Jordan Measure

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.