ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.rnDeriv_tilted_left_self
lemma rnDeriv_tilted_left_self [SigmaFinite μ] (hf : AEMeasurable f μ) :
    (μ.tilted f).rnDeriv μ =ᵐ[μ] fun x ↦ ENNReal.ofReal (exp (f x) / ∫ x, exp (f x) ∂μ) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.SigmaFinite μ] (hf : AEMeasurable f μ)
Goal: (μ.tilted f).rnDeriv μ =ᵐ[μ] fun x => ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)
-- MeasureTheory.log_rnDeriv_tilted_left_self
lemma log_rnDeriv_tilted_left_self [SigmaFinite μ] (hf : Integrable (fun x ↦ exp (f x)) μ) :
    (fun x ↦ log ((μ.tilted f).rnDeriv μ x).toReal)
      =ᵐ[μ] fun x ↦ f x - log (∫ x, exp (f x) ∂μ) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.SigmaFinite μ] (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ)
Goal: (fun x => Real.log ((μ.tilted f).rnDeriv μ x).toReal) =ᵐ[μ] fun x => f x - Real.log (∫ (x : α), rexp (f x) ∂μ)
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.smulInvariantMeasure_quotient
@[to_additive]
lemma MeasureTheory.QuotientMeasureEqMeasurePreimage.smulInvariantMeasure_quotient
    [IsMulLeftInvariant ν] [hasFun : HasFundamentalDomain Γ.op G ν] :
    SMulInvariantMeasure G (G ⧸ Γ) μ 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [PolishSpace G] (ν : MeasureTheory.Measure G) {Γ : Subgroup G} [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] [ν.IsMulLeftInvariant] [hasFun : MeasureTheory.HasFundamentalDomain (↥Γ.op) G ν]
Goal: MeasureTheory.SMulInvariantMeasure G (G ⧸ Γ) μ
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.vaddInvariantMeasure_quotient
@[to_additive]
lemma MeasureTheory.QuotientMeasureEqMeasurePreimage.smulInvariantMeasure_quotient
    [IsMulLeftInvariant ν] [hasFun : HasFundamentalDomain Γ.op G ν] :
    SMulInvariantMeasure G (G ⧸ Γ) μ 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [PolishSpace G] (ν : MeasureTheory.Measure G) {Γ : AddSubgroup G} [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] [ν.IsAddLeftInvariant] [hasFun : MeasureTheory.HasAddFundamentalDomain (↥Γ.op) G ν]
Goal: MeasureTheory.VAddInvariantMeasure G (G ⧸ Γ) μ
-- measurePreserving_quotientGroup_mk_of_QuotientMeasureEqMeasurePreimage
@[to_additive]
theorem measurePreserving_quotientGroup_mk_of_QuotientMeasureEqMeasurePreimage
    {𝓕 : Set G} (h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) (μ : Measure (G ⧸ Γ))
    [QuotientMeasureEqMeasurePreimage ν μ] :
    MeasurePreserving (@QuotientGroup.mk G _ Γ) (ν.restrict 𝓕) μ 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] (ν : MeasureTheory.Measure G) {Γ : Subgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 ν) (μ : MeasureTheory.Measure (G ⧸ Γ)) [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ]
Goal: MeasureTheory.MeasurePreserving QuotientGroup.mk (ν.restrict 𝓕) μ
-- measurePreserving_quotientAddGroup_mk_of_AddQuotientMeasureEqMeasurePreimage
@[to_additive]
theorem measurePreserving_quotientGroup_mk_of_QuotientMeasureEqMeasurePreimage
    {𝓕 : Set G} (h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) (μ : Measure (G ⧸ Γ))
    [QuotientMeasureEqMeasurePreimage ν μ] :
    MeasurePreserving (@QuotientGroup.mk G _ Γ) (ν.restrict 𝓕) μ 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] (ν : MeasureTheory.Measure G) {Γ : AddSubgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsAddFundamentalDomain (↥Γ.op) 𝓕 ν) (μ : MeasureTheory.Measure (G ⧸ Γ)) [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ]
Goal: MeasureTheory.MeasurePreserving QuotientAddGroup.mk (ν.restrict 𝓕) μ
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
@[to_additive "If `μ` on `G ⧸ Γ` satisfies `AddQuotientMeasureEqMeasurePreimage` relative to a both
  left- and right-invariant measure on `G` and `Γ` is a normal subgroup, then `μ` is a
  left-invariant measure."]
