Proof of Continuous Periodic Functions are Power-Integrable and Dense on the Torus

Each result cited below is universally quantified over the data appearing in its own statement, and is applied to the data named here. The measure space is $(Q,\mathcal{B}_{Q},\lambda_{Q})$ of clause 3, with $\lambda_{Q}(Q)=1$, unless $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$ is named instead. We use repeatedly that a map $u\in C_{\mathrm{per}}$, being continuous on $\mathbb{R}^{n}$, is measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R})$, by claim 3(a) of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets together with claim 5 there.

Claim 1. For a real $c$ we have

$$\{x\in Q:c<u(x)\}=Q\cap\{x\in\mathbb{R}^{n}:c<u(x)\},$$

which belongs to $\mathcal{B}(\mathbb{R}^{n})$ and is contained in $Q$, hence belongs to $\mathcal{B}_{Q}$. By the criterion of clause 3, $u|_{Q}$ is measurable. Next, $|u(x)|\le M$ for every $x$, so by Properties of Real Powers (v) the power $(|u(x)|)^{p}$ is at most $M^{p}$ for every $x\in Q$. The constant function on $Q$ with value $M^{p}$ is $M^{p}\,\mathbf{1}_{Q}$, whose integral is $M^{p}\lambda_{Q}(Q)=M^{p}$ by The Integral of an Indicator Function is the Measure of the Set and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion. Monotonicity of the integral of nonnegative measurable functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion) therefore gives

$$\int_{Q}\bigl|u|_{Q}\bigr|^{p}\,d\lambda_{Q}\le M^{p}<\infty ,$$

so $u|_{Q}\in\mathcal{L}^{p}(\mathbb{T}^{n})$ by Power-Integrable Functions and the p-Seminorm (ii). Applying the monotone power $t\mapsto t^{1/p}$ (Properties of Real Powers (v)) and the identity $(M^{p})^{1/p}=M$ (Properties of Real Powers (vi)) gives $\lVert u|_{Q}\rVert_{p}\le M$ for the $p$-seminorm, and $\lVert[u|_{Q}]\rVert_{L^{p}(\mathbb{T}^{n})}=\lVert u|_{Q}\rVert_{p}$ by The Lebesgue Space Lᵖ(μ) (iii).

Claim 2. Let $M$ be as in claim 1. The product $\mathbf{1}_{Q}u$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$: for a real $c$ the set $\{x:c<\mathbf{1}_{Q}(x)u(x)\}$ equals $Q\cap\{x:c<u(x)\}$ when $0\le c$, and equals the union of that set with the complement of $Q$ when $c<0$; both lie in $\mathcal{B}(\mathbb{R}^{n})$. Moreover $|\mathbf{1}_{Q}u|\le M\mathbf{1}_{Q}$ pointwise, and $\int_{\mathbb{R}^{n}}M\mathbf{1}_{Q}\,d\lambda_{n}=M\lambda_{n}(Q)=M$ is finite by The Integral of an Indicator Function is the Measure of the Set, The Half-Open Unit Cell Tiles Euclidean Space (i) and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion; so $\mathbf{1}_{Q}u$ is integrable by the criterion recorded in Integrable Functions and Their Lebesgue Integral.

Let $u^{+}$ and $u^{-}$ be the positive and negative parts of $u$ in the sense of Integrable Functions and Their Lebesgue Integral. The zero extension of $(u|_{Q})^{+}$ off $Q$ is $\mathbf{1}_{Q}u^{+}$, and likewise for the negative parts. Claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, applied to the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$ and to $Q\in\mathcal{B}(\mathbb{R}^{n})$, therefore gives

$$\int_{Q}(u|_{Q})^{\pm}\,d\lambda_{Q}=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}u^{\pm}\,d\lambda_{n},$$

both sides being finite. Subtracting the two identities and using Integrable Functions and Their Lebesgue Integral on each side yields $\int_{\mathbb{T}^{n}}u|_{Q}\,dx=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}u\,d\lambda_{n}$.

