ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Ioo_ae_eq_Icc'
theorem Ioo_ae_eq_Icc' (ha : μ {a} = 0) (hb : μ {b} = 0) : Ioo a b =ᵐ[μ] Icc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0) (hb : μ {b} = 0)
Goal: Set.Ioo a b =ᵐ[μ] Set.Icc a b
-- MeasureTheory.Ico_ae_eq_Icc'
theorem Ico_ae_eq_Icc' (hb : μ {b} = 0) : Ico a b =ᵐ[μ] Icc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a b : α} (hb : μ {b} = 0)
Goal: Set.Ico a b =ᵐ[μ] Set.Icc a b
-- MeasureTheory.Ico_ae_eq_Ioc'
theorem Ico_ae_eq_Ioc' (ha : μ {a} = 0) (hb : μ {b} = 0) : Ico a b =ᵐ[μ] Ioc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0) (hb : μ {b} = 0)
Goal: Set.Ico a b =ᵐ[μ] Set.Ioc a b
-- MeasurableEmbedding.map_apply
nonrec theorem map_apply (hf : MeasurableEmbedding f) (μ : Measure α) (s : Set β) :
    μ.map f s = μ (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure α) (s : Set β)
Goal: (MeasureTheory.Measure.map f μ) s = μ (f ⁻¹' s)
-- MeasurableEmbedding.comap_add
lemma comap_add (hf : MeasurableEmbedding f) (μ ν : Measure β) :
    (μ + ν).comap f = μ.comap f + ν.comap f 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ ν : MeasureTheory.Measure β)
Goal: MeasureTheory.Measure.comap f (μ + ν) = MeasureTheory.Measure.comap f μ + MeasureTheory.Measure.comap f ν
-- MeasurableEmbedding.absolutelyContinuous_map
lemma absolutelyContinuous_map (hf : MeasurableEmbedding f) (hμν : μ ≪ ν) :
    μ.map f ≪ ν.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} {μ ν : MeasureTheory.Measure α} (hf : MeasurableEmbedding f) (hμν : μ.AbsolutelyContinuous ν)
Goal: (MeasureTheory.Measure.map f μ).AbsolutelyContinuous (MeasureTheory.Measure.map f ν)
-- MeasurableEquiv.map_apply
protected theorem map_apply (f : α ≃ᵐ β) (s : Set β) : μ.map f s = μ (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} (f : α ≃ᵐ β) (s : Set β)
Goal: (MeasureTheory.Measure.map (⇑f) μ) s = μ (⇑f ⁻¹' s)
-- MeasurableEquiv.comap_symm
lemma comap_symm (e : α ≃ᵐ β) : μ.comap e.symm = μ.map e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} (e : α ≃ᵐ β)
Goal: MeasureTheory.Measure.comap (⇑e.symm) μ = MeasureTheory.Measure.map (⇑e) μ
-- MeasurableEquiv.map_symm
lemma map_symm (e : β ≃ᵐ α) : μ.map e.symm = μ.comap e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} (e : β ≃ᵐ α)
Goal: MeasureTheory.Measure.map (⇑e.symm) μ = MeasureTheory.Measure.comap (⇑e) μ
-- MeasurableEquiv.map_symm_map
@[simp]
theorem map_symm_map (e : α ≃ᵐ β) : (μ.map e).map e.symm = μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} (e : α ≃ᵐ β)
Goal: MeasureTheory.Measure.map (⇑e.symm) (MeasureTheory.Measure.map (⇑e) μ) = μ
-- MeasurableEquiv.map_map_symm
@[simp]
theorem map_map_symm (e : α ≃ᵐ β) : (ν.map e.symm).map e = ν 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} (e : α ≃ᵐ β)
Goal: MeasureTheory.Measure.map (⇑e) (MeasureTheory.Measure.map (⇑e.symm) ν) = ν
-- MeasurableEquiv.map_measurableEquiv_injective
theorem map_measurableEquiv_injective (e : α ≃ᵐ β) : Injective (Measure.map e) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: Function.Injective (MeasureTheory.Measure.map ⇑e)
-- MeasurableEquiv.map_apply_eq_iff_map_symm_apply_eq
theorem map_apply_eq_iff_map_symm_apply_eq (e : α ≃ᵐ β) : μ.map e = ν ↔ ν.map e.symm = μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} (e : α ≃ᵐ β)
Goal: MeasureTheory.Measure.map (⇑e) μ = ν ↔ MeasureTheory.Measure.map (⇑e.symm) ν = μ
-- MeasurableEquiv.map_ae
theorem map_ae (f : α ≃ᵐ β) (μ : Measure α) : Filter.map f (ae μ) = ae (map f μ) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (f : α ≃ᵐ β) (μ : MeasureTheory.Measure α)
Goal: Filter.map (⇑f) (MeasureTheory.ae μ) = MeasureTheory.ae (MeasureTheory.Measure.map (⇑f) μ)
-- MeasurableEquiv.quasiMeasurePreserving_symm
theorem quasiMeasurePreserving_symm (μ : Measure α) (e : α ≃ᵐ β) :
    QuasiMeasurePreserving e.symm (map e μ) μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (μ : MeasureTheory.Measure α) (e : α ≃ᵐ β)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (⇑e.symm) (MeasureTheory.Measure.map (⇑e) μ) μ
