ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

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.

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A unit-mass spike has a large maximal superlevel set

Example

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

For $ε > 0$ , define $f_{ε} : = ε^{- 1} 1_{[0, ε]} .$ Then $∥ f_{ε} ∥_{1} = 1$ , and for every $x \geq ε$ , $M f_{ε} (x) \geq \frac{1}{2 x} .$ Consequently, for every $0 < t \leq (2 ε)^{- 1}$ , $λ ({x \in R : M f_{ε} (x) > t}) \geq \frac{1}{2 t} - ε .$

Facts & Assumptions

Given: The Axiom of Countable Choice, a real number $ε > 0$ , and the spike $f_{ε} = ε^{- 1} 1_{[0, ε]}$ .

[L1]

The centered maximal operator is weak type $(1, 1)$ . (The centered Hardy-Littlewood maximal operator is weak type $(1, 1)$ )

Verification

technique · direct

1.1givenalgebra

One has [given, algebra] $∥ f_{ε} ∥_{1} = \int_{0}^{ε} ε^{- 1} d x = 1.$

1.2L1givenalgebra

If $x \geq ε$ , then the centered interval $[0, 2 x]$ contains the [L1, given, algebra] support of $f_{ε}$ , so $M f_{ε} (x) \geq \frac{1}{2 x} \int_{0}^{ε} ε^{- 1} d y = \frac{1}{2 x} .$

2.1step 1.2algebra

If $0 < t \leq (2 ε)^{- 1}$ and $ε \leq x < 1 / (2 t)$ , then [step 1.2, algebra] step 1.2 gives $M f_{ε} (x) > t$ . Therefore $[ε, 1 / (2 t)) \subseteq {M f_{ε} > t},$ so $λ ({M f_{ε} > t}) \geq \frac{1}{2 t} - ε .$

3.1L1step 1.1step 2.1∎

This explicit family matches the weak-type $t^{- 1}$ scale from [L1] on a [L1, step 1.1, step 2.1] unit- $L^{1}$ example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Terence Tao, An Introduction to Measure Theory, Theorem 1.6.20 (standard reference, not scraped)