ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto
theorem tendsto_iff_forall_lintegral_tendsto {γ : Type*} {F : Filter γ}
    {μs : γ → ProbabilityMeasure Ω} {μ : ProbabilityMeasure Ω} :
    Tendsto μs F (𝓝 μ) ↔
      ∀ f : Ω →ᵇ ℝ≥0,
        Tendsto (fun i => ∫⁻ ω, f ω ∂(μs i : Measure Ω)) F (𝓝 (∫⁻ ω, f ω ∂(μ : Measure Ω))) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.ProbabilityMeasure Ω} {μ : MeasureTheory.ProbabilityMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ) ↔
  ∀ (f : Ω →ᵇ ℝ≥0), Filter.Tendsto (fun i => ∫⁻ (ω : Ω), ↑(f ω) ∂↑(μs i)) F (𝓝 (∫⁻ (ω : Ω), ↑(f ω) ∂↑μ))
-- MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_integral_tendsto
theorem tendsto_iff_forall_integral_tendsto {γ : Type*} {F : Filter γ}
    {μs : γ → ProbabilityMeasure Ω} {μ : ProbabilityMeasure Ω} :
    Tendsto μs F (𝓝 μ) ↔
      ∀ f : Ω →ᵇ ℝ,
        Tendsto (fun i => ∫ ω, f ω ∂(μs i : Measure Ω)) F (𝓝 (∫ ω, f ω ∂(μ : Measure Ω))) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.ProbabilityMeasure Ω} {μ : MeasureTheory.ProbabilityMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ), Filter.Tendsto (fun i => ∫ (ω : Ω), f ω ∂↑(μs i)) F (𝓝 (∫ (ω : Ω), f ω ∂↑μ))
-- MeasureTheory.FiniteMeasure.self_eq_mass_mul_normalize
@[simp]
theorem self_eq_mass_mul_normalize (s : Set Ω) : μ s = μ.mass * μ.normalize s 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω) (s : Set Ω)
Goal: μ s = μ.mass * μ.normalize s
-- MeasureTheory.FiniteMeasure.self_eq_mass_smul_normalize
theorem self_eq_mass_smul_normalize : μ = μ.mass • μ.normalize.toFiniteMeasure 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω)
Goal: μ = μ.mass • μ.normalize.toFiniteMeasure
-- MeasureTheory.FiniteMeasure.normalize_eq_of_nonzero
theorem normalize_eq_of_nonzero (nonzero : μ ≠ 0) (s : Set Ω) : μ.normalize s = μ.mass⁻¹ * μ s 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω) (nonzero : μ ≠ 0) (s : Set Ω)
Goal: μ.normalize s = μ.mass⁻¹ * μ s
-- MeasureTheory.FiniteMeasure.normalize_eq_inv_mass_smul_of_nonzero
theorem normalize_eq_inv_mass_smul_of_nonzero (nonzero : μ ≠ 0) :
    μ.normalize.toFiniteMeasure = μ.mass⁻¹ • μ 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω) (nonzero : μ ≠ 0)
Goal: μ.normalize.toFiniteMeasure = μ.mass⁻¹ • μ
-- MeasureTheory.FiniteMeasure.toMeasure_normalize_eq_of_nonzero
theorem toMeasure_normalize_eq_of_nonzero (nonzero : μ ≠ 0) :
    (μ.normalize : Measure Ω) = μ.mass⁻¹ • μ 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω) (nonzero : μ ≠ 0)
Goal: ↑μ.normalize = μ.mass⁻¹ • ↑μ
-- ProbabilityMeasure.toFiniteMeasure_normalize_eq_self
@[simp]
theorem _root_.ProbabilityMeasure.toFiniteMeasure_normalize_eq_self {m0 : MeasurableSpace Ω}
    (μ : ProbabilityMeasure Ω) : μ.toFiniteMeasure.normalize = μ 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: μ.toFiniteMeasure.normalize = μ
-- MeasureTheory.FiniteMeasure.average_eq_integral_normalize
theorem average_eq_integral_normalize {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (nonzero : μ ≠ 0) (f : Ω → E) :
    average (μ : Measure Ω) f = ∫ ω, f ω ∂(μ.normalize : Measure Ω) 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω) {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (nonzero : μ ≠ 0) (f : Ω → E)
Goal: MeasureTheory.average (↑μ) f = ∫ (ω : Ω), f ω ∂↑μ.normalize
-- MeasureTheory.FiniteMeasure.testAgainstNN_eq_mass_mul