-- MeasureTheory.OuterMeasure.toMeasure_zero
theorem OuterMeasure.toMeasure_zero [MeasurableSpace α] :
    (0 : OuterMeasure α).toMeasure (le_top.trans OuterMeasure.zero_caratheodory.symm.le) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.OuterMeasure.toMeasure 0 ⋯ = 0
-- Polynomial.ofReal_eval
theorem ofReal_eval (p : ℝ[X]) (x : ℝ) : (↑(p.eval x) : K) = aeval (↑x) p 
Arguments: {K : Type u_1} [RCLike K] (p : Polynomial ℝ) (x : ℝ)
Goal: ↑(Polynomial.eval x p) = (Polynomial.aeval ↑x) p
-- FiniteDimensional.proper_rclike
theorem proper_rclike [FiniteDimensional K E] : ProperSpace E 
Arguments: (K : Type u_1) (E : Type u_2) [RCLike K] [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]
Goal: ProperSpace E
-- RCLike.reCLM_norm
@[simp, rclike_simps]
theorem reCLM_norm : ‖(reCLM : K →L[ℝ] ℝ)‖ = 1 
Arguments: {K : Type u_1} [RCLike K]
Goal: ‖RCLike.reCLM‖ = 1
-- RCLike.conjCLE_norm
@[simp, rclike_simps]
theorem conjCLE_norm : ‖(@conjCLE K _ : K →L[ℝ] K)‖ = 1 
Arguments: {K : Type u_1} [RCLike K]
Goal: ‖↑RCLike.conjCLE‖ = 1
-- RCLike.ofRealCLM_norm
@[simp, rclike_simps]
theorem ofRealCLM_norm : ‖(ofRealCLM : ℝ →L[ℝ] K)‖ = 1 
Arguments: {K : Type u_1} [RCLike K]
Goal: ‖RCLike.ofRealCLM‖ = 1
-- Polynomial.aeval_conj
lemma aeval_conj (p : ℝ[X]) (z : K) : aeval (conj z) p = conj (aeval z p) 
Arguments: {K : Type u_1} [RCLike K] (p : Polynomial ℝ) (z : K)
Goal: (Polynomial.aeval ((starRingEnd K) z)) p = (starRingEnd K) ((Polynomial.aeval z) p)
-- Polynomial.aeval_ofReal
lemma aeval_ofReal (p : ℝ[X]) (x : ℝ) : aeval (RCLike.ofReal x : K) p = eval x p 
Arguments: {K : Type u_1} [RCLike K] (p : Polynomial ℝ) (x : ℝ)
Goal: (Polynomial.aeval ↑x) p = ↑(Polynomial.eval x p)
-- IsCompact.measure_eq_biInf_integral_hasCompactSupport
lemma IsCompact.measure_eq_biInf_integral_hasCompactSupport
    {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
    {k : Set X} (hk : IsCompact k)
    (μ : Measure X) [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ]
    [LocallyCompactSpace X] [RegularSpace X] :
    μ k = ⨅ (f : X → ℝ) (_ : Continuous f) (_ : HasCompactSupport f) (_ : EqOn f 1 k)
      (_ : 0 ≤ f), ENNReal.ofReal (∫ x, f x ∂μ) 
Arguments: {X : Type u_1} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] {k : Set X} (hk : IsCompact k) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] [LocallyCompactSpace X] [RegularSpace X]
Goal: μ k =
  ⨅ f,
    ⨅ (_ : Continuous f),
      ⨅ (_ : HasCompactSupport f), ⨅ (_ : Set.EqOn f 1 k), ⨅ (_ : 0 ≤ f), ENNReal.ofReal (∫ (x : X), f x ∂μ)
-- MeasureTheory.continuous_integral_apply_neg_add
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) 
Arguments: {G : Type u_1} [TopologicalSpace G] [LocallyCompactSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {g : G → E} (hg : Continuous g) (h'g : HasCompactSupport g)
Goal: Continuous fun x => ∫ (y : G), g (-y + x) ∂μ
-- MeasureTheory.continuous_integral_apply_inv_mul
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) 
Arguments: {G : Type u_1} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {g : G → E} (hg : Continuous g) (h'g : HasCompactSupport g)
Goal: Continuous fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ
-- MeasureTheory.Measure.integral_isMulLeftInvariant_isMulRightInvariant_combo
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] {μ ν : MeasureTheory.Measure G} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [MeasureTheory.IsFiniteMeasureOnCompacts ν] [μ.IsMulLeftInvariant] [ν.IsMulRightInvariant] [ν.IsOpenPosMeasure] {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0)
Goal: ∫ (x : G), f x ∂μ = (∫ (y : G), f y * (∫ (z : G), g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ (x : G), g x ∂μ
-- MeasureTheory.Measure.integral_isAddLeftInvariant_isAddRightInvariant_combo
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] {μ ν : MeasureTheory.Measure G} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [MeasureTheory.IsFiniteMeasureOnCompacts ν] [μ.IsAddLeftInvariant] [ν.IsAddRightInvariant] [ν.IsOpenPosMeasure] {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0)
