l3lab/ntp-mathlib huggingface dataset:
/-- Given a normal subgroup `Γ` of a topological group `G` with Haar measure `μ`, which is also
  right-invariant, and a finite volume fundamental domain `𝓕`, the pushforward to the quotient
  group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on `G ⧸ Γ`. -/
@[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 pushforward
  to the quotient group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on
  `G ⧸ Γ`."]
theorem MeasureTheory.IsFundamentalDomain.map_restrict_quotient  [T2Space (G ⧸ Γ)]
    [SecondCountableTopology (G ⧸ Γ)] (K : PositiveCompacts (G ⧸ Γ)) [Subgroup.Normal Γ]
    [MeasureTheory.Measure.IsHaarMeasure μ] [μ.IsMulRightInvariant] (h𝓕_finite : μ 𝓕 < ⊤) :
    Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕) =
      μ (𝓕 ∩ QuotientGroup.mk' Γ ⁻¹' K) • MeasureTheory.Measure.haarMeasure K := by
  let π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
  have meas_π : Measurable π := continuous_quotient_mk'.measurable
  

Proof state:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝⁷ : Countable ↥Γ
inst✝⁶ : MeasurableSpace (G ⧸ Γ)
inst✝⁵ : BorelSpace (G ⧸ Γ)
inst✝⁴ : T2Space (G ⧸ Γ)
inst✝³ : SecondCountableTopology (G ⧸ Γ)
K : PositiveCompacts (G ⧸ Γ)
inst✝² : Subgroup.Normal Γ
inst✝¹ : IsHaarMeasure μ
inst✝ : IsMulRightInvariant μ
h𝓕_finite : ↑↑μ 𝓕 < ⊤
π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
meas_π : Measurable ⇑π
⊢ map (⇑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ⇑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K

Next tactic:
have 𝓕meas : NullMeasurableSet 𝓕 μ := h𝓕.nullMeasurableSet
/-- Given a normal subgroup `Γ` of a topological group `G` with Haar measure `μ`, which is also
  right-invariant, and a finite volume fundamental domain `𝓕`, the pushforward to the quotient
  group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on `G ⧸ Γ`. -/
@[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 pushforward
  to the quotient group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on
  `G ⧸ Γ`."]
theorem MeasureTheory.IsFundamentalDomain.map_restrict_quotient  [T2Space (G ⧸ Γ)]
    [SecondCountableTopology (G ⧸ Γ)] (K : PositiveCompacts (G ⧸ Γ)) [Subgroup.Normal Γ]
    [MeasureTheory.Measure.IsHaarMeasure μ] [μ.IsMulRightInvariant] (h𝓕_finite : μ 𝓕 < ⊤) :
    Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕) =
      μ (𝓕 ∩ QuotientGroup.mk' Γ ⁻¹' K) • MeasureTheory.Measure.haarMeasure K := by
  let π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
  have meas_π : Measurable π := continuous_quotient_mk'.measurable
  have 𝓕meas : NullMeasurableSet 𝓕 μ := h𝓕.nullMeasurableSet
  

Proof state:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝⁷ : Countable ↥Γ
inst✝⁶ : MeasurableSpace (G ⧸ Γ)
inst✝⁵ : BorelSpace (G ⧸ Γ)
inst✝⁴ : T2Space (G ⧸ Γ)
inst✝³ : SecondCountableTopology (G ⧸ Γ)
K : PositiveCompacts (G ⧸ Γ)
inst✝² : Subgroup.Normal Γ
inst✝¹ : IsHaarMeasure μ
inst✝ : IsMulRightInvariant μ
h𝓕_finite : ↑↑μ 𝓕 < ⊤
π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
meas_π : Measurable ⇑π
𝓕meas : NullMeasurableSet 𝓕
⊢ map (⇑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ⇑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K

Next tactic:
haveI := Fact.mk h𝓕_finite
/-- Given a normal subgroup `Γ` of a topological group `G` with Haar measure `μ`, which is also
  right-invariant, and a finite volume fundamental domain `𝓕`, the pushforward to the quotient
  group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on `G ⧸ Γ`. -/
@[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 pushforward
  to the quotient group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on
  `G ⧸ Γ`."]
theorem MeasureTheory.IsFundamentalDomain.map_restrict_quotient  [T2Space (G ⧸ Γ)]
    [SecondCountableTopology (G ⧸ Γ)] (K : PositiveCompacts (G ⧸ Γ)) [Subgroup.Normal Γ]
    [MeasureTheory.Measure.IsHaarMeasure μ] [μ.IsMulRightInvariant] (h𝓕_finite : μ 𝓕 < ⊤) :
    Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕) =
      μ (𝓕 ∩ QuotientGroup.mk' Γ ⁻¹' K) • MeasureTheory.Measure.haarMeasure K := by
  let π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
  have meas_π : Measurable π := continuous_quotient_mk'.measurable
  have 𝓕meas : NullMeasurableSet 𝓕 μ := h𝓕.nullMeasurableSet
  haveI := Fact.mk h𝓕_finite
  -- the measure is left-invariant, so by the uniqueness of Haar measure it's enough to show that
  -- it has the stated size on the reference compact set `K`.
  

