$.

Prove that a finitely generated $\sigma$-algebra is a finite set.

Prove or disprove: if $\mathcal{A}=\sigma \left(\left\{ A_{n}\right\}_{n \in \mathbb {N}}\right)$ then $\mathcal{A}$ is uncountable.