Claim 3. The maps $u+v$ and $cu$ lie in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i), so their restrictions lie in $\mathcal{L}^{p}(\mathbb{T}^{n})$ by claim 1. Restriction commutes with the pointwise operations, that is $(u+v)|_{Q}=u|_{Q}+v|_{Q}$ and $(cu)|_{Q}=c\,(u|_{Q})$ as maps on $Q$, and the operations on classes are defined by representatives in The Lebesgue Space Lᵖ(μ) (ii). The two identities follow.

Claim 4. We first prove the approximation statement. Let $f\in\mathcal{L}^{p}(\mathbb{T}^{n})$ and let $\varepsilon$ be a positive real.

Step 1: reduction to a simple function. By Simple Functions are Dense in the Lebesgue Space (ii) there is a simple function $s$ on $(Q,\mathcal{B}_{Q})$ lying in $\mathcal{L}^{p}(\mathbb{T}^{n})$ with $\lVert f-s\rVert_{p}\le\varepsilon/2$.

Step 2: approximating one indicator. We show: for every $A\in\mathcal{B}_{Q}$ and every positive real $\eta$ there is $v\in C_{\mathrm{per}}$ with $\lVert\mathbf{1}_{A}-v|_{Q}\rVert_{p}\le\eta$.

Let $\beta$ be a positive real with $\beta\le\tfrac14$, to be fixed at the end, and put

$$V=\{x\in\mathbb{R}^{n}:\beta<x_{i}<1-\beta\ \text{for every}\ i\in[n]\}.$$

$V$ is open: for $x\in V$ let $r$ be the least element of the finite set $\{x_{i}-\beta:i\in[n]\}\cup\{1-\beta-x_{i}:i\in[n]\}$ of positive reals, positive by repeated use of claim 9 of Elementary Order Arithmetic in an Ordered Field; if $\lVert y-x\rVert<r$ then $|y_{i}-x_{i}|<r$ for every $i$ by claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, whence $\beta<y_{i}<1-\beta$. Also $V\subseteq\mathring{Q}$, since $0<\beta$ and $1-\beta<1$.

Put $A_{\beta}=A\cap V$, a member of $\mathcal{B}_{Q}$. Every $x\in A\setminus A_{\beta}$ lies in $Q$ and has some coordinate with $x_{i}\le\beta$ or $1-\beta\le x_{i}$, so

$$A\setminus A_{\beta}\subseteq\bigcup_{i\in[n]}\bigl(S_{i}\cup T_{i}\bigr),\qquad S_{i}=\{x\in Q:x_{i}\le\beta\},\quad T_{i}=\{x\in Q:1-\beta\le x_{i}\}.$$

Each $S_{i}$ is the Borel rectangle whose $i$th factor is the interval $\{t:0\le t\le\beta\}$ and whose other factors are $\{t:0\le t<1\}$, and each $T_{i}$ is the Borel rectangle whose $i$th factor is $\{t:1-\beta\le t<1\}$ and whose other factors are the same; by claim 4 of Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness and the rectangle identity of Lebesgue Measure on $\mathbb{R}^n$ each has measure $\beta$. By the monotonicity and finite subadditivity of a measure recorded in Basic Properties of a Measure, $\lambda_{n}(A\setminus A_{\beta})\le2n\beta$.

Since $\lambda_{n}(A_{\beta})\le\lambda_{n}(Q)=1<\infty$, Outer and Inner Regularity of Lebesgue Measure on $\mathbb{R}^n$ (iii) provides a compact $K\subseteq A_{\beta}$ with $\lambda_{n}(A_{\beta}\setminus K)\le\beta$, and Outer and Inner Regularity of Lebesgue Measure on $\mathbb{R}^n$ (i) provides an open $U_{0}\supseteq K$ with $\lambda_{n}(U_{0}\setminus K)\le\beta$. Put $U=U_{0}\cap V$, an open set with $K\subseteq U$ (because $K\subseteq A_{\beta}\subseteq V$), with $U\subseteq V\subseteq\mathring{Q}$, and with $\lambda_{n}(U\setminus K)\le\lambda_{n}(U_{0}\setminus K)\le\beta$.