Proof state:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝⁷ : Countable ↥Γ
inst✝⁶ : MeasurableSpace (G ⧸ Γ)
inst✝⁵ : BorelSpace (G ⧸ Γ)
inst✝⁴ : T2Space (G ⧸ Γ)
inst✝³ : SecondCountableTopology (G ⧸ Γ)
K : PositiveCompacts (G ⧸ Γ)
inst✝² : Subgroup.Normal Γ
inst✝¹ : IsHaarMeasure μ
inst✝ : IsMulRightInvariant μ
h𝓕_finite : ↑↑μ 𝓕 < ⊤
π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
meas_π : Measurable ⇑π
𝓕meas : NullMeasurableSet 𝓕
this : Fact (↑↑μ 𝓕 < ⊤)
⊢ map (⇑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ⇑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K

Next tactic:
haveI : (Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕)).IsMulLeftInvariant :=
    h𝓕.isMulLeftInvariant_map
/-- Given a normal subgroup `Γ` of a topological group `G` with Haar measure `μ`, which is also
  right-invariant, and a finite volume fundamental domain `𝓕`, the pushforward to the quotient
  group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on `G ⧸ Γ`. -/
@[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 pushforward
  to the quotient group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on
  `G ⧸ Γ`."]
theorem MeasureTheory.IsFundamentalDomain.map_restrict_quotient  [T2Space (G ⧸ Γ)]
    [SecondCountableTopology (G ⧸ Γ)] (K : PositiveCompacts (G ⧸ Γ)) [Subgroup.Normal Γ]
    [MeasureTheory.Measure.IsHaarMeasure μ] [μ.IsMulRightInvariant] (h𝓕_finite : μ 𝓕 < ⊤) :
    Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕) =
      μ (𝓕 ∩ QuotientGroup.mk' Γ ⁻¹' K) • MeasureTheory.Measure.haarMeasure K := by
  let π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
  have meas_π : Measurable π := continuous_quotient_mk'.measurable
  have 𝓕meas : NullMeasurableSet 𝓕 μ := h𝓕.nullMeasurableSet
  haveI := Fact.mk h𝓕_finite
  -- the measure is left-invariant, so by the uniqueness of Haar measure it's enough to show that
  -- it has the stated size on the reference compact set `K`.
  haveI : (Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕)).IsMulLeftInvariant :=
    h𝓕.isMulLeftInvariant_map
  

Proof state:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝⁷ : Countable ↥Γ
inst✝⁶ : MeasurableSpace (G ⧸ Γ)
inst✝⁵ : BorelSpace (G ⧸ Γ)
inst✝⁴ : T2Space (G ⧸ Γ)
inst✝³ : SecondCountableTopology (G ⧸ Γ)
K : PositiveCompacts (G ⧸ Γ)
inst✝² : Subgroup.Normal Γ
inst✝¹ : IsHaarMeasure μ
inst✝ : IsMulRightInvariant μ
h𝓕_finite : ↑↑μ 𝓕 < ⊤
π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
meas_π : Measurable ⇑π
𝓕meas : NullMeasurableSet 𝓕
this✝ : Fact (↑↑μ 𝓕 < ⊤)
this : IsMulLeftInvariant (map (⇑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕))
⊢ map (⇑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ⇑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K

