Math Problems Vault

Problem 3.9.1

Countable superadditivity: If $A_{1}, A_{2}, \ldots \subseteq \Omega$ are disjoint, then $\mu_{*}\left(\bigcup_{n=1}^{\infty } A_{n}\right) \geqslant \sum_{n=1}^{\infty } \mu_{*}\left(A_{n}\right)$ .

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Problem 3.9.2

For any $A \subseteq \Omega , \mu^{*}(A)=\inf \left\{ \mu_{\mathcal{M}\left(\mu^{*}\right)}^{*}(B): A \subseteq B \in \mathcal{M}\left(\mu^{*}\right)\right\} =\mu^{* *}(A)$ .

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Problem 3.9.3

For any $A \subseteq \Omega , \mu_{*}(A)=\sup \left\{ \mu_{\mathcal{M}\left(\mu^{*}\right)}^{*}(B): B \subseteq A, B \in \mathcal{M}\left(\mu^{*}\right)\right\}$ .

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Problem 3.9.4

For any $A \subseteq \Omega$ , there is $E \in \sigma (\mathcal{A})$ such that $E \subseteq A$ and $\mu_{\sigma (\mathscr {A})}^{*}(E)=\mu_{*}(A)$ .

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Problem 3.9.5

The infimum that defines $\mu^{*}(A)$ and the supremum that defines $\mu_{*}(A)$ are achieved for each $A \subseteq \Omega$ . That is, there exist $C, B \in \sigma (\mathcal{A})$ with $C \subseteq A \subseteq B$ and $\mu_{\sigma (\mathscr {A})}^{*}(C)=\mu_{*}(A)$ and $\mu_{\sigma (\mathscr {A})}^{*}(B)=\mu^{*}(A)$ .

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Problem 3.9.6

Let $A \subseteq \Omega$ and let $\left\{ A_{n}\right\}$ denote a disjoint sequence of $\mathcal{M}\left(\mu^{*}\right)$ -sets. Then we have $\mu_{*}\left(\bigcup_{n=1}^{\infty }\left(A \cap A_{n}\right)\right)=\sum_{n=1}^{\infty } \mu_{*}\left(A_{n} \cap A\right)$ .

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Problem 3.9.7

If $A, B \subseteq \Omega$ are disjoint, then $\mu_{*}(A \cup B) \leqslant \mu_{*}(A)+ \mu^{*}(B) \leqslant \mu^{*}(A \cup B)$ .

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Problem 3.9.8

If $A \in \mathcal{M}\left(\mu^{*}\right)$ and $B \subseteq \Omega$ , then $\mu_{\mathcal{M}\left(\mu^{*}\right)}^{*}(A)=\mu_{*}(B \cap A)+\mu^{*}\left(B^{c} \cap A\right)$ .

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Problem 3.9.9

(a)(b)(c)

Let $\Omega =(0,1] \times (0,1]$ , and let $\mathcal{F}=\left\{ B \times (0,1]: B \in \mathcal{B}, B \subseteq (0,1]\right\}$ . Define $\mu : \mathcal{F} \rightarrow \overline{\mathbb {R}}$ by requiring that $\mu (B \times (0,1])=v(B)$ whenever $B \in \mathcal{B}$ and $B \subseteq (0,1]$ , where $v$ is the unique measure on $\left\{ B \in \mathcal{B}: B \subseteq (0,1]\right\}$ that assigns measure $b-a$ to each $(a, b] \subseteq (0,1]$ .

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(a)

$\mathcal{F}$ is a $\sigma$ -field.

(b)

$\mu$ is well-defined and is a measure on $\mathcal{F}$ .

(c)

Let $C=(0,1] \times \left\{ \frac{1}{2}\right\}$ . Then $C \notin \mathcal{F}$ , $v_{*}(C)=0$ , and $v^{*}(C)=1$ .

Problem 3.9.10

Is the hypothesis of finiteness essential in part (b) of Claim 3?

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Remark.

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Claim 3 (Section 3.9): We have the following: (a) if $A \in \mathcal{M}\left(\mu^{*}\right)$ , then $\mu_{*}(A)=\mu^{*}(A)=\mu^{* *}(A)$ , and (b) if $A \subseteq \Omega$ is such that $\mu_{*}(A)=\mu^{*}(A)=\mu^{* *}(A)<+\infty$ , then $A \in \mathcal{M}\left(\mu^{*}\right)$ .