Goal: ∫ (x : G), f x ∂μ = (∫ (y : G), f y * (∫ (z : G), g (-z + y) ∂ν)⁻¹ ∂ν) * ∫ (x : G), g x ∂μ
-- MeasureTheory.Measure.exists_integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
@[to_additive exists_integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport]
lemma exists_integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport (μ' μ : Measure G)
    [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
    ∃ (c : ℝ≥0), ∀ (f : G → ℝ), Continuous f → HasCompactSupport f →
      ∫ x, f x ∂μ' = ∫ x, f x ∂(c • μ) 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant]
Goal: ∃ c, ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
-- MeasureTheory.Measure.exists_integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport
@[to_additive exists_integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport]
lemma exists_integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport (μ' μ : Measure G)
    [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
    ∃ (c : ℝ≥0), ∀ (f : G → ℝ), Continuous f → HasCompactSupport f →
      ∫ x, f x ∂μ' = ∫ x, f x ∂(c • μ) 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant]
Goal: ∃ c, ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
-- MeasureTheory.Measure.integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
@[to_additive integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport
"Two left invariant measures integrate in the same way continuous compactly supported functions,
up to the scalar `addHaarScalarFactor μ' μ`. See also
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which gives the same result for compact
sets, and `measure_isAddHaarMeasure_eq_smul_of_isOpen` for open sets."]
theorem integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) :
    ∫ x, f x ∂μ' = ∫ x, f x ∂(haarScalarFactor μ' μ • μ) 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂μ'.haarScalarFactor μ • μ
-- MeasureTheory.Measure.integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport
@[to_additive integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport
"Two left invariant measures integrate in the same way continuous compactly supported functions,
up to the scalar `addHaarScalarFactor μ' μ`. See also
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which gives the same result for compact
sets, and `measure_isAddHaarMeasure_eq_smul_of_isOpen` for open sets."]
theorem integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) :
    ∫ x, f x ∂μ' = ∫ x, f x ∂(haarScalarFactor μ' μ • μ) 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂μ'.addHaarScalarFactor μ • μ
-- MeasureTheory.Measure.addHaarScalarFactor_eq_integral_div
@[to_additive addHaarScalarFactor_eq_integral_div]
lemma haarScalarFactor_eq_integral_div (μ' μ : Measure G) [IsHaarMeasure μ]
    [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] {f : G → ℝ} (hf : Continuous f)
    (h'f : HasCompactSupport f) (int_nonzero : ∫ x, f x ∂μ ≠ 0) :
    haarScalarFactor μ' μ = (∫ x, f x ∂μ') / ∫ x, f x ∂μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) (int_nonzero : ∫ (x : G), f x ∂μ ≠ 0)
Goal: ↑(μ'.addHaarScalarFactor μ) = (∫ (x : G), f x ∂μ') / ∫ (x : G), f x ∂μ
-- MeasureTheory.Measure.haarScalarFactor_eq_integral_div
@[to_additive addHaarScalarFactor_eq_integral_div]
lemma haarScalarFactor_eq_integral_div (μ' μ : Measure G) [IsHaarMeasure μ]
    [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] {f : G → ℝ} (hf : Continuous f)
    (h'f : HasCompactSupport f) (int_nonzero : ∫ x, f x ∂μ ≠ 0) :
    haarScalarFactor μ' μ = (∫ x, f x ∂μ') / ∫ x, f x ∂μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) (int_nonzero : ∫ (x : G), f x ∂μ ≠ 0)
Goal: ↑(μ'.haarScalarFactor μ) = (∫ (x : G), f x ∂μ') / ∫ (x : G), f x ∂μ
-- MeasureTheory.Measure.addHaarScalarFactor_smul
@[to_additive (attr := simp) addHaarScalarFactor_smul]
lemma haarScalarFactor_smul [LocallyCompactSpace G] (μ' μ : Measure G) [IsHaarMeasure μ]
    [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] {c : ℝ≥0} :
