7.11.E: Problems on Vitali Coverings
Prove Theorem 1 for globes, filling in all details.
[Hint: Use Problem 16 in §8.]
\(\Rightarrow\) 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.]
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 \(I\).
Fill in all proof details in Lemmas 1 and 2. Do it also for \(\overline{\mathcal{K}}=\){globes}.
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.]
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,
\[m^{\prime} A \leq \sum\left(m G_{k}+m G_{k}^{*}\right) \leq m G+\varepsilon \leq m^{*} A+2 \varepsilon.]\]
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.]
From Problem 7 infer that Lebesgue measure in \(E^{n}\) is rotation invariant. (A rotation about \(\overline{p}\) is an isometry \(f\) such that \(f(\overline{p})=\overline{p}\).)
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,
\[m^{*}(A-U) \leq \lim _{i \rightarrow \infty} \frac{1}{i}=0.]\]
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 \(\overline{\mathcal{K}} \cup \overline{\mathcal{K}}^{*}\).]
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.
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
\[\underline{D} s(\overline{p})=\min D_{\overline{p}} \text { and } \overline{D} s(\overline{p})=\max D_{\overline{p}}.\]
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.]