7.11.E: Problems on Vitali Coverings

Exercise $\PageIndex{1}$

Prove Theorem 1 for globes, filling in all details.
[Hint: Use Problem 16 in §8.]

Exercise $\PageIndex{2}$

$\Rightarrow$ Show that any (even uncountable) union of globes or nondegenerate cubes $J_{i} \subset E^{n}$ is L-measurable.
[Hint: Include in $\mathcal{K}$ each globe (cube) that lies in some $J_{i}.$ Then Theorem 1 represents $\cup J_{I}$ as a countable union plus a null set.]

Exercise $\PageIndex{3}$

Supplement Theorem 1 by proving that
$$m^{*}\left(A-\bigcup I_{k}^{o}\right)=0$$
and
$$m^{*} A=m^{*}\left(A \cap \bigcup I_{k}^{o}\right);$$
here $I^{o}=$ interior of $I$.

Exercise $\PageIndex{4}$

Fill in all proof details in Lemmas 1 and 2. Do it also for $\overline{\mathcal{K}}=${globes}.

Exercise $\PageIndex{5}$

Given $m Z=0$ and $\varepsilon>0,$ prove that there are open globes
$$G_{k}^{*} \subseteq E^{n},$$
with
$$Z \subset \bigcup_{k=1}^{\infty} G_{k}^{*}$$
and
$$\sum_{k=1}^{\infty} m G_{k}^{*}<\varepsilon.$$
[Hint: Use Problem 3(f) in §5 and Problem 16(iii) from §8.]

Exercise $\PageIndex{6}$

Do Problem 3 in §5 for
(i) $\mathcal{C}^{\prime}=\{\text {open globes}\},$ and
(ii) $\mathcal{C}^{\prime}=\left\{\text {all globes in } E^{n}\right\}$.
[Hints for (i): Let $m^{\prime}=$ outer measure induced by $v^{\prime}: \mathcal{C}^{\prime} \rightarrow E^{1}.$ From Problem 3(e) in §5, show that
$$\left(\forall A \subseteq E^{n}\right) \quad m^{\prime} A \geq m^{*} A.$$
To prove $m^{\prime} A \leq m^{*} A$ also, fix $\varepsilon>0$ and an open set $G \supseteq A$ with
$$m^{*} A+\varepsilon \geq m G \text { (Theorem 3 of §8).}$$
Globes inside $G$ cover $A$ in the $V$-sense (why?); so
$$A \subseteq Z \cup \bigcup G_{k} \text { (disjoint)}$$
for some globes $G_{k}$ and null set $Z.$ With $G_{k}^{*}$ as in Problem 5,
$$m^{\prime} A \leq \sum\left(m G_{k}+m G_{k}^{*}\right) \leq m G+\varepsilon \leq m^{*} A+2 \varepsilon.]$$

Exercise $\PageIndex{7}$

Suppose $f : E^{n} \stackrel{\text { onto }}{\longleftrightarrow} E^{n}$ is an isometry, i.e., satisfies
$$|f(\overline{x})-f(\overline{y})|=|\overline{x}-\overline{y}| \quad \text { for } \overline{x}, \overline{y} \in E^{n}.$$
Prove that
(i) $\left(\forall A \subseteq E^{n}\right) m^{*} A=m^{*} f[A],$ and
(ii) $A \in \mathcal{M}^{*}$ iff $f[A] \in \mathcal{M}^{*}$.
[Hints: If $A$ is a globe of radius $r,$ so is $f[A]$ (verify!); thus Problems 14 and 16 in §8 apply. In the general case, argue as in Theorem 4 of §8 , replacing intervals by globes (see Problem 6). Note that $f^{-1}$ is an isometry, too.]

Exercise $\PageIndex{7'}$

From Problem 7 infer that Lebesgue measure in $E^{n}$ is rotation invariant. (A rotation about $\overline{p}$ is an isometry $f$ such that $f(\overline{p})=\overline{p}$.)

Exercise $\PageIndex{8}$

A $V$-covering $\mathcal{K}$ of $A \subseteq E^{n}$ is called normal iff
(i) $(\forall I \in K) 0<m \overline{I}=m I^{o},$ and
(ii) for every $\overline{p} \in A,$ there is some $c \in(0, \infty)$ and a sequence
$$I_{k} \rightarrow \overline{p} \quad\left(\left\{I_{k}\right\} \subseteq \mathcal{K}\right)$$
such that
$$(\forall k)\left(\exists \text { cube } J_{k} \supseteq I_{k}\right) \quad c \cdot m^{*} I_{k} \geq m J_{k}.$$
(We then say that $\overline{p}$ and $\left\{I_{k}\right\}$ are normal; specifically, $c$-normal.)
Prove Theorems 1 and 2 for any normal $\mathcal{K}$.
[Hints: By Problem 21 of Chapter 3, §16, $d I=d \overline{I}$.
First, suppose $\mathcal{K}$ is uniformly normal, i.e., all $\overline{p} \in A$ are $c$-normal for the same $c.$
In the general case, let
$$A_{i}=\{\overline{x} \in A | \overline{x} \text { is } i \text {-normal}\}, \quad i=1,2, \ldots;$$
so $\mathcal{K}$ is uniform for $A_{i}.$ Verify that $A_{i} \nearrow A$.
Then select, step by step, as in Theorem 1, a disjoint sequence $\left\{I_{k}\right\} \subseteq \mathcal{K}$ and naturals $n_{1}<n_{2}<\cdots<n_{i}<\cdots$ such that
$$(\forall i) \quad m^{*}\left(A_{i}-\bigcup_{k=1}^{n_{i}} I_{k}\right)<\frac{1}{i}.$$
Let
$$U=\bigcup_{k=1}^{\infty} I_{k}.$$
Then
$$(\forall i) \quad m^{*}\left(A_{i}-U\right)<\frac{1}{i}$$
and
$$A_{i}-U \nearrow A-U.$$
(Why?) Thus by Problems 7 and 8 in §6,
$$m^{*}(A-U) \leq \lim _{i \rightarrow \infty} \frac{1}{i}=0.]$$

