Exercise 1.3.9 ($L^p$ is separable)

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Suppose that $\mu$ is $\sigma$-finite and $\mathcal{X}$ is separable (i.e., countably generated). Show that $L^p$ is separable (i.e., has a countable dense subset) for all $1 \leq p < \infty$. Give a counterexample that shows that $L^{\infty}$ need not be separable.

Elu Thingol · Accepted Answer

\textit{1. Show that $L^p(\mathbb{R})$ is separable for $1 \leq p < \infty$ }\newline The strategy is as follows. \begin{figure}[H] \centering % Define variables \def \rectWidth {9.5cm} \def \rectHeight {1.5cm} \def \verticalDistance {2.5cm} \def \arrowLength {4cm} \begin{tikzpicture} % Draw rectangles with text \draw (0,0) rectangle (\rectWidth,-\rectHeight) node[pos=.5,align=left] {elementary functions with $\mathbb{Q}$-coefficients\ and supports of finite measure from sets of countable basis}; \draw (0,-\verticalDistance) rectangle (\rectWidth,-\verticalDistance-\rectHeight) node[pos=.5,align=left] {elementary functions\with support of finite measure}; \draw (0,-2*\verticalDistance) rectangle (\rectWidth,-2*\verticalDistance-\rectHeight) node[pos=.5,align=left] {bounded functions in $L^p$\ with support of finite dimension}; \draw (0,-3*\verticalDistance) rectangle (\rectWidth,-3*\verticalDistance-\rectHeight) node[pos=.5,align=left] {Bounded functions in $L^p$}; \draw (0,-4*\verticalDistance) rectangle (\rectWidth,-4*\verticalDistance-\rectHeight) node[pos=.5,align=left] {$L^p(X,\mathcal{B},m)$}; % Draw arrows with text \draw[->] (\rectWidth/2,-\rectHeight) -- (\rectWidth/2,-\verticalDistance) node[midway,right] {(i) are countable and dense in}; \draw[->] (\rectWidth/2,-\verticalDistance-\rectHeight) -- (\rectWidth/2,-2*\verticalDistance) node[midway,right] {(ii) are dense in}; \draw[->] (\rectWidth/2,-2*\verticalDistance-\rectHeight) -- (\rectWidth/2,-3*\verticalDistance) node[midway,right] {(iii) are dense in}; \draw[->] (\rectWidth/2,-3*\verticalDistance-\rectHeight) -- (\rectWidth/2,-4*\verticalDistance) node[midway,right] {(iv) are dense in}; \end{tikzpicture} \end{figure} \paragraph{(i)} Let $g$ be an elementary function with $\mu(A) < \infty$ for $A:=\operatorname{supp}g$. By definition, we write\footnote{Technically speaking, the elements of $L^{p}$ are not functions, but as soon as the differences do not cause any problems except for the zero sets, we can write them like this}. \begin{equation*} g = \sum_{i=1}^{m} c_i \cdot 1_{A_i} \end{equation*} for $c_i \in \mathbb{R}$. Our goal is to approximate $g$ by functions $(g_{\varepsilon})_{\varepsilon > 0}$ in the form \begin{equation*} g_{\varepsilon} = \sum_{i=1}^{m} \widetilde{c}_i \cdot 1_{\widetilde{A}_i} \end{equation*} (denseness) where $\widetilde{c}_i$ should be rational and $\widetilde{A}_i$ should come from a countable basis of $\mathcal{B}$ (countability).\newline For each set $A_i$, we find the approximations $\widetilde{A}_i$ from a countable basis of $\mathcal{B}$ with the property that the symmetric difference remains small\footnote{For example, as follows. By construction of $L^p$ we can assume that $A_i$'s do not contain their boundary and are therefore countable unions of open intervals $A_i = \bigcup_{j \in \mathbb{N}} (a_i^j,b_i^j)$ are representable. For each $(a_i^j,b_i^j)$ let $(\widetilde{a}_i^j,\widetilde{b}_i^j)_{n\in\mathbb{N}}$ be the numbers approximating from the inside (i.e. $a_i^j$ from the right and $b_i^j$ from the left) from $\mathbb{Q}$ with the difference of most $\varepsilon/2^{j+1}$. Thus we define $$\widetilde{A}_i := \bigcup_{j \in \mathbb{N}} (\widetilde{a}_i^j,\widetilde{b}_i^j).$$ and obtain the summed estimate $m(A_i \setminus \widetilde{A}_i) < \varepsilon$ plus $\widetilde{A}_i \subseteq A_i$.