no goals
G : Type u_1
H : Type ?u.271746
α : Type u_2
β : Type ?u.271752
E : Type ?u.271755
inst✝¹⁰ : Group G
inst✝⁹ : Group H
inst✝⁸ : MulAction G α
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MulAction H β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
hs : IsFundamentalDomain G s
ht : IsFundamentalDomain G t
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A
this : (∫⁻ (x : α) in s, indicator A 1 x ∂μ) = ∫⁻ (x : α) in t, indicator A 1 x ∂μ
⊢ ↑↑μ (A ∩ s) = ↑↑μ (A ∩ t)
simpa [Measure.restrict_apply hA₀, lintegral_indicator _ hA₀] using this
G : Type u_1
H : Type ?u.271746
α : Type u_2
β : Type ?u.271752
E : Type ?u.271755
inst✝¹⁰ : Group G
inst✝⁹ : Group H
inst✝⁸ : MulAction G α
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MulAction H β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
hs : IsFundamentalDomain G s
ht : IsFundamentalDomain G t
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A
g : G
x : α
⊢ indicator A (fun x => 1) (g • x) = indicator A (fun x => 1) x
G : Type u_1
H : Type ?u.271746
α : Type u_2
β : Type ?u.271752
E : Type ?u.271755
inst✝¹⁰ : Group G
inst✝⁹ : Group H
inst✝⁸ : MulAction G α
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MulAction H β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
hs : IsFundamentalDomain G s
ht : IsFundamentalDomain G t
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A
⊢ (∫⁻ (x : α) in s, indicator A 1 x ∂μ) = ∫⁻ (x : α) in t, indicator A 1 x ∂μ
refine hs.set_lintegral_eq ht (Set.indicator A fun _ => 1) fun g x ↦ ?_
no goals
case h.e'_3.h.e'_4
G : Type u_1
H : Type ?u.271746
α : Type u_2
β : Type ?u.271752
E : Type ?u.271755
inst✝¹⁰ : Group G
inst✝⁹ : Group H
inst✝⁸ : MulAction G α
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MulAction H β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
hs : IsFundamentalDomain G s
ht : IsFundamentalDomain G t
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A
g : G
x : α
⊢ A = (fun x => g • x) ⁻¹' A
rw [hA g]