theorem testAgainstNN_eq_mass_mul (f : Ω →ᵇ ℝ≥0) :
    μ.testAgainstNN f = μ.mass * μ.normalize.toFiniteMeasure.testAgainstNN f 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω) [TopologicalSpace Ω] (f : Ω →ᵇ ℝ≥0)
Goal: μ.testAgainstNN f = μ.mass * μ.normalize.toFiniteMeasure.testAgainstNN f
-- MeasureTheory.FiniteMeasure.normalize_testAgainstNN
theorem normalize_testAgainstNN (nonzero : μ ≠ 0) (f : Ω →ᵇ ℝ≥0) :
    μ.normalize.toFiniteMeasure.testAgainstNN f = μ.mass⁻¹ * μ.testAgainstNN f 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} (μ : MeasureTheory.FiniteMeasure Ω) [TopologicalSpace Ω] (nonzero : μ ≠ 0) (f : Ω →ᵇ ℝ≥0)
Goal: μ.normalize.toFiniteMeasure.testAgainstNN f = μ.mass⁻¹ * μ.testAgainstNN f
-- MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass
theorem tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass {γ : Type*}
    {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    (μs_lim : Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize))
    (mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass)) (f : Ω →ᵇ ℝ≥0) :
    Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f)) 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.FiniteMeasure Ω} [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} (μs_lim : Filter.Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize)) (mass_lim : Filter.Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass)) (f : Ω →ᵇ ℝ≥0)
Goal: Filter.Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f))
-- MeasureTheory.FiniteMeasure.tendsto_normalize_testAgainstNN_of_tendsto
theorem tendsto_normalize_testAgainstNN_of_tendsto {γ : Type*} {F : Filter γ}
    {μs : γ → FiniteMeasure Ω} (μs_lim : Tendsto μs F (𝓝 μ)) (nonzero : μ ≠ 0) (f : Ω →ᵇ ℝ≥0) :
    Tendsto (fun i => (μs i).normalize.toFiniteMeasure.testAgainstNN f) F
      (𝓝 (μ.normalize.toFiniteMeasure.testAgainstNN f)) 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.FiniteMeasure Ω} [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} (μs_lim : Filter.Tendsto μs F (𝓝 μ)) (nonzero : μ ≠ 0) (f : Ω →ᵇ ℝ≥0)
Goal: Filter.Tendsto (fun i => (μs i).normalize.toFiniteMeasure.testAgainstNN f) F
  (𝓝 (μ.normalize.toFiniteMeasure.testAgainstNN f))
-- MeasureTheory.FiniteMeasure.tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass
theorem tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass {γ : Type*} {F : Filter γ}
    {μs : γ → FiniteMeasure Ω} (μs_lim : Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize))
    (mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass)) : Tendsto μs F (𝓝 μ) 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.FiniteMeasure Ω} [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} (μs_lim : Filter.Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize)) (mass_lim : Filter.Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass))
Goal: Filter.Tendsto μs F (𝓝 μ)
-- MeasureTheory.FiniteMeasure.tendsto_normalize_of_tendsto
theorem tendsto_normalize_of_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    (μs_lim : Tendsto μs F (𝓝 μ)) (nonzero : μ ≠ 0) :
    Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize) 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.FiniteMeasure Ω} [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} (μs_lim : Filter.Tendsto μs F (𝓝 μ)) (nonzero : μ ≠ 0)
Goal: Filter.Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize)