lemma MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
    [hasFun : HasFundamentalDomain Γ.op G ν] [QuotientMeasureEqMeasurePreimage ν μ] :
    μ.IsMulLeftInvariant 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [PolishSpace G] {Γ : Subgroup G} [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsMulLeftInvariant] [hasFun : MeasureTheory.HasFundamentalDomain (↥Γ.op) G ν] [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ]
Goal: μ.IsMulLeftInvariant
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.addInvariantMeasure_quotient
@[to_additive "If `μ` on `G ⧸ Γ` satisfies `AddQuotientMeasureEqMeasurePreimage` relative to a both
  left- and right-invariant measure on `G` and `Γ` is a normal subgroup, then `μ` is a
  left-invariant measure."]
lemma MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
    [hasFun : HasFundamentalDomain Γ.op G ν] [QuotientMeasureEqMeasurePreimage ν μ] :
    μ.IsMulLeftInvariant 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [PolishSpace G] {Γ : AddSubgroup G} [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsAddLeftInvariant] [hasFun : MeasureTheory.HasAddFundamentalDomain (↥Γ.op) G ν] [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ]
Goal: μ.IsAddLeftInvariant
-- MeasureTheory.Measure.IsAddLeftInvariant.addQuotientMeasureEqMeasurePreimage_of_set
@[to_additive
"Assume that a measure `μ` is `IsAddLeftInvariant`, that the action of `Γ` on `G` has a
measurable fundamental domain `s` with positive finite volume, and that there is a single measurable
set `V ⊆ G ⧸ Γ` along which the pullback of `μ` and `ν` agree (so the scaling is right). Then
`μ` satisfies `AddQuotientMeasureEqMeasurePreimage`. The main tool of the proof is the uniqueness of
left invariant measures, if normalized by a single positive finite-measured set."]
theorem MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set {s : Set G}
    (fund_dom_s : IsFundamentalDomain Γ.op s ν) {V : Set (G ⧸ Γ)}
    (meas_V : MeasurableSet V) (neZeroV : μ V ≠ 0) (hV : μ V = ν (π ⁻¹' V ∩ s))
    (neTopV : μ V ≠ ⊤) : QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [PolishSpace G] {Γ : AddSubgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsAddLeftInvariant] [ν.IsAddRightInvariant] [MeasureTheory.SigmaFinite ν] [μ.IsAddLeftInvariant] [MeasureTheory.SigmaFinite μ] {s : Set G} (fund_dom_s : MeasureTheory.IsAddFundamentalDomain (↥Γ.op) s ν) {V : Set (G ⧸ Γ)} (meas_V : MeasurableSet V) (neZeroV : μ V ≠ 0) (hV : μ V = ν (QuotientAddGroup.mk ⁻¹' V ∩ s)) (neTopV : μ V ≠ ⊤)
Goal: MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ
-- MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set
@[to_additive
"Assume that a measure `μ` is `IsAddLeftInvariant`, that the action of `Γ` on `G` has a
measurable fundamental domain `s` with positive finite volume, and that there is a single measurable
set `V ⊆ G ⧸ Γ` along which the pullback of `μ` and `ν` agree (so the scaling is right). Then
`μ` satisfies `AddQuotientMeasureEqMeasurePreimage`. The main tool of the proof is the uniqueness of
left invariant measures, if normalized by a single positive finite-measured set."]
