LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-04

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Step functions on one period are dense in L^1 on the torus

Statement

Assume the Axiom of Countable Choice.

Let $f$ be integrable on one period. For every $ε > 0$ there is a step function $s$ on $[0, 1)$ such that

$\int_{0}^{1} ∣ f (t) - s (t) ∣ d t < ε .$

Equivalently, one-period step functions are dense in $L^{1} (T)$ .

Facts & Assumptions

Given: The Axiom of Countable Choice, a one-period integrable function $f$ , and a real $ε > 0$ .

[L1]

The torus conventions identify one-period integrable functions with the $L^{1} (T)$ objects used on this page (Period-one Fourier coefficients, partial sums, and convolution on the torus).

[L2]

Assuming the Axiom of Countable Choice, finite linear combinations of interval indicators are dense in $L^{1} (R)$ (Finite linear combinations of box indicators are dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$ ).

Proof

technique · direct

1.1

Define $g : R \to R$ by $g (t) = f (t)$ for $0 \leq t < 1$ and $g (t) = 0$ otherwise. Then $g \in L^{1} (R)$ . By [L2], choose a finite linear combination of interval indicators $u$ with $\int_{R} ∣ g (t) - u (t) ∣ d t < ε .$

givenL2choose

2.1

Restrict $u$ to $[0, 1)$ and call the restriction $s$ . Intersecting each interval in $u$ with $[0, 1)$ produces only finitely many subintervals, so $s$ is a step function on $[0, 1)$ . Since $g = f$ on $[0, 1)$ , $\int_{0}^{1} ∣ f (t) - s (t) ∣ d t = \int_{0}^{1} ∣ g (t) - u (t) ∣ d t \leq \int_{R} ∣ g (t) - u (t) ∣ d t < ε .$

L1step 1.1algebra

3.1

Step 2.1 is exactly the claimed density statement on one period.

step 2.1∎

Depends on

Used by

Riemann-Lebesgue lemma for Fourier coefficients Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)