7.8.E: Problems on Lebesgue Measure
Fill in all details in the proof of Theorems 3 and 4.
Prove Note 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.]
Review Problem 3 in §5.
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 \(y+p \in[0,1]\).]
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 \(Q+r=Q+r^{\prime}\).]
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).]
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 \(T\).]
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.]
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.]
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.]
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).]
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.]
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.]
Show that globes of equal radius have the same Lebesgue measure.
[Hint: Use Theorem 4.]
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.]
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 \(c \rightarrow 1\).]
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 \(m P=\varepsilon>0\).
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.]
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.]