Apply Continuous Partition of Unity Subordinate to a Finite Open Cover of a Compact Set in a Metric Space in the metric space $(\mathbb{R}^{n},d_{E})$ to the compact set $K$ and the one-member family consisting of the open set $U$, which covers $K$. It yields a continuous $g:\mathbb{R}^{n}\to\mathbb{R}$ with $0\le g\le1$ and a closed $D\subseteq U$ such that $g$ vanishes off $D$ and $g(x)=1$ for every $x\in K$. In particular $g$ vanishes off $U$, and $U\subseteq\mathring{Q}$, so Elementary Properties of Lattice-Periodic Functions (iv) applies with $W=U$: the map $v=g\circ\pi$ lies in $C_{\mathrm{per}}$, agrees with $g$ on $Q$, and satisfies $0\le v\le1$.

Put $E=\bigl((A\setminus K)\cup(U\setminus K)\bigr)\cap Q$, a member of $\mathcal{B}_{Q}$. Note first that $(\mathbf{1}_{E})^{p}=\mathbf{1}_{E}$: indeed $0^{p}=0$ by Real Powers (i), while $1^{p}=1^{p}\cdot1^{p}$ by Properties of Real Powers (iii) and $1^{p}\ne0$ by Properties of Real Powers (i), so cancelling the nonzero factor gives $1^{p}=1$. Hence $\mathbf{1}_{E}\in\mathcal{L}^{p}(\mathbb{T}^{n})$, its $p$th power having integral $\lambda_{Q}(E)\le\lambda_{Q}(Q)=1<\infty$ by The Integral of an Indicator Function is the Measure of the Set and Basic Properties of a Measure. For $x\in Q$ with $x\notin E$ we have $\mathbf{1}_{A}(x)=v(x)$: if $x\in K$ then $x\in A$ and $g(x)=1$, so both values are $1$; if $x\notin K$ then $x\notin A$ and $x\notin U$, so $\mathbf{1}_{A}(x)=0$ and $g(x)=0$. For $x\in E$ both $\mathbf{1}_{A}(x)$ and $v(x)$ lie between $0$ and $1$, so $|\mathbf{1}_{A}(x)-v(x)|\le1$. Hence $|\mathbf{1}_{A}-v|_{Q}|\le\mathbf{1}_{E}$ pointwise on $Q$, and Elementary Properties of the p-Seminorm (iii) gives

$$\lVert\mathbf{1}_{A}-v|_{Q}\rVert_{p}\le\lVert\mathbf{1}_{E}\rVert_{p}=\bigl(\lambda_{Q}(E)\bigr)^{1/p},$$

using $(\mathbf{1}_{E})^{p}=\mathbf{1}_{E}$ as just noted, together with The Integral of an Indicator Function is the Measure of the Set. Finally

$$\lambda_{Q}(E)\le\lambda_{n}(A\setminus K)+\lambda_{n}(U\setminus K)\le\bigl(\lambda_{n}(A\setminus A_{\beta})+\lambda_{n}(A_{\beta}\setminus K)\bigr)+\beta\le2n\beta+\beta+\beta=(2n+2)\beta,$$

again by Basic Properties of a Measure. Choosing $\beta$ to be the least of $\tfrac14$ and $\eta^{p}/(2n+2)$ makes $\lambda_{Q}(E)\le\eta^{p}$ and hence $\lVert\mathbf{1}_{A}-v|_{Q}\rVert_{p}\le(\eta^{p})^{1/p}=\eta$, by Properties of Real Powers (v) and Properties of Real Powers (vi).

