Prod.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 981 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned Carleson blueprint

2 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.MeasureTheory.Measure.Prod

Imported by 1 file

Carleson.lean

theorem MeasureTheory.AEMeasurable.lintegral_prod_right' Proof copied from MeasureTheory.AEStronglyMeasurable.integral_prod_right' Was it intentional that there's no left version? line 21 source
Source comment
Proof copied from `MeasureTheory.AEStronglyMeasurable.integral_prod_right'` Was it intentional that there's no left version?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, MeasureTheory.SFinite, Prod, ENNReal, AEMeasurable, MeasureTheory.Measure.prod, MeasureTheory.lintegral, Prod.mk
Proof uses: Exists.intro, And, Measurable, Filter.EventuallyEq, MeasureTheory.ae, AEMeasurable.mk, And.intro, Measurable.lintegral_prod_right', AEMeasurable.measurable_mk, Filter.mp_mem, Set.ofPred, Filter.Eventually, Eq, MeasureTheory.Measure.ae_ae_of_ae_prod, AEMeasurable.ae_eq_mk, Filter.univ_mem', Membership.mem, Set, id, MeasureTheory.lintegral_congr_ae
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem AEMeasurable.lintegral_prod_right' [SFinite ν] ⦃f : α × β → ℝ≥0∞⦄
    (hf : AEMeasurable f (μ.prod ν)) : AEMeasurable (fun (x : α) ↦ ∫⁻ (y : β), f (x, y) ∂ν) μ :=
…
```
theorem MeasureTheory.AEMeasurable.lintegral_prod_right XXX: the corresponding result for AEStronglyMeasurable doesn't exist; should it be added? line 27 source
Source comment
XXX: the corresponding result for AEStronglyMeasurable doesn't exist; should it be added?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, MeasureTheory.SFinite, ENNReal, AEMeasurable, Prod, Function.uncurry, MeasureTheory.Measure.prod, MeasureTheory.lintegral
Proof uses: AEMeasurable.lintegral_prod_right'
4 instances, coercions and auxiliary constants left out.
```
theorem AEMeasurable.lintegral_prod_right [SFinite ν] {f : α → β → ℝ≥0∞}
    (hf : AEMeasurable (uncurry f) (μ.prod ν)) : AEMeasurable (fun x ↦ ∫⁻ y, f x y ∂ν) μ :=
…
```