MeasurableSpacePadics.lean`

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16 declarations, 0 contain sorry, 0 declared as axioms

Measurability and measures on the p-adics

This file endows ℤ_[p] and ℚ_[p] with their Borel sigma-algebra and their Haar measure that makes ℤ_[p] (or the copy of ℤ_[p] inside ℚ_[p]) have norm 1.

Imports (3)

Imported by 2 files

instance Padic.instMeasurableSpace line 27 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasurableSpace, Padic
Proof uses: borel
7 instances, coercions and auxiliary constants left out.
instance Padic.instBorelSpace line 28 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, BorelSpace, Padic
Proof uses: BorelSpace.mk, rfl, MeasurableSpace
16 instances, coercions and auxiliary constants left out.
def Padic.unitBallPositiveCompact The unit ball as a compact set with nonempty interior line 32 source
Source documentation
The unit ball as a compact set with nonempty interior.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, TopologicalSpace.PositiveCompacts, Padic
Proof uses: TopologicalSpace.PositiveCompacts.mk, TopologicalSpace.Compacts.mk, Set.ofPred, LE.le, Real, Norm.norm, OfNat.ofNat
20 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
def unitBallPositiveCompact : PositiveCompacts ℚ_[p] where
…
```
instance Padic.instMeasureSpace line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasureTheory.MeasureSpace, Padic
Proof uses: MeasureTheory.MeasureSpace.mk, MeasureTheory.Measure.addHaarMeasure, Padic.unitBallPositiveCompact
13 instances, coercions and auxiliary constants left out.
instance Padic.instIsAddHaarMeasure line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasureTheory.Measure.IsAddHaarMeasure, Padic, MeasureTheory.MeasureSpace.volume
Proof uses: Padic.unitBallPositiveCompact
27 instances, coercions and auxiliary constants left out.
lemma Padic.volume_closedBall_one line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Eq, ENNReal, MeasureTheory.Measure, Padic, Set, MeasureTheory.MeasureSpace.volume, Set.ofPred, LE.le, Real, Norm.norm, OfNat.ofNat
Proof uses: MeasureTheory.Measure.addHaarMeasure_self, Padic.unitBallPositiveCompact
22 instances, coercions and auxiliary constants left out.
instance PadicInt.instMeasurableSpace line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasurableSpace, PadicInt
Proof uses: Padic, Membership.mem, Subring, PadicInt.subring
7 instances, coercions and auxiliary constants left out.
instance PadicInt.instBorelSpace line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, BorelSpace, PadicInt
Proof uses: Padic, Membership.mem, Subring, PadicInt.subring
19 instances, coercions and auxiliary constants left out.
lemma PadicInt.isMeasurableEmbedding_coe line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasurableEmbedding, Subtype, Padic, Membership.mem, Subring, PadicInt.subring
Proof uses: Eq.mpr, Eq.refl, Topology.IsOpenEmbedding.measurableEmbedding, PadicInt.isOpenEmbedding_coe
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma PadicInt.isMeasurableEmbedding_coeRingHom line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasurableEmbedding, PadicInt, Padic, RingHom, PadicInt.Coe.ringHom
Proof uses: Eq.rec, Membership.mem, Subring, PadicInt.subring, Eq, Subtype, PadicInt.isMeasurableEmbedding_coe, Eq.symm, PadicInt.coe_coeRingHom
28 instances, coercions and auxiliary constants left out.
instance PadicInt.instMeasureSpace line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasureTheory.MeasureSpace, PadicInt
Proof uses: MeasureTheory.MeasureSpace.mk, MeasureTheory.Measure.addHaarMeasure, Top.top, TopologicalSpace.PositiveCompacts
16 instances, coercions and auxiliary constants left out.
instance PadicInt.instIsAddHaarMeasure line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasureTheory.Measure.IsAddHaarMeasure, PadicInt, MeasureTheory.MeasureSpace.volume
Proof uses: Top.top, TopologicalSpace.PositiveCompacts
30 instances, coercions and auxiliary constants left out.
lemma PadicInt.volume_univ line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Eq, ENNReal, MeasureTheory.Measure, PadicInt, Set, MeasureTheory.MeasureSpace.volume, Set.univ, OfNat.ofNat
Proof uses: MeasureTheory.Measure.addHaarMeasure_self, Top.top, TopologicalSpace.PositiveCompacts
22 instances, coercions and auxiliary constants left out.
instance PadicInt.instIsFiniteMeasure line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, MeasureTheory.IsFiniteMeasure, PadicInt, MeasureTheory.MeasureSpace.volume
Proof uses: MeasureTheory.IsFiniteMeasure.mk, of_eq_true, LT.lt, ENNReal, MeasureTheory.Measure, Set, Set.univ, Top.top, Eq.trans, OfNat.ofNat, True, congrFun', congrArg, PadicInt.volume_univ
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma PadicInt.volume_coe_univ line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Eq, ENNReal, MeasureTheory.Measure, Padic, Set, MeasureTheory.MeasureSpace.volume, Set.image, PadicInt, Membership.mem, Subring, PadicInt.subring, Set.univ
Proof uses: Eq.mpr, Set.ofPred, LE.le, Real, Norm.norm, OfNat.ofNat, id, congrArg, Eq.trans, PadicInt.volume_univ, Eq.symm, Padic.volume_closedBall_one, Subtype, Subtype.coe_image_univ, Eq.refl
29 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma PadicInt.volume_coe https://github.com/ImperialCollegeLondon/FLT/issues/278 line 86 source
Source comment
https://github.com/ImperialCollegeLondon/FLT/issues/278

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Set, PadicInt, Eq, ENNReal, MeasureTheory.Measure, Padic, MeasureTheory.MeasureSpace.volume, Set.image, Membership.mem, Subring, PadicInt.subring
Proof uses: Set.univ, PadicInt.volume_coe_univ, Eq.mpr, RingHom, PadicInt.Coe.ringHom, id, congrArg, Eq.symm, PadicInt.coe_coeRingHom, MeasureTheory.Measure.comap, MeasurableEmbedding.comap_apply, PadicInt.isMeasurableEmbedding_coeRingHom, MeasureTheory.Measure.IsAddLeftInvariant, AddMonoidHom, RingHom.toAddMonoidHom, MeasureTheory.Measure.IsAddLeftInvariant.comap, MeasureTheory.IsFiniteMeasureOnCompacts, TopologicalSpace.induced, MeasureTheory.IsFiniteMeasureOnCompacts.comap', Iff.mpr and 20 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
@[simp] lemma volume_coe (s : Set ℤ_[p]) : volume ((↑) '' s : Set ℚ_[p]) = volume s := by
…
```