ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IsFundamentalDomain.hasFiniteIntegral_on_iff
@[to_additive]
protected theorem hasFiniteIntegral_on_iff (hs : IsFundamentalDomain G s μ)
    (ht : IsFundamentalDomain G t μ) {f : α → E} (hf : ∀ (g : G) (x), f (g • x) = f x) :
    HasFiniteIntegral f (μ.restrict s) ↔ HasFiniteIntegral f (μ.restrict t) 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsFundamentalDomain G s μ) (ht : MeasureTheory.IsFundamentalDomain G t μ) {f : α → E} (hf : ∀ (g : G) (x : α), f (g • x) = f x)
Goal: MeasureTheory.HasFiniteIntegral f (μ.restrict s) ↔ MeasureTheory.HasFiniteIntegral f (μ.restrict t)
-- MeasureTheory.IsFundamentalDomain.integrableOn_iff
@[to_additive]
protected theorem integrableOn_iff (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    {f : α → E} (hf : ∀ (g : G) (x), f (g • x) = f x) : IntegrableOn f s μ ↔ IntegrableOn f t μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsFundamentalDomain G s μ) (ht : MeasureTheory.IsFundamentalDomain G t μ) {f : α → E} (hf : ∀ (g : G) (x : α), f (g • x) = f x)
Goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn f t μ
-- MeasureTheory.IsAddFundamentalDomain.integrableOn_iff
@[to_additive]
protected theorem integrableOn_iff (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    {f : α → E} (hf : ∀ (g : G) (x), f (g • x) = f x) : IntegrableOn f s μ ↔ IntegrableOn f t μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsAddFundamentalDomain G s μ) (ht : MeasureTheory.IsAddFundamentalDomain G t μ) {f : α → E} (hf : ∀ (g : G) (x : α), f (g +ᵥ x) = f x)
Goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn f t μ
-- MeasureTheory.IsFundamentalDomain.integral_eq_tsum_of_ac
@[to_additive]
theorem integral_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → E)
    (hf : Integrable f ν) : ∫ x, f x ∂ν = ∑' g : G, ∫ x in g • s, f x ∂ν 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] {ν : MeasureTheory.Measure α} [NormedSpace ℝ E] (h : MeasureTheory.IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → E) (hf : MeasureTheory.Integrable f ν)
Goal: ∫ (x : α), f x ∂ν = ∑' (g : G), ∫ (x : α) in g • s, f x ∂ν
-- MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum_of_ac
@[to_additive]
theorem integral_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → E)
    (hf : Integrable f ν) : ∫ x, f x ∂ν = ∑' g : G, ∫ x in g • s, f x ∂ν 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] {ν : MeasureTheory.Measure α} [NormedSpace ℝ E] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → E) (hf : MeasureTheory.Integrable f ν)
Goal: ∫ (x : α), f x ∂ν = ∑' (g : G), ∫ (x : α) in g +ᵥ s, f x ∂ν
-- MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum
@[to_additive]
theorem integral_eq_tsum (h : IsFundamentalDomain G s μ) (f : α → E) (hf : Integrable f μ) :
    ∫ x, f x ∂μ = ∑' g : G, ∫ x in g • s, f x ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = ∑' (g : G), ∫ (x : α) in g +ᵥ s, f x ∂μ
-- MeasureTheory.IsFundamentalDomain.integral_eq_tsum
@[to_additive]
theorem integral_eq_tsum (h : IsFundamentalDomain G s μ) (f : α → E) (hf : Integrable f μ) :
    ∫ x, f x ∂μ = ∑' g : G, ∫ x in g • s, f x ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = ∑' (g : G), ∫ (x : α) in g • s, f x ∂μ
-- MeasureTheory.IsFundamentalDomain.integral_eq_tsum'
@[to_additive]
theorem integral_eq_tsum' (h : IsFundamentalDomain G s μ) (f : α → E) (hf : Integrable f μ) :
    ∫ x, f x ∂μ = ∑' g : G, ∫ x in s, f (g⁻¹ • x) ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = ∑' (g : G), ∫ (x : α) in s, f (g⁻¹ • x) ∂μ
-- MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum'
@[to_additive]
theorem integral_eq_tsum' (h : IsFundamentalDomain G s μ) (f : α → E) (hf : Integrable f μ) :
    ∫ x, f x ∂μ = ∑' g : G, ∫ x in s, f (g⁻¹ • x) ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = ∑' (g : G), ∫ (x : α) in s, f (-g +ᵥ x) ∂μ
-- MeasureTheory.IsFundamentalDomain.integral_eq_tsum''
@[to_additive] lemma integral_eq_tsum'' (h : IsFundamentalDomain G s μ)