Step 3: from indicators to $s$. Let $s=\sum_{l=1}^{r}c_{l}\mathbf{1}_{A_{l}}$ be the standard representation of $s$, with distinct values $c_{l}$ and $A_{l}=s^{-1}(\{c_{l}\})\in\mathcal{B}_{Q}$. Let $L=\{l\in[r]:c_{l}\ne0\}$; since the terms with $c_{l}=0$ contribute nothing, $s=\sum_{l\in L}c_{l}\mathbf{1}_{A_{l}}$ pointwise on $Q$. If $L$ is empty then $s$ is the zero map, and the map $u$ with constant value $0$ lies in $C_{\mathrm{per}}$ with $\lVert s-u|_{Q}\rVert_{p}=0\le\varepsilon/2$. Otherwise put $C=\sum_{l\in L}|c_{l}|$, a positive real. Apply Step 2 to each $A_{l}$, $l\in L$, with $\eta=\varepsilon/(2C)$, obtaining $v_{l}\in C_{\mathrm{per}}$ with $\lVert\mathbf{1}_{A_{l}}-v_{l}|_{Q}\rVert_{p}\le\varepsilon/(2C)$, and put $u=\sum_{l\in L}c_{l}v_{l}$, which lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i). Then $s-u|_{Q}=\sum_{l\in L}c_{l}\bigl(\mathbf{1}_{A_{l}}-v_{l}|_{Q}\bigr)$ pointwise on $Q$, so by Minkowski's Inequality (iii) and Elementary Properties of the p-Seminorm (ii),

$$\lVert s-u|_{Q}\rVert_{p}\le\sum_{l\in L}|c_{l}|\,\bigl\lVert\mathbf{1}_{A_{l}}-v_{l}|_{Q}\bigr\rVert_{p}\le C\cdot\frac{\varepsilon}{2C}=\frac{\varepsilon}{2}.$$

Step 4: conclusion. By Minkowski's Inequality (i),

$$\bigl\lVert f-u|_{Q}\bigr\rVert_{p}\le\lVert f-s\rVert_{p}+\bigl\lVert s-u|_{Q}\bigr\rVert_{p}\le\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon ,$$

and $\lVert[f]-[u|_{Q}]\rVert_{L^{p}(\mathbb{T}^{n})}=\lVert f-u|_{Q}\rVert_{p}$ by The Lebesgue Space Lᵖ(μ) (ii) and The Lebesgue Space Lᵖ(μ) (iii). This is the approximation statement.

Density. Let $F\in L^{p}(\mathbb{T}^{n})$ and choose a representative $f\in\mathcal{L}^{p}(\mathbb{T}^{n})$ with $F=[f]$. For each $k\in\mathbb{N}$ the approximation statement, applied with $\varepsilon=1/k$, provides $u_{k}\in C_{\mathrm{per}}$ with $\lVert F-[u_{k}|_{Q}]\rVert_{L^{p}(\mathbb{T}^{n})}\le1/k$; the choice of one such $u_{k}$ for each $k$ uses countable choice. Given a positive real $\varepsilon$, The Archimedean Property of the Real Numbers provides $k_{0}\in\mathbb{N}$ with $1<k_{0}\varepsilon$, and then $\lVert F-[u_{k}|_{Q}]\rVert_{L^{p}(\mathbb{T}^{n})}\le1/k<\varepsilon$ for every $k\ge k_{0}$. So the sequence $([u_{k}|_{Q}])_{k\in\mathbb{N}}$ in $\mathcal{C}$ converges to $F$ in the metric of $L^{p}(\mathbb{T}^{n})$, and $F$ lies in the closure of $\mathcal{C}$ by Sequential Characterization of the Closure in a Metric Space. As $F$ was arbitrary, the closure of $\mathcal{C}$ is all of $L^{p}(\mathbb{T}^{n})$, that is, $\mathcal{C}$ is dense.