Next tactic:
rw [Measure.haarMeasure_unique (Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕)) K,
    Measure.map_apply meas_π, Measure.restrict_apply₀' 𝓕meas, inter_comm]
/-- Given a normal subgroup `Γ` of a topological group `G` with Haar measure `μ`, which is also
  right-invariant, and a finite volume fundamental domain `𝓕`, the pushforward to the quotient
  group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on `G ⧸ Γ`. -/
@[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 pushforward
  to the quotient group `G ⧸ Γ` of the restriction of `μ` to `𝓕` is a multiple of Haar measure on
  `G ⧸ Γ`."]
theorem MeasureTheory.IsFundamentalDomain.map_restrict_quotient  [T2Space (G ⧸ Γ)]
    [SecondCountableTopology (G ⧸ Γ)] (K : PositiveCompacts (G ⧸ Γ)) [Subgroup.Normal Γ]
    [MeasureTheory.Measure.IsHaarMeasure μ] [μ.IsMulRightInvariant] (h𝓕_finite : μ 𝓕 < ⊤) :
    Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕) =
      μ (𝓕 ∩ QuotientGroup.mk' Γ ⁻¹' K) • MeasureTheory.Measure.haarMeasure K := by
  let π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
  have meas_π : Measurable π := continuous_quotient_mk'.measurable
  have 𝓕meas : NullMeasurableSet 𝓕 μ := h𝓕.nullMeasurableSet
  haveI := Fact.mk h𝓕_finite
  -- the measure is left-invariant, so by the uniqueness of Haar measure it's enough to show that
  -- it has the stated size on the reference compact set `K`.
  haveI : (Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕)).IsMulLeftInvariant :=
    h𝓕.isMulLeftInvariant_map
  rw [Measure.haarMeasure_unique (Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕)) K,
    Measure.map_apply meas_π, Measure.restrict_apply₀' 𝓕meas, inter_comm]
  

Proof state:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝⁷ : Countable ↥Γ
inst✝⁶ : MeasurableSpace (G ⧸ Γ)
inst✝⁵ : BorelSpace (G ⧸ Γ)
inst✝⁴ : T2Space (G ⧸ Γ)
inst✝³ : SecondCountableTopology (G ⧸ Γ)
K : PositiveCompacts (G ⧸ Γ)
inst✝² : Subgroup.Normal Γ
inst✝¹ : IsHaarMeasure μ
inst✝ : IsMulRightInvariant μ
h𝓕_finite : ↑↑μ 𝓕 < ⊤
π : G →* G ⧸ Γ := QuotientGroup.mk' Γ
meas_π : Measurable ⇑π
𝓕meas : NullMeasurableSet 𝓕
this✝ : Fact (↑↑μ 𝓕 < ⊤)
this : IsMulLeftInvariant (map (⇑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕))
⊢ MeasurableSet ↑K

Next tactic:
exact K.isCompact.measurableSet
/-- Given a normal subgroup `Γ` of a 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` is
  measure-preserving between appropriate multiples of Haar measure on `G` and `G ⧸ Γ`. -/
@[to_additive MeasurePreservingQuotientAddGroup.mk' "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` is measure-preserving between appropriate
  multiples of Haar measure on `G` and `G ⧸ Γ`."]
theorem MeasurePreservingQuotientGroup.mk' [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)]
    (K : PositiveCompacts (G ⧸ Γ)) [Subgroup.Normal Γ] [μ.IsHaarMeasure]
    [μ.IsMulRightInvariant] (h𝓕_finite : μ 𝓕 < ⊤) (c : ℝ≥0)
    (h : μ (𝓕 ∩ QuotientGroup.mk' Γ ⁻¹' K) = c) :
    MeasurePreserving (QuotientGroup.mk' Γ) (μ.restrict 𝓕)
      (c • MeasureTheory.Measure.haarMeasure K) where
  measurable := continuous_quotient_mk'.measurable
  map_eq := by 

Proof state:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝⁷ : Countable ↥Γ
inst✝⁶ : MeasurableSpace (G ⧸ Γ)
inst✝⁵ : BorelSpace (G ⧸ Γ)
inst✝⁴ : T2Space (G ⧸ Γ)
inst✝³ : SecondCountableTopology (G ⧸ Γ)
K : PositiveCompacts (G ⧸ Γ)
inst✝² : Subgroup.Normal Γ
inst✝¹ : IsHaarMeasure μ
inst✝ : IsMulRightInvariant μ
h𝓕_finite : ↑↑μ 𝓕 < ⊤
c : ℝ≥0
h : ↑↑μ (𝓕 ∩ ⇑(QuotientGroup.mk' Γ) ⁻¹' ↑K) = ↑c
⊢ map (⇑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = c • haarMeasure K

Next tactic:
rw [h𝓕.map_restrict_quotient K h𝓕_finite, h]
/-- Given a normal subgroup `Γ` of a 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` is
  measure-preserving between appropriate multiples of Haar measure on `G` and `G ⧸ Γ`. -/
@[to_additive MeasurePreservingQuotientAddGroup.mk' "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` is measure-preserving between appropriate
  multiples of Haar measure on `G` and `G ⧸ Γ`."]
theorem MeasurePreservingQuotientGroup.mk' [T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)]
    (K : PositiveCompacts (G ⧸ Γ)) [Subgroup.Normal Γ] [μ.IsHaarMeasure]
    [μ.IsMulRightInvariant] (h𝓕_finite : μ 𝓕 < ⊤) (c : ℝ≥0)
    (h : μ (𝓕 ∩ QuotientGroup.mk' Γ ⁻¹' K) = c) :
    MeasurePreserving (QuotientGroup.mk' Γ) (μ.restrict 𝓕)
      (c • MeasureTheory.Measure.haarMeasure K) where
  measurable := continuous_quotient_mk'.measurable
  map_eq := by rw [h𝓕.map_restrict_quotient K h𝓕_finite, h]; 