    (f : α → E) (hf : Integrable f μ) : ∫ x, f x ∂μ = ∑' g : G, ∫ x in s, f (g • x) ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = ∑' (g : G), ∫ (x : α) in s, f (g • x) ∂μ
-- MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum''
@[to_additive] lemma integral_eq_tsum'' (h : IsFundamentalDomain G s μ)
    (f : α → E) (hf : Integrable f μ) : ∫ x, f x ∂μ = ∑' g : G, ∫ x in s, f (g • x) ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = ∑' (g : G), ∫ (x : α) in s, f (g +ᵥ x) ∂μ
-- MeasureTheory.IsAddFundamentalDomain.setIntegral_eq_tsum
@[to_additive]
theorem setIntegral_eq_tsum (h : IsFundamentalDomain G s μ) {f : α → E} {t : Set α}
    (hf : IntegrableOn f t μ) : ∫ x in t, f x ∂μ = ∑' g : G, ∫ x in t ∩ g • s, f x ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsAddFundamentalDomain G s μ) {f : α → E} {t : Set α} (hf : MeasureTheory.IntegrableOn f t μ)
Goal: ∫ (x : α) in t, f x ∂μ = ∑' (g : G), ∫ (x : α) in t ∩ (g +ᵥ s), f x ∂μ
-- MeasureTheory.IsFundamentalDomain.setIntegral_eq_tsum
@[to_additive]
theorem setIntegral_eq_tsum (h : IsFundamentalDomain G s μ) {f : α → E} {t : Set α}
    (hf : IntegrableOn f t μ) : ∫ x in t, f x ∂μ = ∑' g : G, ∫ x in t ∩ g • s, f x ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsFundamentalDomain G s μ) {f : α → E} {t : Set α} (hf : MeasureTheory.IntegrableOn f t μ)
Goal: ∫ (x : α) in t, f x ∂μ = ∑' (g : G), ∫ (x : α) in t ∩ g • s, f x ∂μ
-- MeasureTheory.IsAddFundamentalDomain.setIntegral_eq_tsum'
@[to_additive]
theorem setIntegral_eq_tsum' (h : IsFundamentalDomain G s μ) {f : α → E} {t : Set α}
    (hf : IntegrableOn f t μ) : ∫ x in t, f x ∂μ = ∑' g : G, ∫ x in g • t ∩ s, f (g⁻¹ • x) ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsAddFundamentalDomain G s μ) {f : α → E} {t : Set α} (hf : MeasureTheory.IntegrableOn f t μ)
Goal: ∫ (x : α) in t, f x ∂μ = ∑' (g : G), ∫ (x : α) in (g +ᵥ t) ∩ s, f (-g +ᵥ x) ∂μ
-- MeasureTheory.IsFundamentalDomain.setIntegral_eq_tsum'
@[to_additive]
theorem setIntegral_eq_tsum' (h : IsFundamentalDomain G s μ) {f : α → E} {t : Set α}
    (hf : IntegrableOn f t μ) : ∫ x in t, f x ∂μ = ∑' g : G, ∫ x in g • t ∩ s, f (g⁻¹ • x) ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (h : MeasureTheory.IsFundamentalDomain G s μ) {f : α → E} {t : Set α} (hf : MeasureTheory.IntegrableOn f t μ)
Goal: ∫ (x : α) in t, f x ∂μ = ∑' (g : G), ∫ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ
-- MeasureTheory.IsFundamentalDomain.setIntegral_eq
@[to_additive]
protected theorem setIntegral_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    {f : α → E} (hf : ∀ (g : G) (x), f (g • x) = f x) : ∫ x in s, f x ∂μ = ∫ x in t, f x ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [Group G] [MulAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (hs : MeasureTheory.IsFundamentalDomain G s μ) (ht : MeasureTheory.IsFundamentalDomain G t μ) {f : α → E} (hf : ∀ (g : G) (x : α), f (g • x) = f x)
Goal: ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in t, f x ∂μ
-- MeasureTheory.IsAddFundamentalDomain.setIntegral_eq
@[to_additive]
protected theorem setIntegral_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    {f : α → E} (hf : ∀ (g : G) (x), f (g • x) = f x) : ∫ x in s, f x ∂μ = ∫ x in t, f x ∂μ 
Arguments: {G : Type u_1} {α : Type u_3} {E : Type u_5} [AddGroup G] [AddAction G α] [MeasurableSpace α] [NormedAddCommGroup E] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] [NormedSpace ℝ E] (hs : MeasureTheory.IsAddFundamentalDomain G s μ) (ht : MeasureTheory.IsAddFundamentalDomain G t μ) {f : α → E} (hf : ∀ (g : G) (x : α), f (g +ᵥ x) = f x)
Goal: ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in t, f x ∂μ
-- MeasureTheory.IsFundamentalDomain.measure_le_of_pairwise_disjoint
@[to_additive "If the additive action of a countable group `G` admits an invariant measure `μ` with
  a fundamental domain `s`, then every null-measurable set `t` such that the sets `g +ᵥ t ∩ s` are
  pairwise a.e.-disjoint has measure at most `μ s`."]