theorem MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set {s : Set G}
    (fund_dom_s : IsFundamentalDomain Γ.op s ν) {V : Set (G ⧸ Γ)}
    (meas_V : MeasurableSet V) (neZeroV : μ V ≠ 0) (hV : μ V = ν (π ⁻¹' V ∩ s))
    (neTopV : μ V ≠ ⊤) : QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [PolishSpace G] {Γ : Subgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsMulLeftInvariant] [ν.IsMulRightInvariant] [MeasureTheory.SigmaFinite ν] [μ.IsMulLeftInvariant] [MeasureTheory.SigmaFinite μ] {s : Set G} (fund_dom_s : MeasureTheory.IsFundamentalDomain (↥Γ.op) s ν) {V : Set (G ⧸ Γ)} (meas_V : MeasurableSet V) (neZeroV : μ V ≠ 0) (hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)) (neTopV : μ V ≠ ⊤)
Goal: MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ
-- MeasureTheory.leftInvariantIsQuotientMeasureEqMeasurePreimage
@[to_additive "If a measure `μ` is
left-invariant and satisfies the right scaling condition, then it satisfies
`AddQuotientMeasureEqMeasurePreimage`."]
theorem MeasureTheory.leftInvariantIsQuotientMeasureEqMeasurePreimage [IsFiniteMeasure μ]
    [hasFun : HasFundamentalDomain Γ.op G ν]
    (h : covolume Γ.op G ν = μ univ) : QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [PolishSpace G] {Γ : Subgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsMulLeftInvariant] [ν.IsMulRightInvariant] [MeasureTheory.SigmaFinite ν] [μ.IsMulLeftInvariant] [MeasureTheory.SigmaFinite μ] [MeasureTheory.IsFiniteMeasure μ] [hasFun : MeasureTheory.HasFundamentalDomain (↥Γ.op) G ν] (h : MeasureTheory.covolume (↥Γ.op) G ν = μ Set.univ)
Goal: MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ
-- MeasureTheory.leftInvariantIsAddQuotientMeasureEqMeasurePreimage
@[to_additive "If a measure `μ` is
left-invariant and satisfies the right scaling condition, then it satisfies
`AddQuotientMeasureEqMeasurePreimage`."]
theorem MeasureTheory.leftInvariantIsQuotientMeasureEqMeasurePreimage [IsFiniteMeasure μ]
    [hasFun : HasFundamentalDomain Γ.op G ν]
    (h : covolume Γ.op G ν = μ univ) : QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [PolishSpace G] {Γ : AddSubgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsAddLeftInvariant] [ν.IsAddRightInvariant] [MeasureTheory.SigmaFinite ν] [μ.IsAddLeftInvariant] [MeasureTheory.SigmaFinite μ] [MeasureTheory.IsFiniteMeasure μ] [hasFun : MeasureTheory.HasAddFundamentalDomain (↥Γ.op) G ν] (h : MeasureTheory.addCovolume (↥Γ.op) G ν = μ Set.univ)
Goal: MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.haarMeasure_quotient
@[to_additive "If a measure `μ` on the quotient `G ⧸ Γ` of an additive group `G` by a discrete
normal subgroup `Γ` having fundamental domain, satisfies `AddQuotientMeasureEqMeasurePreimage`
relative to a standardized choice of Haar measure on `G`, and assuming `μ` is finite, then `μ` is
itself Haar."]
theorem MeasureTheory.QuotientMeasureEqMeasurePreimage.haarMeasure_quotient [LocallyCompactSpace G]
    [QuotientMeasureEqMeasurePreimage ν μ] [i : HasFundamentalDomain Γ.op G ν]
    [IsFiniteMeasure μ] : IsHaarMeasure μ 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [PolishSpace G] {Γ : Subgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsHaarMeasure] [ν.IsMulRightInvariant] [LocallyCompactSpace G] [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] [i : MeasureTheory.HasFundamentalDomain (↥Γ.op) G ν] [MeasureTheory.IsFiniteMeasure μ]
Goal: μ.IsHaarMeasure
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.addHaarMeasure_quotient
@[to_additive "If a measure `μ` on the quotient `G ⧸ Γ` of an additive group `G` by a discrete
normal subgroup `Γ` having fundamental domain, satisfies `AddQuotientMeasureEqMeasurePreimage`
relative to a standardized choice of Haar measure on `G`, and assuming `μ` is finite, then `μ` is
itself Haar."]