-- MeasureTheory.FiniteMeasure.tendsto_normalize_iff_tendsto
theorem tendsto_normalize_iff_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    (nonzero : μ ≠ 0) :
    Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize) ∧
        Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass) ↔
      Tendsto μs F (𝓝 μ) 
Arguments: {Ω : Type u_1} [Nonempty Ω] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.FiniteMeasure Ω} [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} (nonzero : μ ≠ 0)
Goal: Filter.Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize) ∧ Filter.Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass) ↔
  Filter.Tendsto μs F (𝓝 μ)
-- MeasureTheory.ProbabilityMeasure.toMeasure_map
@[simp] lemma toMeasure_map (ν : ProbabilityMeasure Ω) {f : Ω → Ω'} (hf : AEMeasurable f ν) :
    (ν.map hf).toMeasure = ν.toMeasure.map f 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.ProbabilityMeasure Ω) {f : Ω → Ω'} (hf : AEMeasurable f ↑ν)
Goal: ↑(ν.map hf) = MeasureTheory.Measure.map f ↑ν
-- MeasureTheory.ProbabilityMeasure.map_apply'
lemma map_apply' (ν : ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)
    {A : Set Ω'} (A_mble : MeasurableSet A) :
    (ν.map f_aemble : Measure Ω') A = (ν : Measure Ω) (f ⁻¹' A) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ↑ν) {A : Set Ω'} (A_mble : MeasurableSet A)
Goal: ↑(ν.map f_aemble) A = ↑ν (f ⁻¹' A)
-- MeasureTheory.ProbabilityMeasure.map_apply_of_aemeasurable
lemma map_apply_of_aemeasurable (ν : ProbabilityMeasure Ω) {f : Ω → Ω'}
    (f_aemble : AEMeasurable f ν) {A : Set Ω'} (A_mble : MeasurableSet A) :
    (ν.map f_aemble) A = ν (f ⁻¹' A) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ↑ν) {A : Set Ω'} (A_mble : MeasurableSet A)
Goal: (ν.map f_aemble) A = ν (f ⁻¹' A)
-- MeasureTheory.ProbabilityMeasure.map_apply
lemma map_apply (ν : ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)
    {A : Set Ω'} (A_mble : MeasurableSet A) :
    (ν.map f_aemble) A = ν (f ⁻¹' A) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ↑ν) {A : Set Ω'} (A_mble : MeasurableSet A)
Goal: (ν.map f_aemble) A = ν (f ⁻¹' A)
-- MeasureTheory.ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous
lemma tendsto_map_of_tendsto_of_continuous {ι : Type*} {L : Filter ι}
    (νs : ι → ProbabilityMeasure Ω) (ν : ProbabilityMeasure Ω) (lim : Tendsto νs L (𝓝 ν))
    {f : Ω → Ω'} (f_cont : Continuous f) :
    Tendsto (fun i ↦ (νs i).map f_cont.measurable.aemeasurable) L
      (𝓝 (ν.map f_cont.measurable.aemeasurable)) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [TopologicalSpace Ω'] [BorelSpace Ω'] {ι : Type u_3} {L : Filter ι} (νs : ι → MeasureTheory.ProbabilityMeasure Ω) (ν : MeasureTheory.ProbabilityMeasure Ω) (lim : Filter.Tendsto νs L (𝓝 ν)) {f : Ω → Ω'} (f_cont : Continuous f)
Goal: Filter.Tendsto (fun i => (νs i).map ⋯) L (𝓝 (ν.map ⋯))
-- MeasureTheory.ProbabilityMeasure.continuous_map
lemma continuous_map {f : Ω → Ω'} (f_cont : Continuous f) :
    Continuous (fun ν ↦ ProbabilityMeasure.map ν f_cont.measurable.aemeasurable) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [TopologicalSpace Ω'] [BorelSpace Ω'] {f : Ω → Ω'} (f_cont : Continuous f)
Goal: Continuous fun ν => ν.map ⋯
-- AddAction.aestabilizer_coe
@[to_additive (attr := simps) "A.e. stabilizer of a set under an additive group action."]