Exercise $\PageIndex{9}$

A $V$-covering $\overline{K}^{*}$ of $E^{n}$ is called universal iff
(i) $(\forall I \in \overline{\mathcal{K}}^{*}\right) 0<m \overline{I}=m I^{o}<\infty,$ and
(ii) whenever a subfamily $\mathcal{K} \subseteq \overline{\mathcal{K}}^{*}$ covers a set $A \subseteq E^{n}$ in the $V$-sense, we have
$$m^{*}\left(A-\bigcup I_{k}\right)=0$$
for a disjoint sequence
$$\left\{I_{k}\right\} \subseteq \mathcal{K}.$$
Show the following.
(a) $\overline{\mathcal{K}}^{*} \subseteq \mathcal{M}^{*}$.
(b) Lemmas 1 and 2 are true with $\overline{\mathcal{K}}$ replaced by any universal $\overline{\mathcal{K}}^{*}.$ (In this case, write $\underline{D}^{*} s$ and $\overline{D}^{*} s$ for the analogues of $\underline{D} s$ and $\overline{D}_{s}$.)
(c) $\underline{D s}=\underline{D}^{*} s=\overline{D}^{*} s=\overline{D} s$ a.e.
[Hints: (a) By (i), $I=\overline{I}$ minus a null set $Z \subseteq \overline{I}-I^{o}$.
(c) Argue as in Lemma 2, but set
$$Q=J\left(\underline{D}^{*} s>u>v>\underline{D} s\right)$$
and
$$\mathcal{K}^{\prime}=\left\{I \in \overline{\mathcal{K}}^{*} | I \subseteq G^{\prime}, \frac{s I}{m I}>v\right\}$$
to prove a.e. that $\underline{D}^{*} s \leq \underline{D} s;$ similarly for $\underline{D} s \leq D^{*} s$.
Throughout assume that $s : \mathcal{M}^{\prime} \rightarrow E^{*}\left(\mathcal{M}^{\prime} \supseteq \overline{\mathcal{K}} \cup \overline{\mathcal{K}}^{*}\right)$ is a measure in $E^{n},$ finite on $\overline{\mathcal{K}} \cup \overline{\mathcal{K}}^{*}$.]

Exercise $\PageIndex{10}$

Continuing Problems 8 and 9, verify that
(a) \(\overline{\mathcal{K}}=\{\text {nondegenerate cubes \}\}\) is a normal and universal $V$-covering of $E^{n}$;
(b) so also is $\overline{\mathcal{K}}^{o}=\left\{\text {all globes in} E^{n}\right\}$;
(c) $\overline{\mathcal{C}}=\{\text {nondegenerate intervals}\}$ is normal.
Note that $\overline{\mathcal{C}}$ is not universal.

Exercise $\PageIndex{11}$

Continuing Definition 3, we call $q$ a derivate of $s,$ and write $q \sim D s(\overline{p}),$ iff
$$q=\lim _{k \rightarrow \infty} \frac{s I_{k}}{m I_{k}}$$
for some sequence $I_{k} \rightarrow \overline{p},$ with $I_{k} \in \overline{\mathcal{K}}$.
Set
$$D_{\overline{p}}=\left\{q \in E^{*} | q \sim D s(\overline{p})\right\}$$
and prove that
$$\underline{D} s(\overline{p})=\min D_{\overline{p}} \text { and } \overline{D} s(\overline{p})=\max D_{\overline{p}}.$$

Exercise $\PageIndex{12}$

Let $\mathcal{K}^{*}$ be a normal $V$-covering of $E^{n}$ (see Problem 8). Given a measure $s$ in $E^{n},$ finite on $\mathcal{K}^{*} \cup \overline{\mathcal{K}},$ write
$$q \sim D^{*} s(\overline{p})$$
iff
$$q=\lim _{k \rightarrow \infty} \frac{s I_{k}}{m I_{k}}$$
for some normal sequence $I_{k} \rightarrow \overline{p},$ with $I_{k} \in \mathcal{K}^{*}$.
Set
$$D_{\overline{p}}^{*}=\left\{q \in E^{*} | q \sim D^{*} s(\overline{p})\right\},$$
and then
$$\underline{D}^{*} s(\overline{p})=\inf D_{\overline{p}}^{*} \text { and } \overline{D}^{*} s(\overline{p})=\sup D_{\overline{p}}^{*}.$$
Prove that
$$\underline{D} s=\underline{D}^{*} s=\overline{D}^{*} s=\overline{D} s \text { a.e. on } E^{n}.$$
[Hint: $E^{n}=\bigcup_{i=1}^{\infty} E_{i},$ where
$$E_{i}=\left\{\overline{x} \in E^{n} | \overline{x} \text { is } i \text {-normal}\right\}.$$
On each $E_{i}, \mathcal{K}^{*}$ is uniformly normal. To prove $\underline{D} s=\underline{D}^{*} s$ a.e. on $E_{i},$ "imitate" Problem 9(c). Proceed.]