ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IsAddFundamentalDomain.restrict_restrict
@[to_additive]
theorem restrict_restrict (h : IsFundamentalDomain G s μ) (g : G) (t : Set α) :
    (μ.restrict t).restrict (g • s) = μ.restrict (g • s ∩ t) 
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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (g : G) (t : Set α)
Goal: (μ.restrict t).restrict (g +ᵥ s) = μ.restrict ((g +ᵥ s) ∩ t)
-- MeasureTheory.IsFundamentalDomain.restrict_restrict
@[to_additive]
theorem restrict_restrict (h : IsFundamentalDomain G s μ) (g : G) (t : Set α) :
    (μ.restrict t).restrict (g • s) = μ.restrict (g • s ∩ t) 
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] (h : MeasureTheory.IsFundamentalDomain G s μ) (g : G) (t : Set α)
Goal: (μ.restrict t).restrict (g • s) = μ.restrict (g • s ∩ t)
-- MeasureTheory.IsFundamentalDomain.smul
@[to_additive]
theorem smul (h : IsFundamentalDomain G s μ) (g : G) : IsFundamentalDomain G (g • s) 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] (h : MeasureTheory.IsFundamentalDomain G s μ) (g : G)
Goal: MeasureTheory.IsFundamentalDomain G (g • s) μ
-- MeasureTheory.IsAddFundamentalDomain.vadd
@[to_additive]
theorem smul (h : IsFundamentalDomain G s μ) (g : G) : IsFundamentalDomain G (g • s) 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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (g : G)
Goal: MeasureTheory.IsAddFundamentalDomain G (g +ᵥ s) μ
-- MeasureTheory.IsAddFundamentalDomain.sum_restrict_of_ac
@[to_additive]
theorem sum_restrict_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
    (sum fun g : G => ν.restrict (g • s)) = ν 
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] {ν : MeasureTheory.Measure α} (h : MeasureTheory.IsAddFundamentalDomain G s μ) (hν : ν ≪ μ)
Goal: (MeasureTheory.Measure.sum fun g => ν.restrict (g +ᵥ s)) = ν
-- MeasureTheory.IsFundamentalDomain.sum_restrict_of_ac
@[to_additive]
theorem sum_restrict_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
    (sum fun g : G => ν.restrict (g • s)) = ν 
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] {ν : MeasureTheory.Measure α} (h : MeasureTheory.IsFundamentalDomain G s μ) (hν : ν ≪ μ)
Goal: (MeasureTheory.Measure.sum fun g => ν.restrict (g • s)) = ν
-- MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum_of_ac
@[to_additive]
theorem lintegral_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂ν = ∑' g : G, ∫⁻ x in g • s, f x ∂ν 
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] {ν : MeasureTheory.Measure α} (h : MeasureTheory.IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂ν = ∑' (g : G), ∫⁻ (x : α) in g • s, f x ∂ν
-- MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum_of_ac
@[to_additive]
theorem lintegral_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂ν = ∑' g : G, ∫⁻ x in g • s, f x ∂ν 
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] {ν : MeasureTheory.Measure α} (h : MeasureTheory.IsAddFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂ν = ∑' (g : G), ∫⁻ (x : α) in g +ᵥ s, f x ∂ν
-- MeasureTheory.IsAddFundamentalDomain.sum_restrict
@[to_additive]
theorem sum_restrict (h : IsFundamentalDomain G s μ) : (sum fun g : G => μ.restrict (g • s)) = 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] (h : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: (MeasureTheory.Measure.sum fun g => μ.restrict (g +ᵥ s)) = μ
-- MeasureTheory.IsFundamentalDomain.sum_restrict
@[to_additive]
theorem sum_restrict (h : IsFundamentalDomain G s μ) : (sum fun g : G => μ.restrict (g • s)) = 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] (h : MeasureTheory.IsFundamentalDomain G s μ)
Goal: (MeasureTheory.Measure.sum fun g => μ.restrict (g • s)) = μ
-- MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum
@[to_additive]
theorem lintegral_eq_tsum (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑' g : G, ∫⁻ x in g • s, f x ∂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] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g • s, f x ∂μ
-- MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum
@[to_additive]
theorem lintegral_eq_tsum (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑' g : G, ∫⁻ x in g • s, f x ∂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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g +ᵥ s, f x ∂μ
-- MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum'
@[to_additive]
theorem lintegral_eq_tsum' (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑' g : G, ∫⁻ x in s, f (g⁻¹ • x) ∂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] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in s, f (g⁻¹ • x) ∂μ
-- MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum'
@[to_additive]
theorem lintegral_eq_tsum' (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑' g : G, ∫⁻ x in s, f (g⁻¹ • x) ∂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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in s, f (-g +ᵥ x) ∂μ
-- MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum''
@[to_additive] lemma lintegral_eq_tsum'' (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑' g : G, ∫⁻ x in s, f (g • x) ∂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] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in s, f (g • x) ∂μ
-- MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum''
@[to_additive] lemma lintegral_eq_tsum'' (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑' g : G, ∫⁻ x in s, f (g • x) ∂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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in s, f (g +ᵥ x) ∂μ
-- MeasureTheory.IsFundamentalDomain.setLIntegral_eq_tsum
@[to_additive]
theorem setLIntegral_eq_tsum (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α) :
    ∫⁻ 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} [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] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α)
Goal: ∫⁻ (x : α) in t, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in t ∩ g • s, f x ∂μ
-- MeasureTheory.IsAddFundamentalDomain.setLIntegral_eq_tsum
@[to_additive]
theorem setLIntegral_eq_tsum (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α) :
    ∫⁻ 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} [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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α)
Goal: ∫⁻ (x : α) in t, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in t ∩ (g +ᵥ s), f x ∂μ
-- MeasureTheory.IsAddFundamentalDomain.setLIntegral_eq_tsum'
@[to_additive]
theorem setLIntegral_eq_tsum' (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α) :
    ∫⁻ 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} [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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α)
Goal: ∫⁻ (x : α) in t, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in (g +ᵥ t) ∩ s, f (-g +ᵥ x) ∂μ
-- MeasureTheory.IsFundamentalDomain.setLIntegral_eq_tsum'
@[to_additive]
theorem setLIntegral_eq_tsum' (h : IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α) :
    ∫⁻ 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} [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] (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α → ℝ≥0∞) (t : Set α)
Goal: ∫⁻ (x : α) in t, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ
-- MeasureTheory.IsFundamentalDomain.measure_eq_tsum_of_ac
@[to_additive]
theorem measure_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (t : Set α) :
    ν t = ∑' g : G, ν (t ∩ g • s) 
