Math Problems Vault

Problem 8.4.1

In this exercise, we deal with the formal definition of $L^{\infty }$ . We define $L^{\infty }$ to consist of those equivalence classes $\mathfrak {A} \in \mathcal{E} / \sim$ such that $\operatorname {ess} \sup f<+\infty$ for some $f \in \mathfrak {A}$ .

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(a)

If $f=g \: \mu$ -a.e. on $\Omega$ , then $\operatorname {ess} \sup f=\operatorname {ess} \sup g$ . Therefore, we may equivalently say that $\mathfrak {A} \in L^{\infty }$ iff $\operatorname {ess} \sup f<+\infty$ for every $f \in \mathfrak {A}$ .

(b)

If $\mathfrak {A}, \mathfrak {B} \in L^{\infty }$ and $c \in \mathbb {R}$ , then $c \mathfrak {A} \in L^{\infty }$ and $\mathfrak {A}+\mathfrak {B} \in L^{\infty }$ . Therefore, $L^{\infty }$ is a vector space.

We define $\left\| \cdot \right\|_{\infty }$ on $L^{\infty }$ by writing $\left\| \mathfrak {A}\right\|_{\infty }=\operatorname {ess} \sup f$ , where $f \in \mathfrak {A}$ .

(c)

The definition of $\left\| \cdot \right\|_{\infty }$ is well defined.

(d)

For $\mathfrak {A}, \mathfrak {B}, \mathfrak {C} \in L^{\infty }$ , define $\rho (\mathfrak {A}, \mathfrak {B})=\left\| \mathfrak {A}-\mathfrak {B}\right\|_{\infty }$ . Then

$0 \leqslant \rho (\mathfrak {A}, \mathfrak {B})<+\infty$ .
$\rho (\mathfrak {A}, \mathfrak {A})=0$ , and $\rho (\mathfrak {A}, \mathfrak {B})=0$ iff $\mathfrak {A}=\mathfrak {B}$ .
$\rho (\mathfrak {A}, \mathfrak {B})=\rho (\mathfrak {B}, \mathfrak {A})$ .
$\rho (\mathfrak {A}, \mathfrak {C}) \leqslant \rho (\mathfrak {A}, \mathfrak {B})+\rho (\mathfrak {B}, \mathfrak {C})$ . Therefore, $\rho$ is a metric on $L^{\infty }$ .

(e)

$L^{\infty }$ is complete in the metric $\rho$ . Therefore $L^{\infty }$ is a Banach space.

(f)

Remark.

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All parts of this exercise can be handled quickly by using the results of this section.

Problem 8.4.2

(a)(b)

Consider the $\sigma$ -finite measure space $(\Omega , \mathcal{F}, \mu )$ .

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(a)

$f \in L^{\infty }$ iff there is a bounded $\mathcal{F} / \mathfrak {B}^{*}$ -measurable function $g$ on $\Omega$ such that $f=g \mu -$ a. e. on $\Omega$ .

(b)

If $f \in L^{\infty }$ , then $\left\| f\right\|_{\infty }=\inf \left\{ \sup_{\omega \in \Omega }\left|g(\omega )\right|: g \text{ is as in (a)}\right\}$ .

Problem 8.4.3

Consider the measure space $\left(\mathbb {R}^{k}, \mathcal{B}^{k}, \lambda_{k}\right)$ . We have $\operatorname {ess} \sup f=\sup_{\boldsymbol {x} \in \mathbb {R}^{k}}\left|f(\boldsymbol {x})\right|$ if $f$ is continuous.

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Problem 8.4.4

Quickly prove the $p=\infty$ version of Hölder's Inequality.

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Problem 8.4.5

(a)(b)

Let ( $\Omega , \mathcal{F}, \mu$ ) be such that $\mu (\Omega )<\infty$ , and let $f$ denote a bounded $\mathcal{F} / \mathcal{B}^{*}$ -measurable function on $\Omega$ .

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(a)

For every $1 \leqslant p \leqslant \infty$ we have $f \in L^{p}$ , hence $\left\| f\right\|_{p}$ exists.

(b)

$\lim_{p \rightarrow \infty }\left\| f\right\|_{p}=\left\| f\right\|_{\infty }$ .

Problem 8.4.6

For each $k \in \mathbb {N}$ , $L^{\infty }\left(\mathbb {R}^{k}, \mathcal{B}^{k}, \lambda_{k}\right)$ fails to contain an unbounded continuous function. Also, for each $1 \leqslant p<+\infty$ , $L^{p}(\mathbb {R}, \mathcal{B}, \lambda )$ contains an unbounded continuous function.

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Remark.

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See Exercise 8.4.3.

Problem 8.4.7

Let $\Omega =(0,1]$ , let $\mathcal{F}$ denote the Borel subsets of $(0,1]$ , and let $\mu$ denote Lebesgue measure restricted to the Borel subsets of $(0,1]$ . Show in this case that $L^{\infty }$ is not separable, that is, there does not exist a countable dense subset of $L^{\infty }$ .

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