ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- QuotientGroup.integral_eq_integral_automorphize
@[to_additive "This is a simple version of the **Unfolding Trick**: Given a subgroup `Γ` of an
  additive  group `G`, the integral of a function `f` on `G` with respect to a right-invariant
  measure `μ` is equal to the integral over the quotient `G ⧸ Γ` of the automorphization of `f`."]
lemma QuotientGroup.integral_eq_integral_automorphize {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [μ.IsMulRightInvariant] {f : G → E}
    (hf₁ : Integrable f μ) (hf₂ : AEStronglyMeasurable (automorphize f) μ_𝓕) :
    ∫ x : G, f x ∂μ = ∫ x : G ⧸ Γ, automorphize f x ∂μ_𝓕 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : Subgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 μ) [Countable ↥Γ] [MeasurableSpace (G ⧸ Γ)] [BorelSpace (G ⧸ Γ)] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [μ.IsMulRightInvariant] {f : G → E} (hf₁ : MeasureTheory.Integrable f μ) (hf₂ :
  MeasureTheory.AEStronglyMeasurable (QuotientGroup.automorphize f)
    (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)))
Goal: ∫ (x : G), f x ∂μ =
  ∫ (x : G ⧸ Γ), QuotientGroup.automorphize f x ∂MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)
-- QuotientGroup.integral_mul_eq_integral_automorphize_mul
lemma QuotientGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]
    [NormedSpace ℝ K] [μ.IsMulRightInvariant] {f : G → K}
    (f_ℒ_1 : Integrable f μ) {g : G ⧸ Γ → K} (hg : AEStronglyMeasurable g μ_𝓕)
    (g_ℒ_infinity : essSup (fun x ↦ ↑‖g x‖₊) μ_𝓕 ≠ ∞)
    (F_ae_measurable : AEStronglyMeasurable (QuotientGroup.automorphize f) μ_𝓕) :
    ∫ x : G, g (x : G ⧸ Γ) * (f x) ∂μ
      = ∫ x : G ⧸ Γ, g x * (QuotientGroup.automorphize f x) ∂μ_𝓕 
Arguments: {G : Type u_1} [Group G] [MeasurableSpace G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : Subgroup G} {𝓕 : Set G} (h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 μ) [Countable ↥Γ] [MeasurableSpace (G ⧸ Γ)] [BorelSpace (G ⧸ Γ)] {K : Type u_2} [NormedField K] [NormedSpace ℝ K] [μ.IsMulRightInvariant] {f : G → K} (f_ℒ_1 : MeasureTheory.Integrable f μ) {g : G ⧸ Γ → K} (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕))) (g_ℒ_infinity : essSup (fun x => ↑‖g x‖₊) (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)) ≠ ⊤) (F_ae_measurable :
  MeasureTheory.AEStronglyMeasurable (QuotientGroup.automorphize f)
    (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)))
Goal: ∫ (x : G), g ↑x * f x ∂μ =
  ∫ (x : G ⧸ Γ), g x * QuotientGroup.automorphize f x ∂MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)
-- QuotientAddGroup.integral_mul_eq_integral_automorphize_mul
lemma QuotientAddGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]
    [NormedSpace ℝ K] [μ'.IsAddRightInvariant] {f : G' → K}
    (f_ℒ_1 : Integrable f μ') {g : G' ⧸ Γ' → K} (hg : AEStronglyMeasurable g μ_𝓕)
    (g_ℒ_infinity : essSup (fun x ↦ (‖g x‖₊ : ℝ≥0∞)) μ_𝓕 ≠ ∞)
    (F_ae_measurable : AEStronglyMeasurable (QuotientAddGroup.automorphize f) μ_𝓕)
    (h𝓕 : IsAddFundamentalDomain Γ'.op 𝓕' μ') :
    ∫ x : G', g (x : G' ⧸ Γ') * (f x) ∂μ'
      = ∫ x : G' ⧸ Γ', g x * (QuotientAddGroup.automorphize f x) ∂μ_𝓕 
