TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Finite linear combinations of box indicators are dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$

Statement

Assume the Axiom of Countable Choice.

Let $1 \leq p < \infty$ . Finite linear combinations of indicator functions of boxes are dense in $L^{p} (R^{n})$ .

Facts & Assumptions

Given: The Axiom of Countable Choice, $1 \leq p < \infty$ , $ε > 0$ , and $f \in L^{p} (R^{n})$ .

[L1]

Simple functions with finite-measure support are dense in $L^{p}$ (Simple functions with finite-measure support are dense in $L^{p} (μ)$ for $1 \leq p < \infty$ ).

[L2]

Every finite-measure measurable set can be approximated in symmetric difference by a finite union of boxes (A finite-measure measurable set in $R^{n}$ is approximable in measure by a finite union of boxes).

[L3]

Minkowski's inequality is available in $L^{p}$ (Minkowski's inequality for integrals, including $p = \infty$ ).

Proof

technique · direct

1.1L1L2givenchoose

By [L1], choose a simple function [L1, L2, given, choose] $s = \sum_{j = 1}^{m} a_{j} 1_{E_{j}}$ with each $E_{j}$ of finite measure and $∥ f - s ∥_{p} < ε / 2$ . For each $j$ , choose a finite union of boxes $B_{j}$ with $λ_{n} (E_{j} △ B_{j}) < {(\frac{ε}{2 m (1 + \sum_{j = 1}^{m} ∣ a_{j} ∣)})}^{p}$ by [L2].

2.1L3step 1.1algebra

Put [L3, step 1.1, algebra] $t : = \sum_{j = 1}^{m} a_{j} 1_{B_{j}} .$ Then $∥ s - t ∥_{p} \leq \sum_{j = 1}^{m} ∣ a_{j} ∣ ∥ 1_{E_{j}} - 1_{B_{j}} ∥_{p} = \sum_{j = 1}^{m} ∣ a_{j} ∣ λ_{n} (E_{j} △ B_{j})^{1 / p} < ε / 2$ by [L3] and the choice of the $B_{j}$ .

3.1step 1.1step 2.1algebra∎

Therefore [step 1.1, step 2.1, algebra] $∥ f - t ∥_{p} \leq ∥ f - s ∥_{p} + ∥ s - t ∥_{p} < ε .$ Since $t$ is a finite linear combination of box indicators, these functions are dense in $L^{p} (R^{n})$ .

Depends on

Used by

Dependency tree · two levels

20 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

Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)
Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)

Finite linear combinations of box indicators are dense in Lp(Rn)Lp(Rn) for 1≤p<∞1≤p<∞