theorem MeasureTheory.QuotientMeasureEqMeasurePreimage.haarMeasure_quotient [LocallyCompactSpace G]
    [QuotientMeasureEqMeasurePreimage ν μ] [i : HasFundamentalDomain Γ.op G ν]
    [IsFiniteMeasure μ] : IsHaarMeasure μ 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [PolishSpace G] {Γ : AddSubgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [ν.IsAddHaarMeasure] [ν.IsAddRightInvariant] [LocallyCompactSpace G] [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] [i : MeasureTheory.HasAddFundamentalDomain (↥Γ.op) G ν] [MeasureTheory.IsFiniteMeasure μ]
Goal: μ.IsAddHaarMeasure
-- IsFundamentalDomain.AddQuotientMeasureEqMeasurePreimage_AddHaarMeasure
@[to_additive "Given a normal
subgroup `Γ` of an additive topological group `G` with Haar measure `μ`, which is also
right-invariant, and a finite volume fundamental domain `𝓕`, the quotient map to `G ⧸ Γ`,
properly normalized, satisfies `AddQuotientMeasureEqMeasurePreimage`."]
theorem IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_HaarMeasure {𝓕 : Set G}
    (h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) [IsMulLeftInvariant μ] [SigmaFinite μ]
    {V : Set (G ⧸ Γ)} (hV : (interior V).Nonempty) (meas_V : MeasurableSet V)
    (hμK : μ V = ν ((π ⁻¹' V) ∩ 𝓕)) (neTopV : μ V ≠ ⊤) :
    QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [PolishSpace G] {Γ : AddSubgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite ν] [ν.IsAddHaarMeasure] [ν.IsAddRightInvariant] {𝓕 : Set G} (h𝓕 : MeasureTheory.IsAddFundamentalDomain (↥Γ.op) 𝓕 ν) [μ.IsAddLeftInvariant] [MeasureTheory.SigmaFinite μ] {V : Set (G ⧸ Γ)} (hV : (interior V).Nonempty) (meas_V : MeasurableSet V) (hμK : μ V = ν (QuotientAddGroup.mk ⁻¹' V ∩ 𝓕)) (neTopV : μ V ≠ ⊤)
Goal: MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ
-- IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_HaarMeasure
@[to_additive "Given a normal
subgroup `Γ` of an additive topological group `G` with Haar measure `μ`, which is also
right-invariant, and a finite volume fundamental domain `𝓕`, the quotient map to `G ⧸ Γ`,
properly normalized, satisfies `AddQuotientMeasureEqMeasurePreimage`."]
theorem IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_HaarMeasure {𝓕 : Set G}
    (h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) [IsMulLeftInvariant μ] [SigmaFinite μ]
    {V : Set (G ⧸ Γ)} (hV : (interior V).Nonempty) (meas_V : MeasurableSet V)
    (hμK : μ V = ν ((π ⁻¹' V) ∩ 𝓕)) (neTopV : μ V ≠ ⊤) :
    QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [PolishSpace G] {Γ : Subgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] {μ : MeasureTheory.Measure (G ⧸ Γ)} (ν : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite ν] [ν.IsHaarMeasure] [ν.IsMulRightInvariant] {𝓕 : Set G} (h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 ν) [μ.IsMulLeftInvariant] [MeasureTheory.SigmaFinite μ] {V : Set (G ⧸ Γ)} (hV : (interior V).Nonempty) (meas_V : MeasurableSet V) (hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)) (neTopV : μ V ≠ ⊤)
Goal: MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ
-- IsFundamentalDomain.AddQuotientMeasureEqMeasurePreimage_vaddAddHaarMeasure
@[to_additive "Given a
normal subgroup `Γ` of an additive topological group `G` with Haar measure `μ`, which is also
right-invariant, and a finite volume fundamental domain `𝓕`, the quotient map to `G ⧸ Γ`,
properly normalized, satisfies `AddQuotientMeasureEqMeasurePreimage`."]