Arguments: {G' : Type u_1} [AddGroup G'] [MeasurableSpace G'] [TopologicalSpace G'] [TopologicalAddGroup G'] [BorelSpace G'] {μ' : MeasureTheory.Measure G'} {Γ' : AddSubgroup G'} [Countable ↥Γ'] [MeasurableSpace (G' ⧸ Γ')] [BorelSpace (G' ⧸ Γ')] {𝓕' : Set G'} {K : Type u_2} [NormedField K] [NormedSpace ℝ K] [μ'.IsAddRightInvariant] {f : G' → K} (f_ℒ_1 : MeasureTheory.Integrable f μ') {g : G' ⧸ Γ' → K} (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map QuotientAddGroup.mk (μ'.restrict 𝓕'))) (g_ℒ_infinity : essSup (fun x => ↑‖g x‖₊) (MeasureTheory.Measure.map QuotientAddGroup.mk (μ'.restrict 𝓕')) ≠ ⊤) (F_ae_measurable :
  MeasureTheory.AEStronglyMeasurable (QuotientAddGroup.automorphize f)
    (MeasureTheory.Measure.map QuotientAddGroup.mk (μ'.restrict 𝓕'))) (h𝓕 : MeasureTheory.IsAddFundamentalDomain (↥Γ'.op) 𝓕' μ')
Goal: ∫ (x : G'), g ↑x * f x ∂μ' =
  ∫ (x : G' ⧸ Γ'),
    g x * QuotientAddGroup.automorphize f x ∂MeasureTheory.Measure.map QuotientAddGroup.mk (μ'.restrict 𝓕')
-- MeasureTheory.tendsto_lintegral_nn_filter_of_le_const
theorem tendsto_lintegral_nn_filter_of_le_const {ι : Type*} {L : Filter ι} [L.IsCountablyGenerated]
    (μ : Measure Ω) [IsFiniteMeasure μ] {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
    (fs_le_const : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c) {f : Ω → ℝ≥0}
    (fs_lim : ∀ᵐ ω : Ω ∂μ, Tendsto (fun i ↦ fs i ω) L (𝓝 (f ω))) :
    Tendsto (fun i ↦ ∫⁻ ω, fs i ω ∂μ) L (𝓝 (∫⁻ ω, f ω ∂μ)) 
Arguments: {Ω : Type u_1} [TopologicalSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} [L.IsCountablyGenerated] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0} (fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c) {f : Ω → ℝ≥0} (fs_lim : ∀ᵐ (ω : Ω) ∂μ, Filter.Tendsto (fun i => (fs i) ω) L (𝓝 (f ω)))
Goal: Filter.Tendsto (fun i => ∫⁻ (ω : Ω), ↑((fs i) ω) ∂μ) L (𝓝 (∫⁻ (ω : Ω), ↑(f ω) ∂μ))
-- MeasureTheory.measure_of_cont_bdd_of_tendsto_filter_indicator
theorem measure_of_cont_bdd_of_tendsto_filter_indicator {ι : Type*} {L : Filter ι}
    [L.IsCountablyGenerated] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : Measure Ω)
    [IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E) (fs : ι → Ω →ᵇ ℝ≥0)
    (fs_bdd : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c)
    (fs_lim : ∀ᵐ ω ∂μ, Tendsto (fun i ↦ fs i ω) L (𝓝 (indicator E (fun _ ↦ (1 : ℝ≥0)) ω))) :
    Tendsto (fun n ↦ lintegral μ fun ω ↦ fs n ω) L (𝓝 (μ E)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} [L.IsCountablyGenerated] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E) (fs : ι → Ω →ᵇ ℝ≥0) (fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c) (fs_lim : ∀ᵐ (ω : Ω) ∂μ, Filter.Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω)))
Goal: Filter.Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) L (𝓝 (μ E))
-- MeasureTheory.measure_of_cont_bdd_of_tendsto_indicator
theorem measure_of_cont_bdd_of_tendsto_indicator [OpensMeasurableSpace Ω]
    (μ : Measure Ω) [IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E)
    (fs : ℕ → Ω →ᵇ ℝ≥0) (fs_bdd : ∀ n ω, fs n ω ≤ c)
    (fs_lim : Tendsto (fun n ω ↦ fs n ω) atTop (𝓝 (indicator E fun _ ↦ (1 : ℝ≥0)))) :
    Tendsto (fun n ↦ lintegral μ fun ω ↦ fs n ω) atTop (𝓝 (μ E)) 
Arguments: {Ω : Type u_1} [TopologicalSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E) (fs : ℕ → Ω →ᵇ ℝ≥0) (fs_bdd : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c) (fs_lim : Filter.Tendsto (fun n ω => (fs n) ω) Filter.atTop (𝓝 (E.indicator fun x => 1)))
