Problem 5
Suppose \(0
Problem 6
Suppose \(0
Problem 9
Suppose \(1 \leq p<\infty\). If \(\left\|f_{n}-f\right\|_{p} \rightarrow 0\), then \(f_{n} \rightarrow f\) in measure, and hence some subsequence converges to \(f\) a.e. On the other hand, if \(f_{n} \rightarrow f\) in measure and \(\left|f_{n}\right| \leq g \in L^{P}\) for all \(n\), then \(\left\|f_{n}-f\right\|_{p} \rightarrow 0\).
Problem 11
If \(f\) is a measurable function on \(X\), define the essential range \(R_{f}\) of \(f\) to be the set of all \(z \in \mathbb{C}\) such that \(\\{x:|f(x)-z|<\epsilon\\}\) has positive measure for all \(\epsilon>0\). a. \(R_{f}\) is closed. b. If \(f \in L^{\infty}\), then \(R_{f}\) is compact and \(\|f\|_{\infty}=\max \left\\{|z|: z \in R_{f}\right\\}\).
Problem 12
If \(P \neq 2\), the \(L^{P}\) norm does not arise from an inner product on \(L^{P}\), except in trivial cases when \(\operatorname{dim}\left(L^{p}\right) \leq 1\). (Show that the parallelogram law fails.)
Problem 13
\(L^{p}\left(\mathbb{R}^{n}, m\right)\) is separable for \(1 \leq p<\infty\). However, \(L^{\infty}\left(\mathbb{R}^{n}, m\right)\) is not separable. (There is an uncountable set \(\mathcal{F} \subset L^{\infty}\) such that \(\|f-g\|_{\infty} \geq 1\) for all \(f, g \in \mathcal{F}\) with \(f \neq g .)\)
Problem 14
If \(g \in L^{\infty}\), the operator \(T\) defined by \(T f=f g\) is bounded on \(L^{p}\) for \(1 \leq p \leq \infty\). Its operator norm is at most \(\|g\|_{\infty}\), with equality if \(\mu\) is semifinite.
Problem 15
(The Vitali Convergence Theorem) Suppose \(1 \leq p<\infty\) and \(\left\\{f_{n}\right\\}_{1}^{\infty} \subset L^{p}\). In order for \(\left\\{f_{n}\right\\}\) to be Cauchy in the \(L^{p}\) norm it is necessary and sufficient for the following three conditions to hold: (i) \(\left\\{f_{n}\right\\}\) is Cauchy in measure; (ii) the sequence \(\left\\{\left|f_{n}\right|^{P}\right\\}\) is uniformly integrable (see Exercise 11 in \(\S 3.2\) ); and (iii) for every \(\in>0\) there exists \(E \subset X\) such that \(\mu(E)<\infty\) and \(\int_{E^{c}}\left|f_{n}\right|^{p}<\epsilon\) for all \(n\). (To prove the sufficiency: Given \(\epsilon>0\), let \(E\) be as in (iii), and let \(A_{\operatorname{mn}}=\\{x \in E\) : \(\left.\left|f_{m}(x)-f_{n}(x)\right| \geq \epsilon\right\\}\). Then the integrals of \(\left|f_{n}-f_{m}\right|^{p}\) over \(E \backslash A_{m n}, A_{m n}\), and \(E^{c}\) are small when \(m\) and \(n\) are large - for three different reasons.)
Problem 23
Let \((X, \mathcal{M}, \mu)\) be a measure space. A set \(E \in \mathcal{M}\) is called locally null if \(\mu(E \cap F)=0\) for every \(F \in \mathcal{M}\) such that \(\mu(F)<\infty\). If \(f: X \rightarrow \mathbb{C}\) is a measurable function, define $$ \|f\|_{*}=\inf \\{a:\\{x:|f(x)|>a\\} \text { is locally null }\\}, $$ and let \(\mathcal{L}^{\infty}=\mathcal{L}^{\infty}(X, \mathcal{M}, \mu)\) be the space of all measurable \(f\) such that \(\|f\|_{0}<\infty\). We consider \(f, g \in \mathcal{L}^{\infty}\) to be identical if \(\\{x: f(x) \neq g(x)\\}\) is locally null. a. If \(E\) is locally null, then \(\mu(E)\) is cither 0 or \(\infty\). If \(\mu\) is semifinite, then every locally null set is null. b. \(\|\cdot\|_{*}\) is a norm on \(\mathcal{L}^{\infty}\) that makes \(\mathcal{L}^{\infty}\) into a Banach space. If \(\mu\) is semifinite, then \(\mathcal{L}^{\infty}=L^{\infty}\).
Problem 30
Suppose that \(K\) is a nonnegative measurable function on \((0, \infty)\) such
that \(\int_{0}^{\infty} K(x) x^{s-1} d x=\phi(s)<\infty\) for \(0
