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Continuous functions are dense in $L^{p}$ of finite tori and of bounded intervals

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ and $1 \leq p < + \infty$ .

The complex continuous functions on the finite torus $T^{n}$ are dense in complex $L^{p} (T^{n})$ with respect to the $L^{p}$ norm (The one-dimensional torus and its normalized Haar integral, Complex Lp classes and Euclidean test-function conventions).
For every bounded interval $(a, b) \subseteq R$ , the complex continuous functions on $[a, b]$ are dense in complex $L^{p} ((a, b))$ for Lebesgue measure.
The real versions of 1 and 2 hold, the approximants being the real parts of the complex ones.

Facts & Assumptions

[A1]

Complex $C_{c}^{\infty} (R^{n})$ is dense in complex $L^{p} (R^{n})$ for finite $p$ , and complex finite simple functions with finite-measure support are dense in $L^{p}$ of any measure space (Complex finite-simple and smooth compact-support density for finite p).

[A2]

The torus integral is represented on the fundamental domain $[0, 1)^{n}$ , so for $G : = [F \circ q_{n}] \cdot 1_{[0, 1)^{n}}$ one has $∥ G ∥_{L^{p} (R^{n})} = ∥ F ∥_{L^{p} (T^{n})}$ ; Minkowski's inequality holds in complex $L^{p}$ and $∥ Re h ∥_{p} \leq ∥ h ∥_{p}$ (The one-dimensional torus and its normalized Haar integral, Complex Holder, Minkowski, and the quotient norm).

[A3]

If $h \in L^{1}$ and $ε > 0$ then some $δ > 0$ has $μ (E) < δ \Rightarrow \int_{E} ∣ h ∣ < ε$ ; the box formula gives $λ_{n} ([0, 1)^{n} ∖ [δ, 1 - δ]^{n}) = 1 - (1 - 2 δ)^{n} \leq 2 n δ$ , and for every real $η > 0$ some $δ > 0$ has $2 n δ < η$ (Absolute continuity of the integral, A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[A4]

A continuous function $t \mapsto dist (t, C)$ to a closed set is continuous, and maxima and minima of continuous functions are continuous, so the cutoff $χ (x) : = \prod_{j < n} \max {0, 1 - dist (x_{j}, [δ, 1 - δ]) / δ}$ is continuous, equals $1$ on $[δ, 1 - δ]^{n}$ and vanishes outside $(0, 1)^{n}$ ; its support meets only finitely many integer translates of the fundamental cube, so the periodisation $\tilde{K} (s) : = \sum_{m \in Z^{n}} χ H (s + m)$ is a finite sum locally, is $n$ -fold periodic, and descends to a continuous function $K$ on $T^{n}$ by the quotient universal property; on $[0, 1)^{n}$ the sum reduces to $χ H$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map, The one-dimensional torus and its normalized Haar integral).

[A5]

A function in $C_{c}^{\infty} (R^{n})$ is continuous, and on a bounded interval a continuous function is Riemann integrable, hence Lebesgue integrable with the same integral (Complex Lp classes and Euclidean test-function conventions, A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral).

Proof

technique · direct

Given: Countable Choice, $n \geq 1$ , $1 \leq p < + \infty$ , and $f \in L^{p} (T^{n}; C)$ represented by the Borel function $F$ .

1.1

Let $G : = F \circ q_{n}$ on $[0, 1)^{n}$ , extended by $0$ to $R^{n}$ . Then $G \in L^{p} (R^{n})$ and $∥ G ∥_{p} = ∥ f ∥_{L^{p} (T^{n})}$ , and for a measurable $E \subseteq R^{n}$ with $λ_{n} (E)$ small the integral of $∣ G ∣^{p}$ over $E$ is small by absolute continuity of the integral.

A2A3

2.1

Given $ε > 0$ choose $δ > 0$ with $λ_{n} ([0, 1)^{n} ∖ [δ, 1 - δ]^{n}) < δ_{0}$ , where $δ_{0}$ is a threshold for $∣ G ∣^{p}$ and $ε^{p}$ from [A3]; put $G_{δ} : = G \cdot 1_{[δ, 1 - δ]^{n}}$ . Then $∥ G - G_{δ} ∥_{p}^{p} = \int_{[0, 1)^{n} ∖ [δ, 1 - δ]^{n}} ∣ G ∣^{p} < ε^{p}$ , so $∥ G - G_{δ} ∥_{p} < ε$ .

step 1.1A3

3.1

By density of $C_{c}^{\infty} (R^{n})$ choose $H \in C_{c}^{\infty} (R^{n})$ with $∥ G_{δ} - H ∥_{p} < ε$ , and let $χ$ be the cutoff of [A4] for this $δ$ . Since $χ = 1$ on the support of $G_{δ}$ and $∣ χ ∣ \leq 1$ , one has $∥ G_{δ} - χ H ∥_{p} = ∥ χ (G_{δ} - H) ∥_{p} \leq ∥ G_{δ} - H ∥_{p} < ε$ , and $χ H$ is continuous, compactly supported in $[0, 1]^{n}$ , and vanishes on the boundary of that cube.

step 2.1A1A4

4.1

Let $K$ be the continuous function on $T^{n}$ obtained by periodising $χ H$ , as in [A4]; on the fundamental domain $K$ agrees with $χ H$ . Therefore, using that the torus $L^{p}$ integral is represented on $[0, 1)^{n}$ and Minkowski's inequality, $∥ f - K ∥_{L^{p} (T^{n})} = ∥ G - χ H ∥_{L^{p} ([0, 1)^{n})} \leq ∥ G - G_{δ} ∥_{p} + ∥ G_{δ} - χ H ∥_{p} < 2 ε$ .

step 2.1step 3.1A2A4

5.1

Since $ε > 0$ was arbitrary, claim 1 follows: every $f \in L^{p} (T^{n}; C)$ is approximated in $L^{p}$ by the continuous functions $K$ . For claim 2, let $f \in L^{p} ((a, b))$ and extend it by $0$ to $R$ ; the density theorem [A1] gives $H \in C_{c}^{\infty} (R)$ with $∥ f - H ∥_{L^{p} (R)} < ε$ , and restricting $H$ to $[a, b]$ gives a continuous function with $∥ f - H ∥_{L^{p} ((a, b))} \leq ∥ f - H ∥_{L^{p} (R)} < ε$ .

step 4.1A1A5

6.1

For claim 3, let $f$ be real valued and let $H$ approximate it complexly within $ε$ ; then $Re H$ is real continuous and $∥ f - Re H ∥_{p} = ∥ Re (f - H) ∥_{p} \leq ∥ f - H ∥_{p} < ε$ , so the real continuous functions are dense in each of the two settings.

step 4.1step 5.1A2

7.1

Steps 5.1 and 6.1 establish the complex and real density statements on finite tori and on bounded intervals.

step 5.1step 6.1∎

Depends on

Used by

Legendre polynomials from Gram–Schmidt Example
The Haar orthonormal basis of L²((0,1)) Example
The Fourier basis and Parseval's identity on the finite torus Theorem
The trigonometric system is complete in L² of the torus Theorem

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Sources

Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, p.68, Problem 2.18 (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis — §2.3.6, pp.87–88 (standard reference, not scraped)

Continuous functions are dense in LpLp of finite tori and of bounded intervals