Math Problems Vault

Problem 7.5.1

(a)(b)(c)(d)

For general elements $f \in L^{1}$ , define $\left\| f\right\|_{1}=\int \left|f\right| \, d\bar{\lambda }_{k}$ . Then

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(a)

$0 \leqslant \left\| f\right\|_{1}<+\infty$ .

(b)

$\left\| f\right\|_{1}=0$ iff $f=0$ $\bar{\lambda }_{k}$ -a.e.

(c)

$\left\| cf\right\|_{1}=\left|c\right|\left\| f\right\|_{1}$ for every real $c$ .

(d)

$\left\| f+g\right\|_{1} \leqslant \left\| f\right\|_{1}+\left\| g\right\|_{1}$ [where $g \in L^{1}$ as well].

Remark.

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Parts (a)-(c) are nearly trivial. Use the triangle inequality in (d).

Problem 7.5.2

For elements $f, g \in L^{1}$ write $f \sim g$ iff $f=g$ $\bar{\lambda }_{k}$ -a.e. Then $\sim$ is an equivalence relation.

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Remark.

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We can't formally say that $\left\| \cdot \right\|_{1}$ is a norm on $L^{1}$ because it is not the case that $\left\| f\right\|_{1}=0$ forces $f$ to be the zero element --- the function identically zero --- of $L^{1}$ . However, if we agree by fiat to identify elements of $L^{1}$ that are equal $\bar{\lambda }_{k}$ -a.e., and if we view $\left\| \cdot \right\|_{1}$ as being defined on the collection of equivalence classes $L^{1}/\sim$ of $L^{1}$ induced by $\sim$ , then $\left\| \cdot \right\|_{1}$ is a norm on $L^{1}/\sim$ , where the "zero element" is the equivalence class of functions that are equal to zero $\bar{\lambda }_{k}$ -a.e. We ignore this distinction, and instead talk about "the norm $\left\| \cdot \right\|_{1}$ defined on $L^{1}$ " or "the $L^{1}$ -norm $\left\| \cdot \right\|_{1}$ ," applying $\left\| \cdot \right\|_{1}$ to functions instead of equivalence classes. That is, we write $\left\| f\right\|_{1}$ [which is slightly incorrect] in place of the much more cumbersome [but completely correct] $\left\| \left\{ g \in L^{1}: g \sim f\right\} \right\|_{1}$ .

Problem 7.5.3

For $f, g \in L^{1}$ , define the $L^{1}$ -distance between $f$ and $g$ to be $\left\| f-g\right\|_{1}$ . Show that this distance function is a metric if we agree to identify functions that are equal almost everywhere.

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Problem 7.5.4

Here we will show that the collection $C_{c}$ of continuous functions with compact support is dense in $L^{1}$ , where the notion of distance is as given in the previous exercise. That is, given $f \in L^{1}$ and $\varepsilon >0$ , there is $g \in C_{c}$ such that $\left\| f-g\right\|_{1}<\varepsilon$ . Work in the following stages.

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(a)

$f=\; \mathbb {1}_{A}$ , where $A$ is a bounded Lebesgue subset of $\mathbb {R}^{k}$ . [Find compact $K$ and bounded open $G$ with $K \subseteq A \subseteq G$ and $\bar{\lambda }_{k}(G-K)<\varepsilon$ ; use the approximation theorem in Section 4.2. Use Urysohn's Lemma in Section 5.7 to find $g$ on $\mathbb {R}^{k}$ that is continuous with $0 \leqslant g \leqslant 1$ , $g=1$ on $K$ , and $g=0$ on $G^{c}$ . Argue that $g-\; \mathbb {1}_{A}=0$ for all $\boldsymbol {x} \in K \cup G^{c}$ and $\left|g-\; \mathbb {1}_{A}\right| \leqslant 1$ on $G-K$ . Show that this gives $\left\| g-\; \mathbb {1}_{A}\right\|_{1}<\varepsilon$ .]

(b)

$f$ is a nonnegative finite-valued Lebesgue measurable simple function that is zero outside a bounded set, say $f=\sum_{i=1}^{n} c_{i} \; \mathbb {1}_{A_{i}}$ , where $0 \leqslant c_{1}, \ldots , c_{n}<+\infty$ and $A_{1}, \ldots , A_{n}$ are disjoint bounded $\mathcal{L}^{k}$ -sets. [This follows from (a); approximate each $\; \mathbb {1}_{A_{i}}$ by some $g_{i} \in C_{c}$ . Show that $\sum_{i=1}^{n} c_{i} g_{i} \in C_{c}$ approximates $\sum_{i=1}^{n} c_{i} \; \mathbb {1}_{A_{i}}$ .]

