Lebesgue Measure

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 $\cos (x+y) \in \mathbb {Q}$.