/-- This is the **Unfolding Trick**: Given an additive subgroup `Γ'` of an additive group `G'`, the
  integral of a function `f` on `G'` times the lift to `G'` of a function `g` on the quotient
  `G' ⧸ Γ'` with respect to a right-invariant measure `μ` on `G'`, is equal to the integral over
  the quotient of the automorphization of `f` times `g`. -/
lemma QuotientAddGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]
    [NormedSpace ℝ K] [μ'.IsAddRightInvariant] {f : G' → K}
    (f_ℒ_1 : Integrable f μ') {g : G' ⧸ Γ' → K} (hg : AEStronglyMeasurable g μ_𝓕)
    (g_ℒ_infinity : essSup (fun x ↦ (‖g x‖₊ : ℝ≥0∞)) μ_𝓕 ≠ ∞)
    (F_ae_measurable : AEStronglyMeasurable (QuotientAddGroup.automorphize f) μ_𝓕)
    (h𝓕 : IsAddFundamentalDomain Γ'.op 𝓕' μ') :
    ∫ x : G', g (x : G' ⧸ Γ') * (f x) ∂μ'
      = ∫ x : G' ⧸ Γ', g x * (QuotientAddGroup.automorphize f x) ∂μ_𝓕 := by
  let π : G' → G' ⧸ Γ' := QuotientAddGroup.mk
  have meas_π : Measurable π := continuous_quotient_mk'.measurable
  have H₀ : QuotientAddGroup.automorphize ((g ∘ π) * f) = g * (QuotientAddGroup.automorphize f) :=
    by exact QuotientAddGroup.automorphize_smul_left f g
  calc ∫ (x : G'), g (π x) * f x ∂μ' =
    ∫ (x : G' ⧸ Γ'), QuotientAddGroup.automorphize ((g ∘ π) * f) x ∂μ_𝓕 := ?_
    _ = ∫ (x : G' ⧸ Γ'), g x * (QuotientAddGroup.automorphize f x) ∂μ_𝓕 := by simp [H₀]
  have H₁ : Integrable ((g ∘ π) * f) μ'
  · have : AEStronglyMeasurable (fun (x : G') ↦ g (x : (G' ⧸ Γ'))) μ'
    · refine (AEStronglyMeasurable.mono' hg ?_).comp_measurable meas_π
      exact h𝓕.absolutelyContinuous_map
    refine Integrable.essSup_smul f_ℒ_1 this ?_
    have hg' : AEStronglyMeasurable (fun x ↦ (‖g x‖₊ : ℝ≥0∞)) μ_𝓕 :=
      (ENNReal.continuous_coe.comp continuous_nnnorm).comp_aestronglyMeasurable hg
    rw [← essSup_comp_quotientAddGroup_mk h𝓕 hg'.aemeasurable]
    exact g_ℒ_infinity
  have H₂ : AEStronglyMeasurable (QuotientAddGroup.automorphize ((g ∘ π) * f)) μ_𝓕
  · simp_rw [H₀]
    exact hg.mul F_ae_measurable
  

Proof state:
G : Type u_1
inst✝¹⁸ : Group G
inst✝¹⁷ : MeasurableSpace G
inst✝¹⁶ : TopologicalSpace G
inst✝¹⁵ : TopologicalGroup G
inst✝¹⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕✝ : IsFundamentalDomain (↥(Subgroup.op Γ)) 𝓕
inst✝¹³ : Countable ↥Γ
inst✝¹² : MeasurableSpace (G ⧸ Γ)
inst✝¹¹ : BorelSpace (G ⧸ Γ)
G' : Type u_2
inst✝¹⁰ : AddGroup G'
inst✝⁹ : MeasurableSpace G'
inst✝⁸ : TopologicalSpace G'
inst✝⁷ : TopologicalAddGroup G'
inst✝⁶ : BorelSpace G'
μ' : Measure G'
Γ' : AddSubgroup G'
inst✝⁵ : Countable ↥Γ'
inst✝⁴ : MeasurableSpace (G' ⧸ Γ')
inst✝³ : BorelSpace (G' ⧸ Γ')
𝓕' : Set G'
K : Type u_3
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : IsAddRightInvariant μ'
f : G' → K
f_ℒ_1 : Integrable f
g : G' ⧸ Γ' → K
hg : AEStronglyMeasurable g (Measure.map mk (Measure.restrict μ' 𝓕'))
g_ℒ_infinity : essSup (fun x => ↑‖g x‖₊) (Measure.map mk (Measure.restrict μ' 𝓕')) ≠ ⊤
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (Measure.restrict μ' 𝓕'))
h𝓕 : IsAddFundamentalDomain (↥(AddSubgroup.op Γ')) 𝓕'
π : G' → G' ⧸ Γ' := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
H₁ : Integrable (g ∘ π * f)
H₂ : AEStronglyMeasurable (automorphize (g ∘ π * f)) (Measure.map mk (Measure.restrict μ' 𝓕'))
⊢ ∫ (x : G'), g (π x) * f x ∂μ' = ∫ (x : G' ⧸ Γ'), automorphize (g ∘ π * f) x ∂Measure.map mk (Measure.restrict μ' 𝓕')

Next tactic:
apply QuotientAddGroup.integral_eq_integral_automorphize h𝓕 H₁ H₂