theorem measure_le_of_pairwise_disjoint (hs : IsFundamentalDomain G s μ)
    (ht : NullMeasurableSet t μ) (hd : Pairwise (AEDisjoint μ on fun g : G => g • t ∩ s)) :
    μ t ≤ μ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsFundamentalDomain G s μ) (ht : MeasureTheory.NullMeasurableSet t μ) (hd : Pairwise (MeasureTheory.AEDisjoint μ on fun g => g • t ∩ s))
Goal: μ t ≤ μ s
-- MeasureTheory.IsAddFundamentalDomain.measure_le_of_pairwise_disjoint
@[to_additive "If the additive action of a countable group `G` admits an invariant measure `μ` with
  a fundamental domain `s`, then every null-measurable set `t` such that the sets `g +ᵥ t ∩ s` are
  pairwise a.e.-disjoint has measure at most `μ s`."]
theorem measure_le_of_pairwise_disjoint (hs : IsFundamentalDomain G s μ)
    (ht : NullMeasurableSet t μ) (hd : Pairwise (AEDisjoint μ on fun g : G => g • t ∩ s)) :
    μ t ≤ μ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsAddFundamentalDomain G s μ) (ht : MeasureTheory.NullMeasurableSet t μ) (hd : Pairwise (MeasureTheory.AEDisjoint μ on fun g => (g +ᵥ t) ∩ s))
Goal: μ t ≤ μ s
-- MeasureTheory.IsAddFundamentalDomain.exists_ne_zero_vadd_eq
@[to_additive "If the additive action of a countable group `G` admits an invariant measure `μ` with
  a fundamental domain `s`, then every null-measurable set `t` of measure strictly greater than
  `μ s` contains two points `x y` such that `g +ᵥ x = y` for some `g ≠ 0`."]
theorem exists_ne_one_smul_eq (hs : IsFundamentalDomain G s μ) (htm : NullMeasurableSet t μ)
    (ht : μ s < μ t) : ∃ x ∈ t, ∃ y ∈ t, ∃ g, g ≠ (1 : G) ∧ g • x = y 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsAddFundamentalDomain G s μ) (htm : MeasureTheory.NullMeasurableSet t μ) (ht : μ s < μ t)
Goal: ∃ x ∈ t, ∃ y ∈ t, ∃ g, g ≠ 0 ∧ g +ᵥ x = y
-- MeasureTheory.IsFundamentalDomain.exists_ne_one_smul_eq
@[to_additive "If the additive action of a countable group `G` admits an invariant measure `μ` with
  a fundamental domain `s`, then every null-measurable set `t` of measure strictly greater than
  `μ s` contains two points `x y` such that `g +ᵥ x = y` for some `g ≠ 0`."]
theorem exists_ne_one_smul_eq (hs : IsFundamentalDomain G s μ) (htm : NullMeasurableSet t μ)
    (ht : μ s < μ t) : ∃ x ∈ t, ∃ y ∈ t, ∃ g, g ≠ (1 : G) ∧ g • x = y 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s t : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsFundamentalDomain G s μ) (htm : MeasureTheory.NullMeasurableSet t μ) (ht : μ s < μ t)
Goal: ∃ x ∈ t, ∃ y ∈ t, ∃ g, g ≠ 1 ∧ g • x = y
-- MeasureTheory.IsAddFundamentalDomain.essSup_measure_restrict
@[to_additive "If `f` is invariant under the action of a countable additive group `G`, and `μ` is a
  `G`-invariant measure with a fundamental domain `s`, then the `essSup` of `f` restricted to `s`
  is the same as that of `f` on all of its domain."]
