# 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.]

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