Goal: Filter.Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) Filter.atTop (𝓝 (μ E))
-- MeasureTheory.tendsto_lintegral_thickenedIndicator_of_isClosed
theorem tendsto_lintegral_thickenedIndicator_of_isClosed {Ω : Type*} [MeasurableSpace Ω]
    [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] {F : Set Ω}
    (F_closed : IsClosed F) {δs : ℕ → ℝ} (δs_pos : ∀ n, 0 < δs n)
    (δs_lim : Tendsto δs atTop (𝓝 0)) :
    Tendsto (fun n ↦ lintegral μ fun ω ↦ (thickenedIndicator (δs_pos n) F ω : ℝ≥0∞)) atTop
      (𝓝 (μ F)) 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {F : Set Ω} (F_closed : IsClosed F) {δs : ℕ → ℝ} (δs_pos : ∀ (n : ℕ), 0 < δs n) (δs_lim : Filter.Tendsto δs Filter.atTop (𝓝 0))
Goal: Filter.Tendsto (fun n => ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ) Filter.atTop (𝓝 (μ F))
-- HasOuterApproxClosed.exAppr
class HasOuterApproxClosed (X : Type*) [TopologicalSpace X] : Prop where
  exAppr : ∀ (F : Set X), IsClosed F → ∃ (fseq : ℕ → (X →ᵇ ℝ≥0)),
    (∀ n x, fseq n x ≤ 1) ∧ (∀ n x, x ∈ F → 1 ≤ fseq n x) ∧
    Tendsto (fun n : ℕ ↦ (fun x ↦ fseq n x)) atTop (𝓝 (indicator F fun _ ↦ (1 : ℝ≥0)))
Arguments: {X : Type u_1} [TopologicalSpace X] [self : HasOuterApproxClosed X] (F : Set X)
Goal: IsClosed F →
  ∃ fseq,
    (∀ (n : ℕ) (x : X), (fseq n) x ≤ 1) ∧
      (∀ (n : ℕ), ∀ x ∈ F, 1 ≤ (fseq n) x) ∧
        Filter.Tendsto (fun n x => (fseq n) x) Filter.atTop (𝓝 (F.indicator fun x => 1))
-- HasOuterApproxClosed.apprSeq_apply_le_one
lemma apprSeq_apply_le_one (n : ℕ) (x : X) :
    hF.apprSeq n x ≤ 1 
Arguments: {X : Type u_1} [TopologicalSpace X] [HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F) (n : ℕ) (x : X)
Goal: (hF.apprSeq n) x ≤ 1
-- HasOuterApproxClosed.apprSeq_apply_eq_one
lemma apprSeq_apply_eq_one (n : ℕ) {x : X} (hxF : x ∈ F) :
    hF.apprSeq n x = 1 
Arguments: {X : Type u_1} [TopologicalSpace X] [HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F) (n : ℕ) {x : X} (hxF : x ∈ F)
Goal: (hF.apprSeq n) x = 1
-- HasOuterApproxClosed.tendsto_apprSeq
lemma tendsto_apprSeq :
    Tendsto (fun n : ℕ ↦ (fun x ↦ hF.apprSeq n x)) atTop (𝓝 (indicator F fun _ ↦ (1 : ℝ≥0))) 
Arguments: {X : Type u_1} [TopologicalSpace X] [HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F)
Goal: Filter.Tendsto (fun n x => (hF.apprSeq n) x) Filter.atTop (𝓝 (F.indicator fun x => 1))
-- HasOuterApproxClosed.indicator_le_apprSeq
lemma indicator_le_apprSeq (n : ℕ) :
    indicator F (fun _ ↦ 1) ≤ hF.apprSeq n 
Arguments: {X : Type u_1} [TopologicalSpace X] [HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F) (n : ℕ)
Goal: (F.indicator fun x => 1) ≤ ⇑(hF.apprSeq n)
-- HasOuterApproxClosed.measure_le_lintegral
theorem measure_le_lintegral [MeasurableSpace X] [OpensMeasurableSpace X] (μ : Measure X) (n : ℕ) :
    μ F ≤ ∫⁻ x, (hF.apprSeq n x : ℝ≥0∞) ∂μ 
Arguments: {X : Type u_1} [TopologicalSpace X] [HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F) [MeasurableSpace X] [OpensMeasurableSpace X] (μ : MeasureTheory.Measure X) (n : ℕ)
Goal: μ F ≤ ∫⁻ (x : X), ↑((hF.apprSeq n) x) ∂μ
-- HasOuterApproxClosed.tendsto_lintegral_apprSeq
lemma tendsto_lintegral_apprSeq [MeasurableSpace X] [OpensMeasurableSpace X]
    (μ : Measure X) [IsFiniteMeasure μ] :
    Tendsto (fun n ↦ ∫⁻ x, hF.apprSeq n x ∂μ) atTop (𝓝 ((μ : Measure X) F)) 