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] {ν : MeasureTheory.Measure α} (h : MeasureTheory.IsFundamentalDomain G s μ) (hν : ν ≪ μ) (t : Set α)
Goal: ν t = ∑' (g : G), ν (t ∩ g • s)
-- MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum_of_ac
@[to_additive]
theorem measure_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (t : Set α) :
    ν t = ∑' g : G, ν (t ∩ g • s) 
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] {ν : MeasureTheory.Measure α} (h : MeasureTheory.IsAddFundamentalDomain G s μ) (hν : ν ≪ μ) (t : Set α)
Goal: ν t = ∑' (g : G), ν (t ∩ (g +ᵥ s))
-- MeasureTheory.IsFundamentalDomain.measure_eq_tsum'
@[to_additive]
theorem measure_eq_tsum' (h : IsFundamentalDomain G s μ) (t : Set α) :
    μ t = ∑' g : G, μ (t ∩ g • s) 
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] (h : MeasureTheory.IsFundamentalDomain G s μ) (t : Set α)
Goal: μ t = ∑' (g : G), μ (t ∩ g • s)
-- MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum'
@[to_additive]
theorem measure_eq_tsum' (h : IsFundamentalDomain G s μ) (t : Set α) :
    μ t = ∑' g : G, μ (t ∩ g • s) 
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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (t : Set α)
Goal: μ t = ∑' (g : G), μ (t ∩ (g +ᵥ s))
-- MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum
@[to_additive]
theorem measure_eq_tsum (h : IsFundamentalDomain G s μ) (t : Set α) :
    μ t = ∑' g : G, μ (g • t ∩ s) 
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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (t : Set α)
Goal: μ t = ∑' (g : G), μ ((g +ᵥ t) ∩ s)
-- MeasureTheory.IsFundamentalDomain.measure_eq_tsum
@[to_additive]
theorem measure_eq_tsum (h : IsFundamentalDomain G s μ) (t : Set α) :
    μ t = ∑' g : G, μ (g • t ∩ s) 
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] (h : MeasureTheory.IsFundamentalDomain G s μ) (t : Set α)
Goal: μ t = ∑' (g : G), μ (g • t ∩ s)
-- MeasureTheory.IsFundamentalDomain.measure_zero_of_invariant
@[to_additive]
theorem measure_zero_of_invariant (h : IsFundamentalDomain G s μ) (t : Set α)
    (ht : ∀ g : G, g • t = t) (hts : μ (t ∩ s) = 0) : μ t = 0 
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] (h : MeasureTheory.IsFundamentalDomain G s μ) (t : Set α) (ht : ∀ (g : G), g • t = t) (hts : μ (t ∩ s) = 0)
Goal: μ t = 0
-- MeasureTheory.IsAddFundamentalDomain.measure_zero_of_invariant
@[to_additive]
theorem measure_zero_of_invariant (h : IsFundamentalDomain G s μ) (t : Set α)
    (ht : ∀ g : G, g • t = t) (hts : μ (t ∩ s) = 0) : μ t = 0 
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] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (t : Set α) (ht : ∀ (g : G), g +ᵥ t = t) (hts : μ (t ∩ s) = 0)
Goal: μ t = 0
-- MeasureTheory.IsAddFundamentalDomain.measure_eq_card_smul_of_vadd_ae_eq_self
@[to_additive measure_eq_card_smul_of_vadd_ae_eq_self "Given a measure space with an action of a
  finite additive group `G`, the measure of any `G`-invariant set is determined by the measure of
  its intersection with a fundamental domain for the action of `G`."]
theorem measure_eq_card_smul_of_smul_ae_eq_self [Finite G] (h : IsFundamentalDomain G s μ)
    (t : Set α) (ht : ∀ g : G, (g • t : Set α) =ᵐ[μ] t) : μ t = Nat.card G • μ (t ∩ s) 
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] [Finite G] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (t : Set α) (ht : ∀ (g : G), g +ᵥ t =ᵐ[μ] t)
Goal: μ t = Nat.card G • μ (t ∩ s)
-- MeasureTheory.IsFundamentalDomain.measure_eq_card_smul_of_smul_ae_eq_self
@[to_additive measure_eq_card_smul_of_vadd_ae_eq_self "Given a measure space with an action of a
  finite additive group `G`, the measure of any `G`-invariant set is determined by the measure of
  its intersection with a fundamental domain for the action of `G`."]
