7.7.E: Problems on Topologies, Borel Sets, and Regular Measures

Exercise \(\PageIndex{1}\)

Show that \(\mathcal{G}\) is a topology in \(S\) (in \((a)-(c),\) describe \(\mathcal{B}\) also), given
(a) \(\mathcal{G}=2^{S}\);
(b) \(\mathcal{G}=\{\emptyset, S\}\);
(c) \(\mathcal{G}=\{\emptyset \text { and all sets in } S, \text { containing a fixed point } p\};\) or
(d) \(S=E^{*} ; \mathcal{G}\) consists of all possible unions of sets of the form \((a, b), (a, \infty],\) and \([-\infty, b),\) with \(a, b \in E^{1}.\)

Exercise \(\PageIndex{2}\)

\((S, \rho)\) is called a pseudometric space (and \(\rho\) is a pseudometric) iff the metric laws (i)-(iii) of Chapter 3, s 11 hold, but (i) is weakened to
\[\rho(x, x)=0\]
(so that \(\rho(x, y)\) may be \(0\) even if \(x \neq y\)).
(a) Define "globes," "interiors," and "open sets" (i.e., \(\mathcal{G})\) as in Chapter 3, §12; then show that \(\mathcal{G}\) is a topology for \(S.\)
(b) Let \(S=E^{2}\) and
\[\rho(\overline{x}, \overline{y})=\left|x_{1}-y_{1}\right|,\]
where \(\overline{x}=\left(x_{1}, x_{2}\right)\) and \(\overline{y}=\left(y_{1}, y_{2}\right).\) Show that \(\rho\) is a pseudometric but not a metric (the Hausdorff properly fails!).

Exercise \(\PageIndex{3}\)

Define "neighborhood," "interior, "cluster point," "closure," and "function limit" for topological spaces. Specify some notions (e.g., "diameter," "uniform continuity") that do not carry over (they involve distances).

Exercise \(\PageIndex{4}\)

In a topological space \((S, \mathcal{G}),\) detine
\[\mathcal{G}^{0}=\mathcal{G}, \mathcal{G}^{1}=\mathcal{G}_{\delta}, \mathcal{G}^{2}=\mathcal{G}_{\delta \sigma}, \ldots\]
and
\[\mathcal{F}^{0}=\mathcal{F}, \mathcal{F}^{1}=\mathcal{F}_{\sigma}, \mathcal{F}^{2}=\mathcal{F}_{\sigma \delta}, \mathcal{F}^{3}=\mathcal{F}_{\sigma \delta \sigma}, \text { etc.}\]
(Give an inductive definition.) Then prove by induction that
(a) \(\mathcal{G}^{n} \subseteq \mathcal{B}, \mathcal{F}^{n} \subseteq \mathcal{B}\);
(b) \(\mathcal{G}^{n-1} \subseteq \mathcal{G}^{n}, \mathcal{F}^{n-1} \subseteq \mathcal{F}^{n}\);
(c) \((\forall X \subseteq S) X \in \mathcal{F}^{n}\) iff \(-X \in \mathcal{G}^{n}\);
(d) \(\left(\forall X, Y \in \mathcal{F}^{n}\right) X \cap Y \in \mathcal{F}^{n}, X \cup Y \in \mathcal{F}^{n};\) same for \(\mathcal{G}^{n}\);
(e) \(\left(\forall X \in \mathcal{G}^{n}\right)\left(\forall Y \in \mathcal{F}^{n}\right) X-Y \in \mathcal{G}^{n}\) and \(Y-X \in \mathcal{F}^{n}\).
[Hint: \(X-Y=X \cap-Y\).]

Exercise \(\PageIndex{5}\)

For metric and pseudometric spaces (see Problem 2 ) prove that
\[\mathcal{F}^{n} \subseteq \mathcal{G}^{n+1} \text { and } \mathcal{G}^{n} \subseteq \mathcal{F}^{n+1}\]
(cf. Problem 4).
[Hint for \(\mathcal{F} \subseteq \mathcal{G}_{\delta}:\) Let \(F \in \mathcal{F}.\) Set
\[G_{n}=\bigcup_{p \in F} G_{p}\left(\frac{1}{n}\right);\]
so
\[(\forall n) \quad F \subseteq G_{n} \in \mathcal{G}.\]
Hence
\[F \subseteq \bigcap_{n} G_{n} \in \mathcal{G}_{\delta}.\]
Also,
\[\bigcap_{n} G_{n}=\overline{F}=F\]
by Theorem 3 in Chapter 3, §16. Hence deduce that
\[(\forall F \in \mathcal{F}) \quad F \in \mathcal{G}_{\delta},\]
so \(\mathcal{F} \subseteq \mathcal{G}_{\delta};\) hence \(\mathcal{G} \subseteq \mathcal{F}_{\sigma}\) by Problem 4(c). Now use induction.]

Exercise \(\PageIndex{6}\)

If \(m\) is as in Definition 5, then prove the following.
(i) \(m\) is regular.
(ii) \((\forall A \in \mathcal{M}) m A=\sup \{m X | A \supseteq X \in \mathcal{M} \cap \mathcal{F}\}\).
(iii) The latter implies strong regularity if \(m<\infty\) and \(S \in \mathcal{M}\).

