AEMeasurable.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

5 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 3 files

lemma MeasureTheory.aeMeasurable_withDensity_inv TODO: generalize this to other measures and density functions? line 12 source
Source comment
TODO: generalize this to other measures and density functions?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, ENNReal, AEMeasurable, MeasureTheory.MeasureSpace.volume, MeasureTheory.Measure.withDensity, Inv.inv
Proof uses: Eq.mpr, HMul.hMul, id, Eq, congrArg, propext, MeasureTheory.aemeasurable_withDensity_ennreal_iff, MeasurableInv.measurable_inv, AEMeasurable.fun_mul, AEMeasurable.coe_nnreal_ennreal, AEMeasurable.fun_inv, aemeasurable_id, eq_of_heq, MeasurableSpace, autoParam, MeasureTheory.Measure, HEq, Eq.refl, HEq.refl, Eq.symm and 22 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
lemma aeMeasurable_withDensity_inv {f : NNReal → ENNReal} (hf : AEMeasurable f) :
    AEMeasurable f (volume.withDensity (fun t ↦ t⁻¹)) := by
…
```
theorem MeasureTheory.aemeasurable_Ici_of_forall_Icc line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, LinearOrder, Filter.IsCountablyGenerated, Filter.atTop, GE.ge, AEMeasurable, MeasureTheory.Measure.restrict, Set.Icc, Set.Ici
Proof uses: Nonempty, Nonempty.intro, Nat, Filter.Tendsto, Exists, Filter.exists_seq_tendsto, Eq, Set, Set.iUnion, Eq.mpr, id, congrArg, Iff.mpr, LE.le, Set.iUnion_Icc_eq_Ici_self_iff, Filter.Eventually.exists, Filter.Tendsto.eventually, Filter.eventually_ge_atTop, Eq.refl, propext and 11 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
theorem MeasureTheory.aemeasurable_Ici_of_forall_Icc' line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, MeasureTheory.Measure, LinearOrder, Filter.IsCountablyGenerated, Filter.atTop, GT.gt, AEMeasurable, MeasureTheory.Measure.restrict, Set.Icc, Set.Ici
Proof uses: MeasureTheory.aemeasurable_Ici_of_forall_Icc, GE.ge, LT.lt, Eq, Or, LE.le.lt_or_eq, Eq.mpr, id, congrArg, Singleton.singleton, Set, Set.Icc_self, HSMul.hSMul, ENNReal, MeasureTheory.Measure.dirac, MeasureTheory.Measure.restrict_singleton, AEMeasurable.smul_measure, MeasureTheory.aemeasurable_dirac
26 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.AEMeasurable.withDensity line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, ENNReal, AEMeasurable, MeasureTheory.Measure.withDensity
Proof uses: AEMeasurable.mono_ac, MeasureTheory.withDensity_absolutelyContinuous
lemma MeasureTheory.AEStronglyMeasurable.withDensity line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENNReal, MeasureTheory.AEStronglyMeasurable, MeasureTheory.Measure.withDensity
Proof uses: MeasureTheory.AEStronglyMeasurable.mono_ac, MeasureTheory.withDensity_absolutelyContinuous