theorem IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure {𝓕 : Set G}
    (h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) (h𝓕_finite : ν 𝓕 ≠ ⊤) :
    QuotientMeasureEqMeasurePreimage ν
      ((ν ((π ⁻¹' (K : Set (G ⧸ Γ))) ∩ 𝓕)) • haarMeasure K) 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [PolishSpace G] {Γ : AddSubgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] (ν : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite ν] [ν.IsAddHaarMeasure] [ν.IsAddRightInvariant] (K : TopologicalSpace.PositiveCompacts (G ⧸ Γ)) {𝓕 : Set G} (h𝓕 : MeasureTheory.IsAddFundamentalDomain (↥Γ.op) 𝓕 ν) (h𝓕_finite : ν 𝓕 ≠ ⊤)
Goal: MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν
  (ν (QuotientAddGroup.mk ⁻¹' ↑K ∩ 𝓕) • MeasureTheory.Measure.addHaarMeasure K)
-- IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure
@[to_additive "Given a
normal subgroup `Γ` of an additive topological group `G` with Haar measure `μ`, which is also
right-invariant, and a finite volume fundamental domain `𝓕`, the quotient map to `G ⧸ Γ`,
properly normalized, satisfies `AddQuotientMeasureEqMeasurePreimage`."]
theorem IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure {𝓕 : Set G}
    (h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) (h𝓕_finite : ν 𝓕 ≠ ⊤) :
    QuotientMeasureEqMeasurePreimage ν
      ((ν ((π ⁻¹' (K : Set (G ⧸ Γ))) ∩ 𝓕)) • haarMeasure K) 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [PolishSpace G] {Γ : Subgroup G} [Countable ↥Γ] [Γ.Normal] [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)] (ν : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite ν] [ν.IsHaarMeasure] [ν.IsMulRightInvariant] (K : TopologicalSpace.PositiveCompacts (G ⧸ Γ)) {𝓕 : Set G} (h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 ν) (h𝓕_finite : ν 𝓕 ≠ ⊤)
Goal: MeasureTheory.QuotientMeasureEqMeasurePreimage ν (ν (QuotientGroup.mk ⁻¹' ↑K ∩ 𝓕) • MeasureTheory.Measure.haarMeasure K)
-- essSup_comp_quotientAddGroup_mk
@[to_additive "The `essSup` of a function `g` on the additive quotient space `G ⧸ Γ` with respect
  to the pushforward of the restriction, `μ_𝓕`, of a right-invariant measure `μ` to a fundamental
  domain `𝓕`, is the same as the `essSup` of `g`'s lift to the universal cover `G` with respect
  to `μ`."]
lemma essSup_comp_quotientGroup_mk [μ.IsMulRightInvariant] {g : G ⧸ Γ → ℝ≥0∞}
    (g_ae_measurable : AEMeasurable g μ_𝓕) : essSup g μ_𝓕 = essSup (fun (x : G) ↦ g x) μ 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : AddSubgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsAddFundamentalDomain (↥Γ.op) 𝓕 μ) [Countable ↥Γ] [MeasurableSpace (G ⧸ Γ)] [BorelSpace (G ⧸ Γ)] [μ.IsAddRightInvariant] {g : G ⧸ Γ → ℝ≥0∞} (g_ae_measurable : AEMeasurable g (MeasureTheory.Measure.map QuotientAddGroup.mk (μ.restrict 𝓕)))
Goal: essSup g (MeasureTheory.Measure.map QuotientAddGroup.mk (μ.restrict 𝓕)) = essSup (fun x => g ↑x) μ
-- essSup_comp_quotientGroup_mk
@[to_additive "The `essSup` of a function `g` on the additive quotient space `G ⧸ Γ` with respect
  to the pushforward of the restriction, `μ_𝓕`, of a right-invariant measure `μ` to a fundamental
  domain `𝓕`, is the same as the `essSup` of `g`'s lift to the universal cover `G` with respect
  to `μ`."]
