Extension.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

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2 declarations, 0 contain sorry, 0 declared as axioms

Haar measures on group extensions

In this file, if 1 → A → B → C → 1 is a short exact sequence of topological groups, we construct a Haar measure on B from Haar measures on A and B.

Main definitions

TopologicalGroup.IsSES φ ψ: A predicate stating that φ is a closed embedding, ψ is an open quotient map, and φ.range = ψ.ker.
TopologicalGroup.IsSES.inducedMeasure: The Haar measure on B induced by Haar measures on A and C.

Main results

TopologicalGroup.IsSES.isHaarMeasure_inducedMeasure: inducedMeasure is a Haar measure.
TopologicalGroup.IsSES.not_injOn_of_inducedMeasure_gt: A sufficiently large open subset of B cannot be a fundamental domain.

Imports (3)

Imported by 2 files

theorem TopologicalGroup.IsSES.not_injOn_of_inducedMeasure_gt A sufficiently large open subset of B cannot be a fundamental domain line 55 source
Source documentation
A sufficiently large open subset of B cannot be a fundamental domain.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, MonoidHom, TopologicalGroup.IsSES, IsTopologicalGroup, MeasurableSpace, BorelSpace, MeasureTheory.Measure, MeasureTheory.Measure.IsHaarMeasure, LocallyCompactSpace, T2Space, Set, IsOpen, DiscreteTopology, LT.lt, ENNReal, HMul.hMul, Set.univ, Singleton.singleton, OfNat.ofNat and 3 more
Proof uses: Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, LE.le, id, Eq, implies_congr, Eq.refl, TopologicalGroup.IsSES.inducedMeasure_lt_of_injOn
48 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem not_injOn_of_inducedMeasure_gt (U : Set B) (hU : IsOpen U) [DiscreteTopology A]
    (h : μC Set.univ * μA {1} < inducedMeasure H μA μC U) :
    ¬ U.InjOn ψ := by
…
```
theorem TopologicalGroup.IsSES.not_injOn_of_measure_gt line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, IsTopologicalGroup, PolishSpace, MeasurableSpace, BorelSpace, LocallyCompactSpace, Subgroup, Subgroup.Normal, DiscreteTopology, Subtype, Membership.mem, CompactSpace, HasQuotient.Quotient, Exists, NNReal, Set, IsOpen, LT.lt, ENNReal and 6 more
Proof uses: IsClosed, TopologicalGroup.IsSES, Subgroup.subtype, TopologicalGroup.IsSES.ofClosedSubgroup, Exists.intro, ENNReal.toNNReal, HSMul.hSMul, Inv.inv, MeasureTheory.Measure.haarScalarFactor, TopologicalGroup.IsSES.inducedMeasure, Classical.arbitrary, TopologicalSpace.PositiveCompacts, HMul.hMul, Set.univ, Singleton.singleton, OfNat.ofNat, TopologicalGroup.IsSES.not_injOn_of_inducedMeasure_gt, Eq.mpr, id, Eq and 12 more
103 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.