BoundedFiniteSupport.lean`

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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

11 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Carleson.Defs

Imported by 4 files

theorem BoundedFiniteSupport.aestronglyMeasurable line 30 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENorm, Zero, BoundedFiniteSupport, MeasureTheory.AEStronglyMeasurable
Proof uses: MeasureTheory.MemLp.aestronglyMeasurable, Top.top, ENNReal, BoundedFiniteSupport.memLp_top
1 instances, coercions and auxiliary constants left out.
theorem BoundedFiniteSupport.aemeasurable line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENorm, Zero, BoundedFiniteSupport, TopologicalSpace.PseudoMetrizableSpace, BorelSpace, AEMeasurable
Proof uses: MeasureTheory.AEStronglyMeasurable.aemeasurable, BoundedFiniteSupport.aestronglyMeasurable
theorem BoundedFiniteSupport.aestronglyMeasurable_restrict line 39 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENorm, Zero, BoundedFiniteSupport, Set, MeasureTheory.AEStronglyMeasurable, MeasureTheory.Measure.restrict
Proof uses: MeasureTheory.AEStronglyMeasurable.restrict, BoundedFiniteSupport.aestronglyMeasurable
theorem BoundedFiniteSupport.aemeasurable_restrict line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENorm, Zero, BoundedFiniteSupport, TopologicalSpace.PseudoMetrizableSpace, BorelSpace, Set, AEMeasurable, MeasureTheory.Measure.restrict
Proof uses: AEMeasurable.restrict, BoundedFiniteSupport.aemeasurable
theorem BoundedFiniteSupport.eLpNorm_lt_top line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENorm, Zero, BoundedFiniteSupport, LT.lt, ENNReal, MeasureTheory.eLpNorm, Top.top
Proof uses: MeasureTheory.MemLp.eLpNorm_lt_top, BoundedFiniteSupport.memLp_top
5 instances, coercions and auxiliary constants left out.
theorem BoundedFiniteSupport.memLp Bounded finitely supported functions are in all Lᵖ spaces line 58 source
Source documentation
Bounded finitely supported functions are in all Lᵖ spaces.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ESeminormedAddMonoid, BoundedFiniteSupport, ENNReal, MeasureTheory.MemLp
Proof uses: MeasureTheory.MemLp.mono_exponent_of_measure_support_ne_top, Top.top, BoundedFiniteSupport.memLp_top, Function.support, Iff.mp, Not, Membership.mem, Set, Eq, OfNat.ofNat, Function.notMem_support, LT.lt.ne, autoParam, BoundedFiniteSupport.measure_support_lt, le_top
22 instances, coercions and auxiliary constants left out.
```
theorem memLp (hf : BoundedFiniteSupport f μ) (p : ℝ≥0∞) : MemLp f p μ :=
…
```
Mentions: BoundedFiniteSupport
theorem BoundedFiniteSupport.integrable Bounded finitely supported functions are integrable line 70 source
Source documentation
Bounded finitely supported functions are integrable.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, NormedAddCommGroup, BoundedFiniteSupport, MeasureTheory.Integrable
Proof uses: Iff.mp, MeasureTheory.MemLp, OfNat.ofNat, ENNReal, MeasureTheory.memLp_one_iff_integrable, BoundedFiniteSupport.memLp
33 instances, coercions and auxiliary constants left out.
```
theorem integrable (hf : BoundedFiniteSupport f μ) : Integrable f μ :=
…
```
Mentions: BoundedFiniteSupport
theorem BoundedFiniteSupport.indicator line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, NormedAddCommGroup, BoundedFiniteSupport, Set, MeasurableSet, Set.indicator
Proof uses: BoundedFiniteSupport.mk, MeasureTheory.MemLp.indicator, Top.top, ENNReal, BoundedFiniteSupport.memLp_top, Eq.mpr, LT.lt, autoParam, Function.support, Inter.inter, id, Eq, congrArg, Set.support_indicator, MeasureTheory.measure_inter_lt_top_of_right_ne_top, Ne, Eq.symm, propext, lt_top_iff_ne_top, BoundedFiniteSupport.measure_support_lt
42 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem BoundedFiniteSupport.neg line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, NormedAddCommGroup, BoundedFiniteSupport, Neg.neg
Proof uses: BoundedFiniteSupport.mk, Iff.mpr, MeasureTheory.MemLp, Top.top, ENNReal, MeasureTheory.memLp_neg_iff, BoundedFiniteSupport.memLp_top, Eq.rec, Set, Function.support, Eq, LT.lt, autoParam, BoundedFiniteSupport.measure_support_lt, Eq.symm, Function.support_neg
39 instances, coercions and auxiliary constants left out.
theorem BoundedFiniteSupport.add line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, NormedAddCommGroup, BoundedFiniteSupport, HAdd.hAdd
Proof uses: BoundedFiniteSupport.mk, MeasureTheory.MemLp.add, Top.top, ENNReal, BoundedFiniteSupport.memLp_top, LE.le.trans_lt, autoParam, Set, Function.support, Union.union, MeasureTheory.measure_mono, Function.support_add, MeasureTheory.measure_union_le, Iff.mpr, LT.lt, And, ENNReal.add_lt_top, And.intro, BoundedFiniteSupport.measure_support_lt
62 instances, coercions and auxiliary constants left out.
theorem BoundedFiniteSupport.sub line 95 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, NormedAddCommGroup, BoundedFiniteSupport, HSub.hSub
Proof uses: Eq.rec, HAdd.hAdd, Neg.neg, Eq, BoundedFiniteSupport.add, BoundedFiniteSupport.neg, Eq.symm, sub_eq_add_neg
44 instances, coercions and auxiliary constants left out.