} $$m\left(A_i \, \triangle\, \widetilde{A}_i(n)\right) \leq \frac{\varepsilon}{2 \cdot m \cdot(|c_i|^p+1)}.$$ We also approximate $|c_i|^p$'s from below with $\widetilde{c}_i$: $$|c_i - \widetilde{c}_i| \leq \sqrt[p]{\frac{\varepsilon}{2 \cdot m \cdot \mu(A_i \cap \widetilde{A}_i)}} < 1.$$ We use this to calculate: \begin{align*} \|g-g_n\|^p &= \int_X \left\vert \sum_{i=1}^{m} c_i \cdot 1_{A_i} - \widetilde{c}_i \cdot 1_{\widetilde{A}_i} \right\vert^p \ \mathrm{d}\mu\[5pt] &\leq \sum_{i=1}^{m} \int_{A_i \cap \widetilde{A}_i} \left\vert c_i \cdot 1_{A_i} - \widetilde{c}_i \cdot 1_{\widetilde{A}_i} \right\vert^p \ \mathrm{d}\mu + \int_{A_i \triangle \widetilde{A}_i} \left\vert c_i \cdot 1_{A_i} - \widetilde{c}_i \cdot 1_{\widetilde{A}_i} \right\vert^p \ \mathrm{d}\mu\\ &\leq \sum_{i=1}^{m} \left\vert c_i - \widetilde{c}_i\right|^p \cdot \mu(A_i \cap \widetilde{A}_i) + (|c_i|^p+1) \cdot \mu(A_i \triangle \widetilde{A}_i)\ &\leq \varepsilon. %&= \int_X \left\vert \sum_{i=1}^{m} c_i \cdot 1_{A_i} - \widetilde{c}_i \cdot \left(1_{A_i} - 1_{A_i \setminus \widetilde{A}_i}\right) \right\vert^p \ \mathrm{d}\mu\[5pt] %&= \int_X \left\vert \sum_{i=1}^{m} \left(c_i-\widetilde{c}_i\right) \cdot 1_{A_i} + \widetilde{c}_i \cdot 1_{A_i \setminus \widetilde{A}_i} \right\vert^p \ \mathrm{d}\mu\[5pt] %&\leq \int_X \left\vert \sum_{i=1}^{m} \varepsilon \cdot 1_{A_i} - \widetilde{c}_i \cdot 1_{A_i \setminus \widetilde{A}_i} \right\vert^p \ \mathrm{d}\mu\[5pt] %&\leq \sum_{i=1}^{m} \varepsilon \cdot m({A_i}) + \left\vert \widetilde{c}_i\right\vert^p \cdot \varepsilon % \leq \varepsilon \cdot \left(\sum_{i=1}^{m} m({A_i}) + \left\vert {c}\right\vert^p\right) \end{align*} \paragraph{(ii)} Let $f$ be a bounded function in $L^p(X)$ with $\mu(A) < \infty$ for $A:=\operatorname{supp}f$. We discretize the domain of $f$, e.g. with the usual ``dyadic mesh'': $$E_i^n := f^{-1}\left[\frac{i}{2^n},\frac{i+1}{2^n}\right), \quad i\in\mathbb{Z}.$$ Then we define $$g_n := \sum \frac{i}{2^n} \cdot 1_{E_i^n}.$$ and obtain \begin{align*} \|f-g_n\|^p = \int_A | f - g_n|^p \,\mathrm{d}\mu \leq \frac{1}{2^n} \mu(A) \stackrel{n\to\infty}{\to} 0 \end{align*} \paragraph{(iii)} This is the direct consequence of the theorem of monotonic convergence (e.g. if the support is gradually extended) \paragraph{(iv)} Let $f \in L^p(X,\mathcal{B},\mu)$, o.b.d.A. $f \geq 0$\footnote{after $f = f^+ - f^-$}. From measure theory, we know that there is a sequence of increasing elementary functions $(f_n)_{n\in\mathbb{N}}$ that approximates $f$ from below. We have $\|f-f_n\|^p \leq \|f\|^p$, the last one absolutely convergent by assumption. The statement then follows with the theorem of majorized convergence $$\int_X |f-f_n|^p \ \mathrm{d}\mu \to 0.$$ \textit{2. Find an example of a non-separable Hilbert space}\newline The space $\ell^2(\mathbb R)$ of functions $f:\mathbb R\to\mathbb R$, for which $f(x)\ne0$ only for countably many $x$, and $$ \sum_{x\in \mathbb R}f(x)^2 <\infty. $$ It is easy to see that this is a Hilbert space, the crucial argument is that the countable union of countable sets is countable. The functions $f_y$ are defined by $$ f_y(x) = \begin{cases} 1 &\text{ if } x=y\ 0 & \text{ else}\end{cases} $$ Then the supremum norm is $d_\infty(f_y,f_z)=\sqrt2$ if $y\neq z$. Thus $$ \mathcal{C}=\left\{B\left(f_x,\frac12\right) \mid x\in\mathbb{R}\right\} $$ is an uncountable collection of disjoint open balls. Now let $S$ be any dense subset, then every ball in the family $\mathcal{C}$ must contain at least one element of $S$, and these elements must all be distinct, so $S$ must be uncountably infinite. This shows that $\ell^2$ is not separable.

Exercise 1.3.9 ($L^p$ is separable)

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Exercise 1.3.9 ($L^p$ is separable)

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