Countable.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 9 curated AI-written annotations and 485 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, Polynomial Freiman-Ruzsa proof paper, arXiv v2

4 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Imported by 2 files

lemma MeasureTheory.lintegral_eq_sum TODO: Change RHS of lintegral_fintype line 14 source
Source comment
TODO: Change RHS of `lintegral_fintype`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, MeasureTheory.Measure, ENNReal, Fintype, Eq, MeasureTheory.lintegral, Finset.sum, Finset.univ, HMul.hMul, Set, Singleton.singleton
Proof uses: Eq.mpr, id, congrFun', congrArg, MeasureTheory.lintegral_fintype, of_eq_true, Eq.trans, True, Finset.sum_congr, Eq.refl, Finset, Membership.mem, mul_comm, eq_self
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma lintegral_eq_sum (μ : Measure α) (f : α → ℝ≥0∞) [Fintype α] :
    ∫⁻ x, f x ∂μ = ∑ x, μ {x} * f x := by
…
```
lemma MeasureTheory.lintegral_eq_tsum line 18 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, MeasureTheory.Measure, ENNReal, Countable, Eq, MeasureTheory.lintegral, tsum, HMul.hMul, Set, Singleton.singleton, SummationFilter.unconditional
Proof uses: Eq.mpr, id, congrFun', congrArg, MeasureTheory.lintegral_countable', of_eq_true, Eq.trans, True, SummationFilter, optParam, funext, mul_comm, eq_self
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.setLIntegral_eq_sum TODO: Change RHS of lintegral_finset line 23 source
Source comment
TODO: Change RHS of `lintegral_finset`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, MeasureTheory.Measure, Finset, ENNReal, Eq, MeasureTheory.lintegral, MeasureTheory.Measure.restrict, Finset.sum, HMul.hMul, Set, Singleton.singleton
Proof uses: Eq.mpr, id, congrArg, Finset.sum_congr, Eq.refl, Membership.mem, mul_comm, of_eq_true, Eq.trans, True, congrFun', MeasureTheory.lintegral_finset, eq_self
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma setLIntegral_eq_sum (μ : Measure α) (s : Finset α) (f : α → ℝ≥0∞) :
    ∫⁻ x in s, f x ∂μ = ∑ x ∈ s, μ {x} * f x := by
…
```
lemma MeasureTheory.lintegral_eq_single line 27 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, MeasureTheory.Measure, ENNReal, Ne, Eq, OfNat.ofNat, MeasureTheory.lintegral, HMul.hMul, Set, Singleton.singleton
Proof uses: Eq.mpr, HAdd.hAdd, MeasureTheory.Measure.restrict, Compl.compl, id, congrArg, Eq.symm, MeasureTheory.lintegral_add_compl, MeasurableSet.singleton, MeasureTheory.lintegral_singleton, MeasureTheory.setLIntegral_congr_fun, MeasurableSet.compl, of_eq_true, Membership.mem, Eq.trans, True, forall_congr, Not, implies_congr_ctx, congrFun' and 8 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.