    haarScalarFactor (c • μ') μ = c • haarScalarFactor μ' μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] {c : ℝ≥0}
Goal: (c • μ').addHaarScalarFactor μ = c • μ'.addHaarScalarFactor μ
-- MeasureTheory.Measure.haarScalarFactor_smul
@[to_additive (attr := simp) addHaarScalarFactor_smul]
lemma haarScalarFactor_smul [LocallyCompactSpace G] (μ' μ : Measure G) [IsHaarMeasure μ]
    [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] {c : ℝ≥0} :
    haarScalarFactor (c • μ') μ = c • haarScalarFactor μ' μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] {c : ℝ≥0}
Goal: (c • μ').haarScalarFactor μ = c • μ'.haarScalarFactor μ
-- MeasureTheory.Measure.addHaarScalarFactor_self
@[to_additive (attr := simp)]
lemma haarScalarFactor_self (μ : Measure G) [IsHaarMeasure μ] :
    haarScalarFactor μ μ = 1 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure]
Goal: μ.addHaarScalarFactor μ = 1
-- MeasureTheory.Measure.haarScalarFactor_self
@[to_additive (attr := simp)]
lemma haarScalarFactor_self (μ : Measure G) [IsHaarMeasure μ] :
    haarScalarFactor μ μ = 1 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure]
Goal: μ.haarScalarFactor μ = 1
-- MeasureTheory.Measure.addHaarScalarFactor_eq_add
@[to_additive]
lemma haarScalarFactor_eq_mul (μ' μ ν : Measure G)
    [IsHaarMeasure μ] [IsHaarMeasure ν] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
    haarScalarFactor μ' ν = haarScalarFactor μ' μ * haarScalarFactor μ ν 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ ν : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [ν.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant]
Goal: μ'.addHaarScalarFactor ν = μ'.addHaarScalarFactor μ * μ.addHaarScalarFactor ν
-- MeasureTheory.Measure.haarScalarFactor_eq_mul
@[to_additive]
lemma haarScalarFactor_eq_mul (μ' μ ν : Measure G)
    [IsHaarMeasure μ] [IsHaarMeasure ν] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
    haarScalarFactor μ' ν = haarScalarFactor μ' μ * haarScalarFactor μ ν 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ ν : MeasureTheory.Measure G) [μ.IsHaarMeasure] [ν.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant]
Goal: μ'.haarScalarFactor ν = μ'.haarScalarFactor μ * μ.haarScalarFactor ν
-- MeasureTheory.Measure.addHaarScalarFactor_pos_of_isAddHaarMeasure
@[to_additive]
lemma haarScalarFactor_pos_of_isHaarMeasure (μ' μ : Measure G) [IsHaarMeasure μ]
    [IsHaarMeasure μ'] : 0 < haarScalarFactor μ' μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [μ'.IsAddHaarMeasure]
Goal: 0 < μ'.addHaarScalarFactor μ
-- MeasureTheory.Measure.haarScalarFactor_pos_of_isHaarMeasure
@[to_additive]
lemma haarScalarFactor_pos_of_isHaarMeasure (μ' μ : Measure G) [IsHaarMeasure μ]
    [IsHaarMeasure μ'] : 0 < haarScalarFactor μ' μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [μ'.IsHaarMeasure]
Goal: 0 < μ'.haarScalarFactor μ
-- MeasureTheory.Measure.measure_preimage_isAddLeftInvariant_eq_smul_of_hasCompactSupport
@[to_additive measure_preimage_isAddLeftInvariant_eq_smul_of_hasCompactSupport
"Two left invariant measures give the same mass to level sets of continuous compactly supported
functions, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which works for any set with
compact closure."]
lemma measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) :
    μ' (f ⁻¹' {1}) = haarScalarFactor μ' μ • μ (f ⁻¹' {1}) 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: μ' (f ⁻¹' {1}) = μ'.addHaarScalarFactor μ • μ (f ⁻¹' {1})
-- MeasureTheory.Measure.measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport
@[to_additive measure_preimage_isAddLeftInvariant_eq_smul_of_hasCompactSupport
"Two left invariant measures give the same mass to level sets of continuous compactly supported
functions, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which works for any set with
compact closure."]
lemma measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) :
    μ' (f ⁻¹' {1}) = haarScalarFactor μ' μ • μ (f ⁻¹' {1}) 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: μ' (f ⁻¹' {1}) = μ'.haarScalarFactor μ • μ (f ⁻¹' {1})