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

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## 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
$$n^{*} A=m(H \cap K)=m^{*} A$$.]

7.7.E: Problems on Topologies, Borel Sets, and Regular Measures is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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