Theorem: MeasureTheory.IsFundamentalDomain.fundamentalInterior
File: Mathlib/MeasureTheory/Group/FundamentalDomain.lean

tactic: simp_rw [ae_iff, not_exists, ← mem_inv_smul_set_iff, setOf_forall, ← compl_setOf,
  setOf_mem_eq, ← compl_iUnion]
state_before: G : Type u_1
H : Type ?u.546154
α : Type u_2
β : Type ?u.546160
E : Type ?u.546163
inst✝⁶ : Countable G
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
inst✝ : SMulInvariantMeasure G α μ
⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ fundamentalInterior G s

tactic: have :
  ((⋃ g : G, g⁻¹ • s) \ ⋃ g : G, g⁻¹ • fundamentalFrontier G s) ⊆
    ⋃ g : G, g⁻¹ • fundamentalInterior G s := by
  simp_rw [diff_subset_iff, ← iUnion_union_distrib, ← smul_set_union (α := G) (β := α),
    fundamentalFrontier_union_fundamentalInterior]; rfl
state_before: G : Type u_1
H : Type ?u.546154
α : Type u_2
β : Type ?u.546160
E : Type ?u.546163
inst✝⁶ : Countable G
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
inst✝ : SMulInvariantMeasure G α μ
⊢ ↑↑μ ((⋃ (i : G), i⁻¹ • fundamentalInterior G s)ᶜ) = 0

tactic: refine' eq_bot_mono (μ.mono <| compl_subset_compl.2 this) _
state_before: G : Type u_1
H : Type ?u.546154
α : Type u_2
β : Type ?u.546160
E : Type ?u.546163
inst✝⁶ : Countable G
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
inst✝ : SMulInvariantMeasure G α μ
this : ((⋃ (g : G), g⁻¹ • s) \ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s
⊢ ↑↑μ ((⋃ (i : G), i⁻¹ • fundamentalInterior G s)ᶜ) = 0

tactic: simp only [iUnion_inv_smul, compl_sdiff, ENNReal.bot_eq_zero, himp_eq, sup_eq_union,
  @iUnion_smul_eq_setOf_exists _ _ _ _ s]
state_before: G : Type u_1
H : Type ?u.546154
α : Type u_2
β : Type ?u.546160
E : Type ?u.546163
inst✝⁶ : Countable G
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
inst✝ : SMulInvariantMeasure G α μ
this : ((⋃ (g : G), g⁻¹ • s) \ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s
⊢ ↑↑μ (((⋃ (g : G), g⁻¹ • s) \ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s)ᶜ) = ⊥

tactic: exact measure_union_null
  (measure_iUnion_null fun _ => measure_smul_null hs.measure_fundamentalFrontier _) hs.ae_covers
state_before: G : Type u_1
H : Type ?u.546154
α : Type u_2
β : Type ?u.546160
E : Type ?u.546163
inst✝⁶ : Countable G
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
inst✝ : SMulInvariantMeasure G α μ
this : ((⋃ (g : G), g⁻¹ • s) \ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s
⊢ ↑↑μ ((⋃ (g : G), g • fundamentalFrontier G s) ∪ {a | ∃ g, g • a ∈ s}ᶜ) = 0

tactic: simp_rw [diff_subset_iff, ← iUnion_union_distrib, ← smul_set_union (α := G) (β := α),
  fundamentalFrontier_union_fundamentalInterior]
state_before: G : Type u_1
H : Type ?u.546154
α : Type u_2
β : Type ?u.546160
E : Type ?u.546163
inst✝⁶ : Countable G
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
inst✝ : SMulInvariantMeasure G α μ
⊢ ((⋃ (g : G), g⁻¹ • s) \ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s

tactic: rfl
state_before: G : Type u_1
H : Type ?u.546154
α : Type u_2
β : Type ?u.546160
E : Type ?u.546163
inst✝⁶ : Countable G
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
inst✝ : SMulInvariantMeasure G α μ
⊢ (⋃ (g : G), g⁻¹ • s) ⊆ ⋃ (g : G), g⁻¹ • s