Exercise \(\PageIndex{7}\)

Let \(\mu : \mathcal{B} \rightarrow E^{*}\) be a Borel measure in a metric space \((S, \rho).\) Set
\[(\forall A \subseteq S) \quad n^{*} A=\inf \{\mu X | A \subseteq X \in \mathcal{G}\}.\]
Prove that
(i) \(n^{*}\) is an outer measure in \(S\);
(ii) \(n^{*}=\mu\) on \(\mathcal{G}\);
(iii) the \(n^{*}\)-induced measure, \(n : \mathcal{N}^{*} \rightarrow E^{*},\) is topological (so \(\mathcal{B} \subseteq \mathcal{N}^{*}\));
(iv) \(n \geq \mu\) on \(\mathcal{B}\);
(v) \((\forall A \subseteq S)\left(\exists H \in \mathcal{G}_{\delta}\right) A \subseteq H\) and \(\mu H=n^{*} A\).
[Hints: (iii) Using Problem 15 in §5 and Problem 12 in §6, let
\[\rho(X, Y)>\varepsilon>0, \quad U=\bigcup_{x \in X} G_{x}\left(\frac{1}{2} \varepsilon\right), \quad V=\bigcup_{y \in Y} G_{y}\left(\frac{1}{2} \varepsilon\right).\]
Verify that \(U, V \in \mathcal{G}, U \supseteq X, V \supseteq Y, U \cap V=\emptyset\).
By the definition of \(n^{*}\),
\[(\exists G \in \mathcal{G}) \quad G \supseteq X \cup Y \text { and } n^{*} G \leq n^{*}(X \cup Y)+\varepsilon;\]
also, \(X \subseteq G \cap U\) and \(Y \subseteq G \cap V.\) Thus by (ii),
\[n^{*} X \leq \mu(G \cap U) \text { and } n^{*} Y \leq \mu(G \cap V).\]
Hence
\[n^{*} X+n^{*} Y \leq \mu(G \cap U)+\mu(G \cap V)=\mu((G \cap U) \cup(G \cap V)) \leq \mu G=n^{*} G \leq n^{*}(X \cup Y)+\varepsilon.\]
Let \(\varepsilon \rightarrow 0\) to get the \(\mathrm{CP} : n^{*} X+n^{*} Y \leq n^{*}(X \cup Y)\).
(iv) We have \((\forall A \in \mathcal{B})\)
\[n A=n^{*} A=\inf \{\mu X | A \subseteq X \in \mathcal{G}\} \geq \inf \{\mu X | A \subseteq X \in \mathcal{B}\}=\mu A.\]
(Why?)
(v) Use the hint to Problem 11 in §5.]

Exercise \(\PageIndex{8}\)

From Problem 7 with \(m=\mu,\) prove that if
\[A \subseteq G \in \mathcal{G},\]
with \(m G<\infty\) and \(A \in \mathcal{B},\) then \(m A=n A\).
[Hint: \(A, G,\) and \((G-A) \in \mathcal{B}.\) By Problem 7(iii), \(\mathcal{B} \subseteq N^{*}\) and \(n\) is additive on \(\mathcal{B}\); so by Problem 7(ii)(iv),
\[n A=n G-n(G-A) \leq m G-m(G-A)=m A \leq n A.\]
Thus \(m A=n A.\) Explain all!]

Exercise \(\PageIndex{9}\)

Let \(m, n,\) and \(n^{*}\) be as in Problems 7 and 8. Suppose
\[S=\bigcup_{n=1}^{\infty} G_{n},\]
with \(G_{n} \in \mathcal{G}\) and \(m G_{n}<\infty\) (this is called \(\sigma^{0}\)-finiteness).
Prove that
(i) \(m=n\) on \(\mathcal{B},\) and
(ii) \(m\) and \(n\) are strongly regular.
[Hints: Fix \(A \in \mathcal{B}.\) Show that
\[A=\bigcup A_{n} \text { (disjoint)}\]
for some Borel sets \(A_{n} \subseteq G_{n}\) (use Corollary 1 in §1). By Problem 8, \(m A_{n}=n A_{n}\) since
\[A_{n} \subseteq G_{n} \in \mathcal{G}\]
and \(m G_{n}<\infty.\) Now use \(\sigma\)-additivity to find \(m A=n A\).
(ii) Use \(\mathcal{G}\)-regularity, part (i), and Theorem 1.]

Exercise \(\PageIndex{10}\)

Continuing Problems 8 and 9, show that \(n\) is the Lebesgue extension of \(m\) (see Theorem 2 in §6 and Note 3 in §6).
Thus every \(\sigma^{0}\)-finite Borel measure \(m\) in \((S, \rho)\) and its Lebesgue extension are strongly regular.
[Hint: \(m\) induces an outer measure \(m^{*},\) with \(m^{*}=m\) on \(\mathcal{B}.\) It suffices to show that \(m^{*}=n^{*}\) on \(2^{S}.\) (Why?)
So let \(A \subseteq S.\) By Problem 7(v),
\[(\exists H \in \mathcal{B}) A \subseteq H \text { and } n^{*} A=m H=m^{*} H.\]
Also,
\[(\exists K \in \mathcal{B}) A \subseteq K \text { and } m^{*} A=m K\]
(Problem 12 in §5). Deduce that
\[n^{*} A \leq n(H \cap K)=m(H \cap K) \leq m H=n^{*} A\]
and
\(n^{*} A=m(H \cap K)=m^{*} A\).]