7.11.E: Problems on Vitali Coverings

Exercise

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

Exercise

Show that any (even uncountable) union of globes or nondegenerate cubes is L-measurable.
[Hint: Include in each globe (cube) that lies in some Then Theorem 1 represents as a countable union plus a null set.]

Exercise

Supplement Theorem 1 by proving that

and

here interior of .

Exercise

Fill in all proof details in Lemmas 1 and 2. Do it also for {globes}.

Exercise

Given and prove that there are open globes

with

and

[Hint: Use Problem 3(f) in §5 and Problem 16(iii) from §8.]

Exercise

Do Problem 3 in §5 for
(i) and
(ii) .
[Hints for (i): Let outer measure induced by From Problem 3(e) in §5, show that

To prove also, fix and an open set with

Globes inside cover in the -sense (why?); so

for some globes and null set With as in Problem 5,

Exercise

Suppose is an isometry, i.e., satisfies

Prove that
(i) and
(ii) iff .
[Hints: If is a globe of radius so is (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 is an isometry, too.]

Exercise

From Problem 7 infer that Lebesgue measure in is rotation invariant. (A rotation about is an isometry such that .)

Exercise

A -covering of is called normal iff
(i) and
(ii) for every there is some and a sequence

such that

(We then say that and are normal; specifically, -normal.)
Prove Theorems 1 and 2 for any normal .
[Hints: By Problem 21 of Chapter 3, §16, .
First, suppose is uniformly normal, i.e., all are -normal for the same
In the general case, let

so is uniform for Verify that .
Then select, step by step, as in Theorem 1, a disjoint sequence and naturals such that

Let

Then

and

(Why?) Thus by Problems 7 and 8 in §6,

Exercise

A -covering of is called universal iff
(i) and
(ii) whenever a subfamily covers a set in the -sense, we have

for a disjoint sequence

Show the following.
(a) .
(b) Lemmas 1 and 2 are true with replaced by any universal (In this case, write and for the analogues of and .)
(c) a.e.
[Hints: (a) By (i), minus a null set .
(c) Argue as in Lemma 2, but set

and

to prove a.e. that similarly for .
Throughout assume that is a measure in finite on .]

Exercise

Continuing Problems 8 and 9, verify that
(a) \(\overline{\mathcal{K}}=\{\text {nondegenerate cubes \}\}\) is a normal and universal -covering of ;
(b) so also is ;
(c) is normal.
Note that is not universal.

Exercise

Continuing Definition 3, we call a derivate of and write iff

for some sequence with .
Set

and prove that

Exercise

Let be a normal -covering of (see Problem 8). Given a measure in finite on write

iff

for some normal sequence with .
Set

and then

Prove that

[Hint: where

On each is uniformly normal. To prove a.e. on "imitate" Problem 9(c). Proceed.]