
# 7.8.E: Problems on Lebesgue Measure


## Exercise $$\PageIndex{1}$$

Fill in all details in the proof of Theorems 3 and 4.

Prove Note 2.

## Exercise $$\PageIndex{2}$$

From Theorem 3 deduce that
$\left(\forall A \subseteq E^{n}\right)\left(\exists B \in \mathcal{G}_{\delta}\right) \quad A \subseteq B \text { and } m^{*} A=m B.$
[Hint: See the hint to Problem 7 in §5.]

## Exercise $$\PageIndex{3}$$

Review Problem 3 in §5.

## Exercise $$\PageIndex{4}$$

Consider all translates
$R+p \quad\left(p \in E^{1}\right)$
of
$R=\left\{\text {rationals in } E^{1}\right\}.$
Prove the following.
(i) Any two such translates are either disjoint or identical.
(ii) Each $$R+p$$ contains at least one element of $$[0,1]$$.
[Hint for (ii): Fix a rational $$y \in(-p, 1-p),$$ so $$0<y+p<1.$$ Then $$y+p \in R+p$$, and $$y+p \in[0,1]$$.]

## Exercise $$\PageIndex{5}$$

Continuing Problem 4, choose one element $$q \in[0,1]$$ from each $$R+p.$$ Let $$Q$$ be the set of all $$q$$ so chosen.
Call a translate of $$Q, Q+r,$$ "good" iff $$r \in R$$ and $$|r|<1.$$ Let $$U$$ be the union of all "good" translates of $$Q.$$
Prove the following.
(a) There are only countably many "good" $$Q+r$$.
(b) All of them lie in $$[-1,2]$$.
(c) Any two of them are either disjoint or identical.
(d) $$[0,1] \subseteq U \subseteq[-1,2] ;$$ hence $$1 \leq m^{*} U \leq 3$$.
[Hint for (c): Suppose
$y \in(Q+r) \cap\left(Q+r^{\prime}\right).$
Then
$y=q+r=q^{\prime}+r^{\prime} \quad\left(q, q^{\prime} \in Q, r, r^{\prime} \in R\right);$
so $$q=q^{\prime}+\left(r^{\prime}-r\right),$$ with $$\left(r^{\prime}-r\right) \in R$$.
Thus $$q \in R+q^{\prime}$$ and $$q^{\prime}=0+q^{\prime} \in R+q^{\prime}.$$ Deduce that $$q=q^{\prime}$$ and $$r=r^{\prime} =;$$ hence $$Q+r=Q+r^{\prime}$$.]

## Exercise $$\PageIndex{6}$$

Show that $$Q$$ in Problem 5 is not L-measurable.
[Hint: Otherwise, by Theorem 4, each $$Q+r$$ is L-measurable, with $$m(Q+r)=m Q.$$ By 5(a)(c), $$U$$ is a countable disjoint union of "good" translates.
Deduce that $$m U=0$$ if $$m Q=0,$$ or $$m U=\infty,$$ contrary to 5(d).]

## Exercise $$\PageIndex{7}$$

Show that if $$f : S \rightarrow T$$ is continuous, then $$f^{-1}[X]$$ is a Borel set in $$S$$ whenever $$X \in \mathcal{B}$$ in $$T$$.
[Hint: Using Note 1 in §7, show that
$\mathcal{R}=\left\{X \subseteq T | f^{-1}[X] \in \mathcal{B} \text { in } S\right\}$
is a $$\sigma$$-ring in $$T.$$ As $$\mathcal{B}$$ is the least $$\sigma$$-ring $$\supseteq \mathcal{G}, \mathcal{R} \supseteq \mathcal{B}$$ (the Borel field in $$T$$.]

## Exercise $$\PageIndex{8}$$

Prove that every degenerate interval in $$E^{n}$$ has Lebesgue measure $$0,$$ even if it is uncountable. Give an example in $$E^{2}.$$ Prove uncountability.
[Hint: Take $$\overline{a}=(0,0), \overline{b}=(0,1).$$ Define $$f : E^{1} \rightarrow E^{2}$$ by $$f(x)=(0, x).$$ Show that $$f$$ is one-to-one and that $$[\overline{a}, \overline{b}]$$ is the $$f$$-image of $$[0,1].$$ Use Problem 2 of Chapter 1, §9.]

## Exercise $$\PageIndex{9}$$

Show that not all L-measurable sets are Borel sets in $$E^{n}$$.
[Hint for $$E^{2}:$$ With $$[\overline{a}, \overline{b}]$$ and $$f$$ as in Problem 8, show that $$f$$ is continuous (use the sequential criterion). As $$m[\overline{a}, \overline{b}]=0,$$ all subsets of $$[\overline{a}, \overline{b}]$$ are in $$\mathcal{M}^{*}$$ (Theorem 2(i)), hence in $$\mathcal{B}$$ if we assume $$\mathcal{M}^{*}=\mathcal{B}$$. But then by Problem 7 , the same would apply to subsets of $$[0,1],$$ contrary to Problem 6.
Give a similar proof for $$E^{n}(n>1)$$.
Note: In $$E^{1},$$ too, $$\mathcal{B} \neq \mathcal{M}^{*},$$ but a different proof is necessary. We omit it.]

## Exercise $$\PageIndex{10}$$

Show that Cantor's set $$P$$ (Problem 17 in Chapter 3, 14 ) has Lebesgue measure zero, even though it is uncountable.
[Outline: Let
$U=[0,1]-P;$
so $$U$$ is the union of open intervals removed from $$[0,1].$$ Show that
$m U=\frac{1}{2} \sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n}=1$
and use Lemma 1 in §4.]