lemma essSup_comp_quotientGroup_mk [μ.IsMulRightInvariant] {g : G ⧸ Γ → ℝ≥0∞}
    (g_ae_measurable : AEMeasurable g μ_𝓕) : essSup g μ_𝓕 = essSup (fun (x : G) ↦ g x) μ 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : Subgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 μ) [Countable ↥Γ] [MeasurableSpace (G ⧸ Γ)] [BorelSpace (G ⧸ Γ)] [μ.IsMulRightInvariant] {g : G ⧸ Γ → ℝ≥0∞} (g_ae_measurable : AEMeasurable g (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)))
Goal: essSup g (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)) = essSup (fun x => g ↑x) μ
-- MeasureTheory.IsFundamentalDomain.absolutelyContinuous_map
@[to_additive "Given an additive quotient space `G ⧸ Γ` where `Γ` is `Countable`, and the
  restriction, `μ_𝓕`, of a right-invariant measure `μ` on `G` to a fundamental domain `𝓕`, a set
  in the quotient which has `μ_𝓕`-measure zero, also has measure zero under the
  folding of `μ` under the quotient. Note that, if `Γ` is infinite, then the folded map
  will take the value `∞` on any open set in the quotient!"]
lemma _root_.MeasureTheory.IsFundamentalDomain.absolutelyContinuous_map
    [μ.IsMulRightInvariant] :
    map (QuotientGroup.mk : G → G ⧸ Γ) μ ≪ map (QuotientGroup.mk : G → G ⧸ Γ) (μ.restrict 𝓕) 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : Subgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 μ) [Countable ↥Γ] [MeasurableSpace (G ⧸ Γ)] [BorelSpace (G ⧸ Γ)] [μ.IsMulRightInvariant]
Goal: MeasureTheory.Measure.map QuotientGroup.mk μ ≪ MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)
-- MeasureTheory.IsAddFundamentalDomain.absolutelyContinuous_map
@[to_additive "Given an additive quotient space `G ⧸ Γ` where `Γ` is `Countable`, and the
  restriction, `μ_𝓕`, of a right-invariant measure `μ` on `G` to a fundamental domain `𝓕`, a set
  in the quotient which has `μ_𝓕`-measure zero, also has measure zero under the
  folding of `μ` under the quotient. Note that, if `Γ` is infinite, then the folded map
  will take the value `∞` on any open set in the quotient!"]
lemma _root_.MeasureTheory.IsFundamentalDomain.absolutelyContinuous_map
    [μ.IsMulRightInvariant] :
    map (QuotientGroup.mk : G → G ⧸ Γ) μ ≪ map (QuotientGroup.mk : G → G ⧸ Γ) (μ.restrict 𝓕) 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : AddSubgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsAddFundamentalDomain (↥Γ.op) 𝓕 μ) [Countable ↥Γ] [MeasurableSpace (G ⧸ Γ)] [BorelSpace (G ⧸ Γ)] [μ.IsAddRightInvariant]
Goal: MeasureTheory.Measure.map QuotientAddGroup.mk μ ≪ MeasureTheory.Measure.map QuotientAddGroup.mk (μ.restrict 𝓕)
-- QuotientAddGroup.integral_eq_integral_automorphize
@[to_additive "This is a simple version of the **Unfolding Trick**: Given a subgroup `Γ` of an
  additive  group `G`, the integral of a function `f` on `G` with respect to a right-invariant
  measure `μ` is equal to the integral over the quotient `G ⧸ Γ` of the automorphization of `f`."]
lemma QuotientGroup.integral_eq_integral_automorphize {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [μ.IsMulRightInvariant] {f : G → E}
    (hf₁ : Integrable f μ) (hf₂ : AEStronglyMeasurable (automorphize f) μ_𝓕) :
    ∫ x : G, f x ∂μ = ∫ x : G ⧸ Γ, automorphize f x ∂μ_𝓕 
Arguments: {G : Type u_1} [AddGroup G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : AddSubgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsAddFundamentalDomain (↥Γ.op) 𝓕 μ) [Countable ↥Γ] [MeasurableSpace (G ⧸ Γ)] [BorelSpace (G ⧸ Γ)] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [μ.IsAddRightInvariant] {f : G → E} (hf₁ : MeasureTheory.Integrable f μ) (hf₂ :
  MeasureTheory.AEStronglyMeasurable (QuotientAddGroup.automorphize f)
    (MeasureTheory.Measure.map QuotientAddGroup.mk (μ.restrict 𝓕)))
Goal: ∫ (x : G), f x ∂μ =
  ∫ (x : G ⧸ Γ), QuotientAddGroup.automorphize f x ∂MeasureTheory.Measure.map QuotientAddGroup.mk (μ.restrict 𝓕)