File: Mathlib/MeasureTheory/Group/AEStabilizer.lean

full_name: AddAction.aestabilizer_coe
code: /-- A.e. stabilizer of a set under a group action. -/
@[to_additive (attr := simps) "A.e. stabilizer of a set under an additive group action."]
def aestabilizer (s : Set α) : Subgroup G 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) [inst : MeasureTheory.VAddInvariantMeasure G α μ] (s : Set α),
  ↑(AddAction.aestabilizer G μ s) = {g | g +ᵥ s =ᵐ[μ] s}

full_name: MulAction.aestabilizer
code: /-- A.e. stabilizer of a set under a group action. -/
@[to_additive (attr := simps) "A.e. stabilizer of a set under an additive group action."]
def aestabilizer (s : Set α) : Subgroup G 
goal: {x : MeasurableSpace α} →
  (μ : MeasureTheory.Measure α) → [inst : MeasureTheory.SMulInvariantMeasure G α μ] → Set α → Subgroup G

full_name: MulAction.aestabilizer_coe
code: /-- A.e. stabilizer of a set under a group action. -/
@[to_additive (attr := simps) "A.e. stabilizer of a set under an additive group action."]
def aestabilizer (s : Set α) : Subgroup G 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) [inst : MeasureTheory.SMulInvariantMeasure G α μ] (s : Set α),
  ↑(MulAction.aestabilizer G μ s) = {g | g • s =ᵐ[μ] s}

full_name: AddAction.aestabilizer
code: /-- A.e. stabilizer of a set under a group action. -/
@[to_additive (attr := simps) "A.e. stabilizer of a set under an additive group action."]
def aestabilizer (s : Set α) : Subgroup G 
goal: {x : MeasurableSpace α} →
  (μ : MeasureTheory.Measure α) → [inst : MeasureTheory.VAddInvariantMeasure G α μ] → Set α → AddSubgroup G

full_name: MulAction.mem_aestabilizer
code: @[to_additive (attr := simp)]
lemma mem_aestabilizer : g ∈ aestabilizer G μ s ↔ g • s =ᵐ[μ] s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ] {g : G}
  {s : Set α}, g ∈ MulAction.aestabilizer G μ s ↔ g • s =ᵐ[μ] s

full_name: AddAction.mem_aestabilizer
code: @[to_additive (attr := simp)]
lemma mem_aestabilizer : g ∈ aestabilizer G μ s ↔ g • s =ᵐ[μ] s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ] {g : G}
  {s : Set α}, g ∈ AddAction.aestabilizer G μ s ↔ g +ᵥ s =ᵐ[μ] s

full_name: MulAction.stabilizer_le_aestabilizer
code: @[to_additive]
lemma stabilizer_le_aestabilizer (s : Set α) : stabilizer G s ≤ aestabilizer G μ s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ] (s : Set α),
  MulAction.stabilizer G s ≤ MulAction.aestabilizer G μ s

full_name: AddAction.stabilizer_le_aestabilizer
code: @[to_additive]
lemma stabilizer_le_aestabilizer (s : Set α) : stabilizer G s ≤ aestabilizer G μ s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ] (s : Set α),
  AddAction.stabilizer G s ≤ AddAction.aestabilizer G μ s

full_name: AddAction.aestabilizer_empty
code: @[to_additive (attr := simp)]
lemma aestabilizer_empty : aestabilizer G μ ∅ = ⊤ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ],
  AddAction.aestabilizer G μ ∅ = ⊤

full_name: MulAction.aestabilizer_empty
code: @[to_additive (attr := simp)]
lemma aestabilizer_empty : aestabilizer G μ ∅ = ⊤ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ],
  MulAction.aestabilizer G μ ∅ = ⊤

full_name: MulAction.aestabilizer_univ
code: @[to_additive (attr := simp)]
lemma aestabilizer_univ : aestabilizer G μ univ = ⊤ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ],
  MulAction.aestabilizer G μ Set.univ = ⊤

full_name: AddAction.aestabilizer_univ
code: @[to_additive (attr := simp)]
lemma aestabilizer_univ : aestabilizer G μ univ = ⊤ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ],
  AddAction.aestabilizer G μ Set.univ = ⊤

full_name: AddAction.aestabilizer_congr
code: @[to_additive]
lemma aestabilizer_congr (h : s =ᵐ[μ] t) : aestabilizer G μ s = aestabilizer G μ t 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ] {s t : Set α},
  s =ᵐ[μ] t → AddAction.aestabilizer G μ s = AddAction.aestabilizer G μ t

full_name: MulAction.aestabilizer_congr
code: @[to_additive]
lemma aestabilizer_congr (h : s =ᵐ[μ] t) : aestabilizer G μ s = aestabilizer G μ t 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ] {s t : Set α},
  s =ᵐ[μ] t → MulAction.aestabilizer G μ s = MulAction.aestabilizer G μ t

full_name: MulAction.aestabilizer_of_aeconst
code: lemma aestabilizer_of_aeconst (hs : EventuallyConst s (ae μ)) : aestabilizer G μ s = ⊤ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ] {s : Set α},
  Filter.EventuallyConst s (MeasureTheory.ae μ) → MulAction.aestabilizer G μ s = ⊤

full_name: MeasureTheory.smul_ae_eq_self_of_mem_zpowers
code: @[to_additive]
theorem smul_ae_eq_self_of_mem_zpowers (hs : (x • s : Set α) =ᵐ[μ] s)
    (hy : y ∈ Subgroup.zpowers x) : (y • s : Set α) =ᵐ[μ] s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ] {x_1 y : G}
  {s : Set α}, x_1 • s =ᵐ[μ] s → y ∈ Subgroup.zpowers x_1 → y • s =ᵐ[μ] s

full_name: MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples
code: @[to_additive]
theorem smul_ae_eq_self_of_mem_zpowers (hs : (x • s : Set α) =ᵐ[μ] s)
    (hy : y ∈ Subgroup.zpowers x) : (y • s : Set α) =ᵐ[μ] s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ] {x_1 y : G}
  {s : Set α}, x_1 +ᵥ s =ᵐ[μ] s → y ∈ AddSubgroup.zmultiples x_1 → y +ᵥ s =ᵐ[μ] s

full_name: MeasureTheory.inv_smul_ae_eq_self
code: @[to_additive]
theorem inv_smul_ae_eq_self (hs : (x • s : Set α) =ᵐ[μ] s) : (x⁻¹ • s : Set α) =ᵐ[μ] s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ] {x_1 : G}
  {s : Set α}, x_1 • s =ᵐ[μ] s → x_1⁻¹ • s =ᵐ[μ] s

full_name: MeasureTheory.neg_vadd_ae_eq_self
code: @[to_additive]
theorem inv_smul_ae_eq_self (hs : (x • s : Set α) =ᵐ[μ] s) : (x⁻¹ • s : Set α) =ᵐ[μ] s 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ] {x_1 : G}
  {s : Set α}, x_1 +ᵥ s =ᵐ[μ] s → -x_1 +ᵥ s =ᵐ[μ] s