## Exercise $$\PageIndex{11}$$

Let $$\mu : \mathcal{B} \rightarrow E^{*}$$ be the Borel restriction of Lebesgue measure $$m$$ in $$E^{n}$$ (§7). Prove that
(i) $$\mu$$ in incomplete;
(ii) $$m$$ is the Lebesgue extension (* and completion, as in Problem 15 of §6) of $$\mu.$$
[Hints: (i) By Problem 9, some $$\mu$$-null sets are not in $$\mathcal{B}.$$ (ii) See the proof (end) of Theorem 2 in §9 (the next section).]

## Exercise $$\PageIndex{12}$$

Prove the following.
(i) All intervals in $$E^{n}$$ are Borel sets.
(ii) The $$\sigma$$-ring generated by any one of the families $$\mathcal{C}$$ or $$\mathcal{C}^{\prime}$$ in Problem 3 of §5 coincides with the Borel field in $$E^{n}.$$
[Hints: (i) Any interval arises from a closed one by dropping some "faces" (degenerate closed intervals). (ii) Use Lemma 2 from §2 and Problem 7 of §3.]

## Exercise $$\PageIndex{13*}$$

Show that if a measure $$m^{\prime}: \mathcal{M}^{\prime} \rightarrow E^{*}$$ in $$E^{n}$$ agrees on intervals with Lebesgue measure $$m: \mathcal{M}^{*} \rightarrow E^{*},$$ then the following are true.
(i) $$m^{\prime}=m$$ on $$\mathcal{B},$$ the Borel field in $$E^{n}$$.
(ii) If $$m^{\prime}$$ is also complete, then $$m^{\prime}=m$$ on $$\mathcal{M}^{*}$$.
[Hint: (i) Use Problem 13 of §5 and Problem 12 above.]

## Exercise $$\PageIndex{14}$$

Show that globes of equal radius have the same Lebesgue measure.
[Hint: Use Theorem 4.]

## Exercise $$\PageIndex{15}$$

Let $$f : E^{n} \rightarrow E^{n},$$ with
$f(\overline{x})=c \overline{x} \quad(0<c<\infty).$
Prove the following.
(i) $$(\forall A \subseteq E^{n}) m^{*} f[A]=c^{n} m^{*} A$$ ($$m^{*}=$$Lebesgue outer measure).
(ii) $$A \in \mathcal{M}^{*}$$ iff $$f[A] \in \mathcal{M}^{*}$$.
[Hint: If, say, $$A=(\overline{a}, \overline{b}],$$ then $$f[A]=(c \overline{a}, c \overline{b}].$$ (Why?) Proceed as in Theorem 4, using $$f^{-1}$$ also.]

## Exercise $$\PageIndex{16}$$

From Problems 14 and 15 show that
(i) $$m G_{\overline{p}}(c r)=c^{n} \cdot m G_{\overline{p}}(r)$$;
(ii) $$m G_{\overline{p}}(r)=m \overline{G}_{\overline{p}}(r)$$;
(iii) $$m G_{\overline{p}}(r)=a \cdot m I,$$ where $$I$$ is the cube inscribed in $$G_{\overline{p}}(r)$$ and
$a=\left(\frac{1}{2} \sqrt{n}\right)^{n} \cdot m G_{\overline{0}}(1).$
[Hints: (i) $$f\left[G_{\overline{0}}(r)\right]=G_{\overline{0}}(c r).$$ (ii) Prove that
$m G_{\overline{p}} \leq m \overline{G}_{\overline{p}} \leq c^{n} m G_{\overline{p}}$
if $$c>1.$$ Let $$c \rightarrow 1$$.]

## Exercise $$\PageIndex{17}$$

Given $$a<b$$ in $$E^{1},$$ let $$\left\{r_{n}\right\}$$ be the sequence of all rationals in $$A=[a, b].$$
Set $$(\forall n)$$
$\delta_{n}=\frac{b-a}{2^{n+1}}$
and
$G_{n}=\left(a_{n}, b_{n}\right)=(a, b) \cap\left(r_{n}-\frac{1}{2} \delta_{n}, r_{n}+\frac{1}{2} \delta_{n}\right).$
Let
$P=A-\bigcup_{n=1}^{\infty} G_{n}.$
Prove the following.
(i) $$\sum_{n=1}^{\infty} \delta_{n}=\frac{1}{2}(b-a)=\frac{1}{2} m A$$.
(ii) $$P$$ is closed; $$P^{o}=\emptyset,$$ yet $$m P>0$$.
(iii) The $$G_{n}$$ can be made disjoint (see Problem 3 in §2), with $$m P$$ still $$>0.$$
(iv) Construct such a $$P \subseteq A\left(P=\overline{P}, P^{o}=\emptyset\right)$$ of prescribed measure $$m P=\varepsilon>0$$.

## Exercise $$\PageIndex{18}$$

Find an open set $$G \subset E^{1},$$ with $$m G<m \overline{G}<\infty.$$
[Hint: $$G=\cup_{n=1}^{\infty} G_{n}$$ with $$G_{n}$$ as in Problem 17.]

## Exercise $$\PageIndex{19*}$$

If $$A \subseteq E^{n}$$ is open and convex, then $$m A=m \overline{A}$$.
[Hint: Let first $$\overline{0} \in A.$$ Argue as in Problem 16.]