theorem measure_eq_card_smul_of_smul_ae_eq_self [Finite G] (h : IsFundamentalDomain G s μ)
    (t : Set α) (ht : ∀ g : G, (g • t : Set α) =ᵐ[μ] t) : μ t = Nat.card G • μ (t ∩ s) 
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] [Finite G] (h : MeasureTheory.IsFundamentalDomain G s μ) (t : Set α) (ht : ∀ (g : G), g • t =ᵐ[μ] t)
Goal: μ t = Nat.card G • μ (t ∩ s)
-- MeasureTheory.IsFundamentalDomain.setLIntegral_eq
@[to_additive]
protected theorem setLIntegral_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    (f : α → ℝ≥0∞) (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} [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.IsFundamentalDomain G t μ) (f : α → ℝ≥0∞) (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.setLIntegral_eq
@[to_additive]
protected theorem setLIntegral_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    (f : α → ℝ≥0∞) (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} [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.IsAddFundamentalDomain G t μ) (f : α → ℝ≥0∞) (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.IsAddFundamentalDomain.measure_set_eq
@[to_additive]
theorem measure_set_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ) {A : Set α}
    (hA₀ : MeasurableSet A) (hA : ∀ g : G, (fun x => g • x) ⁻¹' A = A) : μ (A ∩ s) = μ (A ∩ t) 
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.IsAddFundamentalDomain G t μ) {A : Set α} (hA₀ : MeasurableSet A) (hA : ∀ (g : G), (fun x => g +ᵥ x) ⁻¹' A = A)
Goal: μ (A ∩ s) = μ (A ∩ t)
-- MeasureTheory.IsFundamentalDomain.measure_set_eq
@[to_additive]
theorem measure_set_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ) {A : Set α}
    (hA₀ : MeasurableSet A) (hA : ∀ g : G, (fun x => g • x) ⁻¹' A = A) : μ (A ∩ s) = μ (A ∩ t) 
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.IsFundamentalDomain G t μ) {A : Set α} (hA₀ : MeasurableSet A) (hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A)
Goal: μ (A ∩ s) = μ (A ∩ t)
-- MeasureTheory.IsFundamentalDomain.measure_eq
@[to_additive "If `s` and `t` are two fundamental domains of the same action, then their measures
  are equal."]
protected theorem measure_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ) :
    μ s = μ t 
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.IsFundamentalDomain G t μ)
Goal: μ s = μ t
-- MeasureTheory.IsAddFundamentalDomain.measure_eq
@[to_additive "If `s` and `t` are two fundamental domains of the same action, then their measures
  are equal."]
protected theorem measure_eq (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ) :
    μ s = μ t 
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.IsAddFundamentalDomain G t μ)
Goal: μ s = μ t
-- MeasureTheory.IsAddFundamentalDomain.aEStronglyMeasurable_on_iff
@[to_additive]
protected theorem aEStronglyMeasurable_on_iff {β : Type*} [TopologicalSpace β]
    [PseudoMetrizableSpace β] (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    {f : α → β} (hf : ∀ (g : G) (x), f (g • x) = f x) :
    AEStronglyMeasurable f (μ.restrict s) ↔ AEStronglyMeasurable f (μ.restrict t) 
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] {β : Type u_6} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] (hs : MeasureTheory.IsAddFundamentalDomain G s μ) (ht : MeasureTheory.IsAddFundamentalDomain G t μ) {f : α → β} (hf : ∀ (g : G) (x : α), f (g +ᵥ x) = f x)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s) ↔ MeasureTheory.AEStronglyMeasurable f (μ.restrict t)
-- MeasureTheory.IsFundamentalDomain.aEStronglyMeasurable_on_iff
@[to_additive]
protected theorem aEStronglyMeasurable_on_iff {β : Type*} [TopologicalSpace β]
    [PseudoMetrizableSpace β] (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
    {f : α → β} (hf : ∀ (g : G) (x), f (g • x) = f x) :
    AEStronglyMeasurable f (μ.restrict s) ↔ AEStronglyMeasurable f (μ.restrict t) 
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] {β : Type u_6} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] (hs : MeasureTheory.IsFundamentalDomain G s μ) (ht : MeasureTheory.IsFundamentalDomain G t μ) {f : α → β} (hf : ∀ (g : G) (x : α), f (g • x) = f x)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s) ↔ MeasureTheory.AEStronglyMeasurable f (μ.restrict t)
-- MeasureTheory.IsAddFundamentalDomain.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} [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.HasFiniteIntegral f (μ.restrict s) ↔ MeasureTheory.HasFiniteIntegral f (μ.restrict t)