(c)

$f$ is a nonnegative measurable function that is zero outside a bounded set. [Find a nonnegative finite-valued nondecreasing sequence $\left\{ s_{n}\right\}_{n=1}^{\infty }$ of Lebesgue measurable simple functions as in (b) that are zero outside of some bounded set with $s_{n} \rightarrow f$ . Show that $\lim_{n} \left\| s_{n}-f\right\|_{1}=0$ by using DCT. Therefore, $f$ may be approximated by the types of functions given in (b).]

(d)

$f$ is a measurable function that is zero outside a bounded set. [Write $f$ as $f^{+}-f^{-}$ ; use (c). This part shows that $f$ may be approximated by functions as given in (c).]

(e)

$f$ a completely general element of $L^{1}$ . [Use DCT to show the limit relation $\lim_{n} \left\| f \; \mathbb {1}_{B(\boldsymbol {0}, n)}-f\right\|_{1}=0$ . Thus $f$ may be approximated by functions of the type given in (d). Here $B(\boldsymbol {0}, n)$ is the open ball centered at the origin with radius $n$ .]

(f)

Work backward from (e) to (a) to prove the claim given at the beginning of this problem.

Remark.

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Approximation theorem for Lebesgue measure (Section 4.2), the part needed here: given $A \in \mathcal{L}^{k}$ and $\varepsilon >0$ , there exist a closed set $C$ and open set $G_{1}$ with $C \subseteq A \subseteq G_{1}$ and $\lambda_{k}\left(G_{1}-C\right)<\varepsilon /2$ ; combined with the analogous statement for $A^{c}$ , this yields compact $K$ and bounded open $G$ with $K \subseteq A \subseteq G$ and $\lambda_{k}(G-K)<\varepsilon$ whenever $A$ is itself bounded. Urysohn's Lemma (Section 5.7): let $\varnothing \subsetneq F \subseteq G \subseteq \mathbb {R}^{k}$ , where $F$ is closed and $G$ is open. Then there exists a continuous function $\phi : \mathbb {R}^{k} \rightarrow \mathbb {R}$ such that (i) $0 \leqslant \phi (\boldsymbol {x}) \leqslant 1$ for each $\boldsymbol {x} \in \mathbb {R}^{k}$ , (ii) $\phi (\boldsymbol {x})=1$ for each $\boldsymbol {x} \in F$ , (iii) $\phi (\boldsymbol {x})=0$ for each $\boldsymbol {x} \in G^{c}$ , and (iv) $0<\phi (\boldsymbol {x})<1$ for each $\boldsymbol {x} \in G-F$ . The support of a function $f$ is the set $\operatorname {supp}(f)=\overline{\left\{ \boldsymbol {\omega } \in \mathbb {R}^{k}: f(\boldsymbol {\omega }) \neq 0\right\} }$ ; if this closed set is bounded, and hence compact, we say that $f$ has compact support.

Problem 7.5.5

We know that if $f \in L^{1}$ , then $\tau_{\boldsymbol {y}} f \in L^{1}$ for every $\boldsymbol {y} \in \mathbb {R}^{k}$ . [ $\tau_{\boldsymbol {y}}$ is the translation operator given in Claim 2.] Show that $\lim_{\boldsymbol {y} \rightarrow \boldsymbol {0}} \left\| \tau_{\boldsymbol {y}} f-f\right\|_{1}=0$ .

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Remark.

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Claim 2 (of this section): let $f: \mathbb {R}^{k} \rightarrow \overline{\mathbb {R}}$ be Lebesgue measurable, and let $\tau_{\boldsymbol {y}}$ denote translation by some $\boldsymbol {y} \in \mathbb {R}^{k}$ : $\tau_{\boldsymbol {y}} f(\boldsymbol {x})=f(\boldsymbol {x}+\boldsymbol {y})$ for every $\boldsymbol {x} \in \mathbb {R}^{k}$ . Then: (a) $\tau_{\boldsymbol {y}} f$ is Lebesgue measurable; (b) if $f$ is nonnegative, then $\int f \, d\lambda_{k}=\int \tau_{\boldsymbol {y}} f \, d\lambda_{k}$ ; (c) if $f \in L^{1}\left(\mathbb {R}^{k}, \mathcal{L}^{k}, \lambda_{k}\right)$ , then $\tau_{\boldsymbol {y}} f \in L^{1}\left(\mathbb {R}^{k}, \mathcal{L}^{k}, \lambda_{k}\right)$ , and again $\int f \, d\lambda_{k}=\int \tau_{\boldsymbol {y}} f \, d\lambda_{k}$ .