def aestabilizer (s : Set α) : Subgroup G 
Arguments: (G : Type u_1) {α : Type u_2} [AddGroup G] [AddAction G α]
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}
-- MulAction.aestabilizer_coe
@[to_additive (attr := simps) "A.e. stabilizer of a set under an additive group action."]
def aestabilizer (s : Set α) : Subgroup G 
Arguments: (G : Type u_1) {α : Type u_2} [Group G] [MulAction G α]
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}
-- MulAction.mem_aestabilizer
@[to_additive (attr := simp)]
lemma mem_aestabilizer : g ∈ aestabilizer G μ s ↔ g • s =ᵐ[μ] s 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
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
-- AddAction.mem_aestabilizer
@[to_additive (attr := simp)]
lemma mem_aestabilizer : g ∈ aestabilizer G μ s ↔ g • s =ᵐ[μ] s 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
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
-- MulAction.stabilizer_le_aestabilizer
@[to_additive]
lemma stabilizer_le_aestabilizer (s : Set α) : stabilizer G s ≤ aestabilizer G μ s 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
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
-- AddAction.stabilizer_le_aestabilizer
@[to_additive]
lemma stabilizer_le_aestabilizer (s : Set α) : stabilizer G s ≤ aestabilizer G μ s 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
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
-- AddAction.aestabilizer_empty
@[to_additive (attr := simp)]
lemma aestabilizer_empty : aestabilizer G μ ∅ = ⊤ 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ],
  AddAction.aestabilizer G μ ∅ = ⊤
-- MulAction.aestabilizer_empty
@[to_additive (attr := simp)]
lemma aestabilizer_empty : aestabilizer G μ ∅ = ⊤ 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ],
  MulAction.aestabilizer G μ ∅ = ⊤
-- MulAction.aestabilizer_univ
@[to_additive (attr := simp)]
lemma aestabilizer_univ : aestabilizer G μ univ = ⊤ 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SMulInvariantMeasure G α μ],
  MulAction.aestabilizer G μ Set.univ = ⊤
-- AddAction.aestabilizer_univ
@[to_additive (attr := simp)]
lemma aestabilizer_univ : aestabilizer G μ univ = ⊤ 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.VAddInvariantMeasure G α μ],
  AddAction.aestabilizer G μ Set.univ = ⊤
-- AddAction.aestabilizer_congr
@[to_additive]
lemma aestabilizer_congr (h : s =ᵐ[μ] t) : aestabilizer G μ s = aestabilizer G μ t 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
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
-- MulAction.aestabilizer_congr
@[to_additive]
lemma aestabilizer_congr (h : s =ᵐ[μ] t) : aestabilizer G μ s = aestabilizer G μ t 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
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
-- MulAction.aestabilizer_of_aeconst
lemma aestabilizer_of_aeconst (hs : EventuallyConst s (ae μ)) : aestabilizer G μ s = ⊤ 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
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 = ⊤
-- MeasureTheory.smul_ae_eq_self_of_mem_zpowers
@[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 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
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
-- MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples
@[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 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
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
-- MeasureTheory.inv_smul_ae_eq_self
@[to_additive]
theorem inv_smul_ae_eq_self (hs : (x • s : Set α) =ᵐ[μ] s) : (x⁻¹ • s : Set α) =ᵐ[μ] s 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
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
-- MeasureTheory.neg_vadd_ae_eq_self
@[to_additive]
theorem inv_smul_ae_eq_self (hs : (x • s : Set α) =ᵐ[μ] s) : (x⁻¹ • s : Set α) =ᵐ[μ] s 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
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
-- MeasurableSet.const_smul
@[to_additive]
theorem MeasurableSet.const_smul {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace G]
    [MeasurableSpace α] [MeasurableSMul G α] {s : Set α} (hs : MeasurableSet s) (a : G) :
