Separability of the Lebesgue Spaces of the Torus

In the setting of The Flat Torus: Standing Notation, let $p$ be a real number with $1\le p$, and let $\mathbb{Q}$ denote the rational numbers. For a natural number $k$ put

$$N_{k}=\{j\in\mathbb{Z}^{n}:0\le j_{i}\le 2^{k}-1\ \text{for every}\ i\in[n]\},$$ $$Q_{k,j}=\{x\in\mathbb{R}^{n}:2^{-k}j_{i}\le x_{i}<2^{-k}(j_{i}+1)\ \text{for every}\ i\in[n]\}\qquad(j\in N_{k}),$$

the powers $2^{k}$ and $2^{-k}$ being those fixed in clause 1 of the Euclidean setting. Let $D_{k}$ be the set of those maps $s:Q\to\mathbb{R}$ for which there is a family $(c_{j})_{j\in N_{k}}$ in $\mathbb{Q}$ with $s(x)=c_{j}$ for every $j\in N_{k}$ and every $x\in Q_{k,j}$, and put $D=\bigcup_{k\in\mathbb{N}}D_{k}$. Then the following hold.

1. (Dyadic partition of the cell)(i) Let $k$ be a natural number. Then $N_{k}$ is a finite set; the sets $Q_{k,j}$, for $j\in N_{k}$, belong to $\mathcal{B}_{Q}$, are pairwise disjoint, and have union $Q$; and any $x,y\in Q_{k,j}$ satisfy $\lVert x-y\rVert\le\sigma_{n}2^{-k}$. Consequently every member of $D_{k}$ is determined by its family $(c_{j})_{j\in N_{k}}$, and conversely every such family determines a member of $D_{k}$.

2. (A countable family)(ii) Every $s\in D$ belongs to $\mathcal{L}^{p}(\mathbb{T}^{n})$, and the set $\{[s]:s\in D\}$ is countable.

3. (Approximation)(iii) For every $f\in\mathcal{L}^{p}(\mathbb{T}^{n})$ and every real $\varepsilon$ with $0<\varepsilon$ there is $s\in D$ with

$$\bigl\lVert[f]-[s]\bigr\rVert_{L^{p}(\mathbb{T}^{n})}\le\varepsilon .$$

4. (Separability)(iv) $\{[s]:s\in D\}$ is a dense subset of $L^{p}(\mathbb{T}^{n})$, and $L^{p}(\mathbb{T}^{n})$ is a separable metric space.