Math Problems Vault

Problem 5.2.1

If $f: \Omega \rightarrow \overline{\mathbb {R}}$ is $\mathcal{F} / \mathcal{B}^{*}$ -measurable, then $\left|f\right|$ is $\mathcal{F} / \mathcal{B}^{*}$ measurable. However, if $\left|f\right|$ is $\mathcal{F} / \mathscr {B}^{*}$ -measurable, then $f$ is not necessarily $\mathcal{F} / \mathfrak {B}^{*}$ -measurable.

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Problem 5.2.2

(a)(b)

Let $n \in \mathbb {N}$ , and let $f_{1}, \ldots , f_{n}$ denote $\mathcal{F} / \mathcal{B}^{*}$ -measurable functions with common domain $A \in \mathscr {F}$ .

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(a)

Both $\max \left\{ f_{1}, f_{2}\right\}$ and $\min \left\{ f_{1}, f_{2}\right\}$ are $\mathcal{F} / \mathscr {B}^{*}$ -measurable functions.

(b)

Both $\max \left\{ f_{1}, \ldots , f_{n}\right\}$ and $\min \left\{ f_{1}, \ldots , f_{n}\right\}$ are $\mathcal{F} / \mathscr {B}^{*}$ -measurable functions.

Problem 5.2.3

Let ( $\Omega , \mathcal{F}, \mu$ ) denote a measure space, and let $f: \Omega \rightarrow \overline{\mathbb {R}}$ denote a $\mathcal{F} / \mathfrak {B}^{*}$ -measurable mapping. Let $\nu : \mathfrak {B}^{*} \rightarrow \overline{\mathbb {R}}$ be such that $\nu (B)= \mu \left(f^{-1}(B)\right)$ for every $B \in \mathscr {B}^{*}$ . That is, $\nu =\mu \circ f^{-1}$ . Then ( $\overline{\mathbb {R}}, \mathscr {B}^{*}, \nu$ ) is a measure space. Furthermore, even if $\mu$ is $\sigma$ -finite, $\nu$ is not necessarily $\sigma$ -finite.

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Problem 5.2.4

This exercise concerns itself with (MF9).

(MF9): Let $y \in \overline{\mathbb {R}}$ , let $A \in \mathcal{F}$ be nonempty, and let $f: A \rightarrow \overline{\mathbb {R}}$ and $g: A \rightarrow \overline{\mathbb {R}}$ be $\mathcal{F} / \mathfrak {B}^{*}$ -measurable. (a) Let $h: A \rightarrow \overline{\mathbb {R}}$ be such that, for every $\omega \in A$ , $h(\omega ) = f(\omega )+g(\omega )$ if $f(\omega )+g(\omega )$ is defined and $h(\omega ) = y$ if $f(\omega )+g(\omega )$ is undefined. Then $h$ is $\mathcal{F} / \mathfrak {B}^{*}$ -measurable. (b) Let $h: A \rightarrow \overline{\mathbb {R}}$ be such that, for every $\omega \in A$ , $h(\omega ) = f(\omega )-g(\omega )$ if $f(\omega )-g(\omega )$ is defined and $h(\omega ) = y$ if $f(\omega )-g(\omega )$ is undefined. Then $h$ is $\mathcal{F} / \mathfrak {B}^{*}$ -measurable.

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(a)

Prove part (b) of (MF9) by suitably adapting the proof of (a).

(b)

Prove part (b) of (MF9) by using (a) of (MF9) and (MF7).

(c)

Show that $\left\{ \omega \in A: f(\omega )+g(\omega )<x\right\} =\bigcup_{r_{1}, r_{2} \in \mathbb {Q} ; r_{1}+r_{2}<x}\left\{ \omega \in A: f(\omega )<r_{1}\right\} \cap \left\{ \omega \in A: g(\omega )<r_{2}\right\}$ .

(d)

Repeat part (c) for $f-g$ by proving an analogous identity.

(e)

Let $y \in \overline{\mathbb {R}}, n \geqslant 2$ , and for $i=1, \ldots , n$ , let $f_{i}: A \rightarrow \overline{\mathbb {R}}$ denote a $\mathcal{F} / \mathcal{B}^{*}$ measurable function. Let $h: A \rightarrow \overline{\mathbb {R}}$ be defined for all $\omega \in A$ by the rule

h(\omega )= \begin{cases} f_{1}(\omega )+\cdots +f_{n}(\omega ) & \text{ if } f_{1}(\omega )+\cdots +f_{n}(\omega ) \text{ is defined } \\ y & \text{ if } f_{1}(\omega )+\cdots +f_{n}(\omega ) \text{ is undefined. }\end{cases}

Show that $h$ is $\mathcal{F} / \mathscr {B}^{*}$ -measurable.

Problem 5.2.5

(a)(b)(c)

This exercise concerns itself with (MF10).

(MF10): Let $A \in \mathcal{F} - \left\{ \varnothing \right\}$ , and let $f: A \rightarrow \overline{\mathbb {R}}$ and $g: A \rightarrow \overline{\mathbb {R}}$ denote $\mathcal{F} / \mathfrak {B}^{*}$ -measurable functions. The following hold: (a) $fg$ is $\mathcal{F} / \mathfrak {B}^{*}$ -measurable. (b) Let $y \in \overline{\mathbb {R}}$ and $h: A \rightarrow \overline{\mathbb {R}}$ be such that, for each $\omega \in A$ , $h(\omega ) = f(\omega )/g(\omega )$ if $\omega \in A$ is such that $f(\omega )/g(\omega )$ is defined, and $h(\omega ) = y$ otherwise. Then $h$ is $\mathcal{F} / \mathfrak {B}^{*}$ -measurable.

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(a)

Directly prove (a) of (MF10) assuming that $f$ and $g$ are real valued instead of extended real valued.

(b)

Prove (b) of (MF10) assuming that $f$ and $g$ are real valued and that $g$ is nonzero on $A$ .

(c)

Using the previous part, now prove (b) of (MF10) in full generality.