theorem essSup_measure_restrict (hs : IsFundamentalDomain G s μ) {f : α → ℝ≥0∞}
    (hf : ∀ γ : G, ∀ x : α, f (γ • x) = f x) : essSup f (μ.restrict s) = essSup f μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsAddFundamentalDomain G s μ) {f : α → ℝ≥0∞} (hf : ∀ (γ : G) (x : α), f (γ +ᵥ x) = f x)
Goal: essSup f (μ.restrict s) = essSup f μ
-- MeasureTheory.IsFundamentalDomain.essSup_measure_restrict
@[to_additive "If `f` is invariant under the action of a countable additive group `G`, and `μ` is a
  `G`-invariant measure with a fundamental domain `s`, then the `essSup` of `f` restricted to `s`
  is the same as that of `f` on all of its domain."]
theorem essSup_measure_restrict (hs : IsFundamentalDomain G s μ) {f : α → ℝ≥0∞}
    (hf : ∀ γ : G, ∀ x : α, f (γ • x) = f x) : essSup f (μ.restrict s) = essSup f μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G] (hs : MeasureTheory.IsFundamentalDomain G s μ) {f : α → ℝ≥0∞} (hf : ∀ (γ : G) (x : α), f (γ • x) = f x)
Goal: essSup f (μ.restrict s) = essSup f μ
-- MeasureTheory.mem_fundamentalFrontier
@[to_additive (attr := simp) MeasureTheory.mem_addFundamentalFrontier]
theorem mem_fundamentalFrontier :
    x ∈ fundamentalFrontier G s ↔ x ∈ s ∧ ∃ g : G, g ≠ 1 ∧ x ∈ g • s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {s : Set α} {x : α}
Goal: x ∈ MeasureTheory.fundamentalFrontier G s ↔ x ∈ s ∧ ∃ g, g ≠ 1 ∧ x ∈ g • s
-- MeasureTheory.mem_addFundamentalFrontier
@[to_additive (attr := simp) MeasureTheory.mem_addFundamentalFrontier]
theorem mem_fundamentalFrontier :
    x ∈ fundamentalFrontier G s ↔ x ∈ s ∧ ∃ g : G, g ≠ 1 ∧ x ∈ g • s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {s : Set α} {x : α}
Goal: x ∈ MeasureTheory.addFundamentalFrontier G s ↔ x ∈ s ∧ ∃ g, g ≠ 0 ∧ x ∈ g +ᵥ s
-- MeasureTheory.mem_addFundamentalInterior
@[to_additive (attr := simp) MeasureTheory.mem_addFundamentalInterior]
theorem mem_fundamentalInterior :
    x ∈ fundamentalInterior G s ↔ x ∈ s ∧ ∀ g : G, g ≠ 1 → x ∉ g • s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {s : Set α} {x : α}
Goal: x ∈ MeasureTheory.addFundamentalInterior G s ↔ x ∈ s ∧ ∀ (g : G), g ≠ 0 → x ∉ g +ᵥ s
-- MeasureTheory.mem_fundamentalInterior
@[to_additive (attr := simp) MeasureTheory.mem_addFundamentalInterior]
theorem mem_fundamentalInterior :
    x ∈ fundamentalInterior G s ↔ x ∈ s ∧ ∀ g : G, g ≠ 1 → x ∉ g • s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {s : Set α} {x : α}
Goal: x ∈ MeasureTheory.fundamentalInterior G s ↔ x ∈ s ∧ ∀ (g : G), g ≠ 1 → x ∉ g • s
-- MeasureTheory.addFundamentalFrontier_subset
@[to_additive MeasureTheory.addFundamentalFrontier_subset]
theorem fundamentalFrontier_subset : fundamentalFrontier G s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {s : Set α}
Goal: MeasureTheory.addFundamentalFrontier G s ⊆ s
-- MeasureTheory.fundamentalFrontier_subset
@[to_additive MeasureTheory.addFundamentalFrontier_subset]
theorem fundamentalFrontier_subset : fundamentalFrontier G s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {s : Set α}
Goal: MeasureTheory.fundamentalFrontier G s ⊆ s
-- MeasureTheory.fundamentalInterior_subset
@[to_additive MeasureTheory.addFundamentalInterior_subset]
theorem fundamentalInterior_subset : fundamentalInterior G s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {s : Set α}
Goal: MeasureTheory.fundamentalInterior G s ⊆ s
-- MeasureTheory.addFundamentalInterior_subset
@[to_additive MeasureTheory.addFundamentalInterior_subset]
theorem fundamentalInterior_subset : fundamentalInterior G s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {s : Set α}
Goal: MeasureTheory.addFundamentalInterior G s ⊆ s
