CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The maximal function of $1_{[0, 1]}$ is not integrable

Statement refuted

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

The centered maximal function of $1_{[0, 1]}$ belongs to $L^{1} (R)$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the function $f = 1_{[0, 1]}$ on $R$ .

[L1]

The centered maximal function is $M f (x) = \sup_{r > 0} \frac{1}{2 r} \int_{x - r}^{x + r} ∣ f (y) ∣ d y .$ (The centered and uncentered Hardy-Littlewood maximal functions)

Counterexample

technique · direct

1.1

Let $x \geq 1$ and take the centered interval $[0, 2 x]$ , which has centre $x$ [L1, given, algebra] and contains $[0, 1]$ . Then [L1] gives $M f (x) \geq \frac{1}{2 x} \int_{0}^{1} 1 d y = \frac{1}{2 x} .$

L1givenalgebra

2.1

Therefore [step 1.1, algebra] $\int_{1}^{R} M f (x) d x \geq \frac{1}{2} \int_{1}^{R} \frac{d x}{x} = \frac{1}{2} \log R (R > 1) .$ Letting $R \to \infty$ shows $\int_{1}^{\infty} M f (x) d x = + \infty$ . Hence $M f \notin L^{1} (R)$ .

step 1.1algebra

3.1

So the displayed claim is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Section I (standard reference, not scraped)

The maximal function of 1[0,1]1[0,1]​ is not integrable