    MeasurableSet (a • s) 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] [MeasurableSpace G] [MeasurableSpace α] [MeasurableSMul G α] {s : Set α} (hs : MeasurableSet s) (a : G)
Goal: MeasurableSet (a • s)
-- MeasurableSet.const_vadd
@[to_additive]
theorem MeasurableSet.const_smul {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace G]
    [MeasurableSpace α] [MeasurableSMul G α] {s : Set α} (hs : MeasurableSet s) (a : G) :
    MeasurableSet (a • s) 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] [MeasurableSpace G] [MeasurableSpace α] [MeasurableVAdd G α] {s : Set α} (hs : MeasurableSet s) (a : G)
Goal: MeasurableSet (a +ᵥ s)
-- MeasurableSet.const_smul_of_ne_zero
theorem MeasurableSet.const_smul_of_ne_zero {G₀ α : Type*} [GroupWithZero G₀] [MulAction G₀ α]
    [MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α] {s : Set α}
    (hs : MeasurableSet s) {a : G₀} (ha : a ≠ 0) : MeasurableSet (a • s) 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [MulAction G₀ α] [MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α] {s : Set α} (hs : MeasurableSet s) {a : G₀} (ha : a ≠ 0)
Goal: MeasurableSet (a • s)
-- MeasurableSet.const_smul₀
theorem MeasurableSet.const_smul₀ {G₀ α : Type*} [GroupWithZero G₀] [Zero α]
    [MulActionWithZero G₀ α] [MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α]
    [MeasurableSingletonClass α] {s : Set α} (hs : MeasurableSet s) (a : G₀) :
    MeasurableSet (a • s) 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Zero α] [MulActionWithZero G₀ α] [MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α] [MeasurableSingletonClass α] {s : Set α} (hs : MeasurableSet s) (a : G₀)
Goal: MeasurableSet (a • s)
-- MeasureTheory.Integrable.comp_neg
@[to_additive]
theorem Integrable.comp_inv [IsInvInvariant μ] {f : G → F} (hf : Integrable f μ) :
    Integrable (fun t => f t⁻¹) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableNeg G] [μ.IsNegInvariant] {f : G → F} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun t => f (-t)) μ
-- MeasureTheory.Integrable.comp_inv
@[to_additive]
theorem Integrable.comp_inv [IsInvInvariant μ] {f : G → F} (hf : Integrable f μ) :
    Integrable (fun t => f t⁻¹) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [Group G] [MeasurableInv G] [μ.IsInvInvariant] {f : G → F} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun t => f t⁻¹) μ
-- MeasureTheory.integral_inv_eq_self
@[to_additive]
theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
    ∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] [Group G] [MeasurableInv G] (f : G → E) (μ : MeasureTheory.Measure G) [μ.IsInvInvariant]
Goal: ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_neg_eq_self
@[to_additive]
theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
    ∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] [AddGroup G] [MeasurableNeg G] (f : G → E) (μ : MeasureTheory.Measure G) [μ.IsNegInvariant]
Goal: ∫ (x : G), f (-x) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_add_left_eq_self
@[to_additive
      "Translating a function by left-addition does not change its integral with respect to a
      left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [μ.IsAddLeftInvariant] (f : G → E) (g : G)
Goal: ∫ (x : G), f (g + x) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_mul_left_eq_self
@[to_additive
      "Translating a function by left-addition does not change its integral with respect to a
      left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [μ.IsMulLeftInvariant] (f : G → E) (g : G)
Goal: ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_add_right_eq_self
@[to_additive
      "Translating a function by right-addition does not change its integral with respect to a
      right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [μ.IsAddRightInvariant] (f : G → E) (g : G)
Goal: ∫ (x : G), f (x + g) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_mul_right_eq_self
@[to_additive
      "Translating a function by right-addition does not change its integral with respect to a
      right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [μ.IsMulRightInvariant] (f : G → E) (g : G)
Goal: ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