Proof state:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝⁷ : Countable ↥Γ
inst✝⁶ : MeasurableSpace (G ⧸ Γ)
inst✝⁵ : BorelSpace (G ⧸ Γ)
inst✝⁴ : T2Space (G ⧸ Γ)
inst✝³ : SecondCountableTopology (G ⧸ Γ)
K : PositiveCompacts (G ⧸ Γ)
inst✝² : Subgroup.Normal Γ
inst✝¹ : IsHaarMeasure μ
inst✝ : IsMulRightInvariant μ
h𝓕_finite : ↑↑μ 𝓕 < ⊤
c : ℝ≥0
h : ↑↑μ (𝓕 ∩ ⇑(QuotientGroup.mk' Γ) ⁻¹' ↑K) = ↑c
⊢ ↑c • haarMeasure K = c • haarMeasure K

Next tactic:
rfl
/-- The `essSup` of a function `g` on the 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`. -/
@[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 := by
  

Proof state:
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : IsMulRightInvariant μ
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g
⊢ essSup g (map QuotientGroup.mk (Measure.restrict μ 𝓕)) = essSup (fun x => g ↑x) μ

Next tactic:
have hπ : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
/-- The `essSup` of a function `g` on the 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`. -/
@[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 := by
  have hπ : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
  

Proof state:
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : IsMulRightInvariant μ
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g
hπ : Measurable QuotientGroup.mk
⊢ essSup g (map QuotientGroup.mk (Measure.restrict μ 𝓕)) = essSup (fun x => g ↑x) μ

Next tactic:
rw [essSup_map_measure g_ae_measurable hπ.aemeasurable]
/-- The `essSup` of a function `g` on the 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`. -/
@[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 := by
  have hπ : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
  rw [essSup_map_measure g_ae_measurable hπ.aemeasurable]
  

Proof state:
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : IsMulRightInvariant μ
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g
hπ : Measurable QuotientGroup.mk
⊢ essSup (g ∘ QuotientGroup.mk) (Measure.restrict μ 𝓕) = essSup (fun x => g ↑x) μ

Next tactic:
refine h𝓕.essSup_measure_restrict ?_
/-- The `essSup` of a function `g` on the 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`. -/
@[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 := by
  have hπ : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
  rw [essSup_map_measure g_ae_measurable hπ.aemeasurable]
  refine h𝓕.essSup_measure_restrict ?_
  

Proof state:
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : IsMulRightInvariant μ
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g
hπ : Measurable QuotientGroup.mk
⊢ ∀ (γ : ↥(Subgroup.op Γ)) (x : G), (g ∘ QuotientGroup.mk) (γ • x) = (g ∘ QuotientGroup.mk) x

Next tactic:
intro ⟨γ, hγ⟩ x
/-- The `essSup` of a function `g` on the 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`. -/
@[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 := by
  have hπ : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
  rw [essSup_map_measure g_ae_measurable hπ.aemeasurable]
  refine h𝓕.essSup_measure_restrict ?_
  intro ⟨γ, hγ⟩ x
  

Proof state:
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : IsMulRightInvariant μ
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g
hπ : Measurable QuotientGroup.mk
γ : Gᵐᵒᵖ
hγ : γ ∈ Subgroup.op Γ
x : G
⊢ (g ∘ QuotientGroup.mk) ({ val := γ, property := hγ } • x) = (g ∘ QuotientGroup.mk) x

Next tactic:
dsimp
/-- The `essSup` of a function `g` on the 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`. -/
@[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 := by
  have hπ : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
  rw [essSup_map_measure g_ae_measurable hπ.aemeasurable]
  refine h𝓕.essSup_measure_restrict ?_
  intro ⟨γ, hγ⟩ x
  dsimp
  

Proof state:
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : IsMulRightInvariant μ
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g
hπ : Measurable QuotientGroup.mk
γ : Gᵐᵒᵖ
hγ : γ ∈ Subgroup.op Γ
x : G
⊢ g ↑(x * MulOpposite.unop γ) = g ↑x

Next tactic:
congr 1