-- MeasureTheory.disjoint_fundamentalInterior_fundamentalFrontier
@[to_additive MeasureTheory.disjoint_addFundamentalInterior_addFundamentalFrontier]
theorem disjoint_fundamentalInterior_fundamentalFrontier :
    Disjoint (fundamentalInterior G s) (fundamentalFrontier G s) 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α)
Goal: Disjoint (MeasureTheory.fundamentalInterior G s) (MeasureTheory.fundamentalFrontier G s)
-- MeasureTheory.disjoint_addFundamentalInterior_addFundamentalFrontier
@[to_additive MeasureTheory.disjoint_addFundamentalInterior_addFundamentalFrontier]
theorem disjoint_fundamentalInterior_fundamentalFrontier :
    Disjoint (fundamentalInterior G s) (fundamentalFrontier G s) 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α)
Goal: Disjoint (MeasureTheory.addFundamentalInterior G s) (MeasureTheory.addFundamentalFrontier G s)
-- MeasureTheory.addFundamentalInterior_union_addFundamentalFrontier
@[to_additive (attr := simp) MeasureTheory.addFundamentalInterior_union_addFundamentalFrontier]
theorem fundamentalInterior_union_fundamentalFrontier :
    fundamentalInterior G s ∪ fundamentalFrontier G s = s 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α)
Goal: MeasureTheory.addFundamentalInterior G s ∪ MeasureTheory.addFundamentalFrontier G s = s
-- MeasureTheory.fundamentalInterior_union_fundamentalFrontier
@[to_additive (attr := simp) MeasureTheory.addFundamentalInterior_union_addFundamentalFrontier]
theorem fundamentalInterior_union_fundamentalFrontier :
    fundamentalInterior G s ∪ fundamentalFrontier G s = s 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α)
Goal: MeasureTheory.fundamentalInterior G s ∪ MeasureTheory.fundamentalFrontier G s = s
-- MeasureTheory.addFundamentalFrontier_union_addFundamentalInterior
@[to_additive (attr := simp) MeasureTheory.addFundamentalFrontier_union_addFundamentalInterior]
theorem fundamentalFrontier_union_fundamentalInterior :
    fundamentalFrontier G s ∪ fundamentalInterior G s = s 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α)
Goal: MeasureTheory.addFundamentalFrontier G s ∪ MeasureTheory.addFundamentalInterior G s = s
-- MeasureTheory.fundamentalFrontier_union_fundamentalInterior
@[to_additive (attr := simp) MeasureTheory.addFundamentalFrontier_union_addFundamentalInterior]
theorem fundamentalFrontier_union_fundamentalInterior :
    fundamentalFrontier G s ∪ fundamentalInterior G s = s 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α)
Goal: MeasureTheory.fundamentalFrontier G s ∪ MeasureTheory.fundamentalInterior G s = s
-- MeasureTheory.sdiff_addFundamentalInterior
@[to_additive (attr := simp) MeasureTheory.sdiff_addFundamentalInterior]
theorem sdiff_fundamentalInterior : s \ fundamentalInterior G s = fundamentalFrontier G s 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α)
Goal: s \ MeasureTheory.addFundamentalInterior G s = MeasureTheory.addFundamentalFrontier G s
-- MeasureTheory.sdiff_fundamentalInterior
@[to_additive (attr := simp) MeasureTheory.sdiff_addFundamentalInterior]
theorem sdiff_fundamentalInterior : s \ fundamentalInterior G s = fundamentalFrontier G s 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α)
Goal: s \ MeasureTheory.fundamentalInterior G s = MeasureTheory.fundamentalFrontier G s
-- MeasureTheory.sdiff_fundamentalFrontier
@[to_additive (attr := simp) MeasureTheory.sdiff_addFundamentalFrontier]
theorem sdiff_fundamentalFrontier : s \ fundamentalFrontier G s = fundamentalInterior G s 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α)
Goal: s \ MeasureTheory.fundamentalFrontier G s = MeasureTheory.fundamentalInterior G s