7.11.E: Problems on Vitali Coverings
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( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercise 7.11.E.1
Prove Theorem 1 for globes, filling in all details.
[Hint: Use Problem 16 in §8.]
Exercise 7.11.E.2
⇒ Show that any (even uncountable) union of globes or nondegenerate cubes Ji⊂En is L-measurable.
[Hint: Include in K each globe (cube) that lies in some Ji. Then Theorem 1 represents ∪JI as a countable union plus a null set.]
Exercise 7.11.E.3
Supplement Theorem 1 by proving that
m∗(A−⋃Iok)=0
and
m∗A=m∗(A∩⋃Iok);
here Io= interior of I.
Exercise 7.11.E.4
Fill in all proof details in Lemmas 1 and 2. Do it also for ¯K={globes}.
Exercise 7.11.E.5
Given mZ=0 and ε>0, prove that there are open globes
G∗k⊆En,
with
Z⊂∞⋃k=1G∗k
and
∞∑k=1mG∗k<ε.
[Hint: Use Problem 3(f) in §5 and Problem 16(iii) from §8.]
Exercise 7.11.E.6
Do Problem 3 in §5 for
(i) C′={open globes}, and
(ii) C′={all globes in En}.
[Hints for (i): Let m′= outer measure induced by v′:C′→E1. From Problem 3(e) in §5, show that
(∀A⊆En)m′A≥m∗A.
To prove m′A≤m∗A also, fix ε>0 and an open set G⊇A with
m∗A+ε≥mG (Theorem 3 of §8).
Globes inside G cover A in the V-sense (why?); so
A⊆Z∪⋃Gk (disjoint)
for some globes Gk and null set Z. With G∗k as in Problem 5,
m′A≤∑(mGk+mG∗k)≤mG+ε≤m∗A+2ε.]
Exercise 7.11.E.7
Suppose f:En onto ⟷En is an isometry, i.e., satisfies
|f(¯x)−f(¯y)|=|¯x−¯y| for ¯x,¯y∈En.
Prove that
(i) (∀A⊆En)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.]