Arguments: {X : Type u_1} [TopologicalSpace X] [HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F) [MeasurableSpace X] [OpensMeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ]
Goal: Filter.Tendsto (fun n => ∫⁻ (x : X), ↑((hF.apprSeq n) x) ∂μ) Filter.atTop (𝓝 (μ F))
-- MeasureTheory.measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure
theorem measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure {Ω : Type*}
    [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω]
    [OpensMeasurableSpace Ω] {μ ν : Measure Ω} [IsFiniteMeasure μ]
    (h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ x, f x ∂μ = ∫⁻ x, f x ∂ν) {F : Set Ω} (F_closed : IsClosed F) :
    μ F = ν F 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω] [OpensMeasurableSpace Ω] {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν) {F : Set Ω} (F_closed : IsClosed F)
Goal: μ F = ν F
-- MeasureTheory.ext_of_forall_lintegral_eq_of_IsFiniteMeasure
theorem ext_of_forall_lintegral_eq_of_IsFiniteMeasure {Ω : Type*}
    [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω]
    [BorelSpace Ω] {μ ν : Measure Ω} [IsFiniteMeasure μ]
    (h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ x, f x ∂μ = ∫⁻ x, f x ∂ν) :
    μ = ν 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω] [BorelSpace Ω] {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν)
Goal: μ = ν
-- MeasureTheory.Content.mono'
structure Content (G : Type w) [TopologicalSpace G] where
  toFun : Compacts G → ℝ≥0
  mono' : ∀ K₁ K₂ : Compacts G, (K₁ : Set G) ⊆ K₂ → toFun K₁ ≤ toFun K₂
  sup_disjoint' :
    ∀ K₁ K₂ : Compacts G, Disjoint (K₁ : Set G) K₂ → IsClosed (K₁ : Set G) → IsClosed (K₂ : Set G)
      → toFun (K₁ ⊔ K₂) = toFun K₁ + toFun K₂
  sup_le' : ∀ K₁ K₂ : Compacts G, toFun (K₁ ⊔ K₂) ≤ toFun K₁ + toFun K₂
Arguments: {G : Type w} [TopologicalSpace G] (self : MeasureTheory.Content G) (K₁ K₂ : TopologicalSpace.Compacts G)
Goal: ↑K₁ ⊆ ↑K₂ → self.toFun K₁ ≤ self.toFun K₂
-- MeasureTheory.Content.sup_le'
structure Content (G : Type w) [TopologicalSpace G] where
  toFun : Compacts G → ℝ≥0
  mono' : ∀ K₁ K₂ : Compacts G, (K₁ : Set G) ⊆ K₂ → toFun K₁ ≤ toFun K₂
  sup_disjoint' :
    ∀ K₁ K₂ : Compacts G, Disjoint (K₁ : Set G) K₂ → IsClosed (K₁ : Set G) → IsClosed (K₂ : Set G)
      → toFun (K₁ ⊔ K₂) = toFun K₁ + toFun K₂
  sup_le' : ∀ K₁ K₂ : Compacts G, toFun (K₁ ⊔ K₂) ≤ toFun K₁ + toFun K₂
Arguments: {G : Type w} [TopologicalSpace G] (self : MeasureTheory.Content G) (K₁ K₂ : TopologicalSpace.Compacts G)
Goal: self.toFun (K₁ ⊔ K₂) ≤ self.toFun K₁ + self.toFun K₂
-- MeasureTheory.Content.sup_disjoint'
structure Content (G : Type w) [TopologicalSpace G] where
  toFun : Compacts G → ℝ≥0
  mono' : ∀ K₁ K₂ : Compacts G, (K₁ : Set G) ⊆ K₂ → toFun K₁ ≤ toFun K₂
  sup_disjoint' :
    ∀ K₁ K₂ : Compacts G, Disjoint (K₁ : Set G) K₂ → IsClosed (K₁ : Set G) → IsClosed (K₂ : Set G)
      → toFun (K₁ ⊔ K₂) = toFun K₁ + toFun K₂
  sup_le' : ∀ K₁ K₂ : Compacts G, toFun (K₁ ⊔ K₂) ≤ toFun K₁ + toFun K₂
Arguments: {G : Type w} [TopologicalSpace G] (self : MeasureTheory.Content G) (K₁ K₂ : TopologicalSpace.Compacts G)
Goal: Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → self.toFun (K₁ ⊔ K₂) = self.toFun K₁ + self.toFun K₂
-- MeasureTheory.Content.apply_eq_coe_toFun
theorem apply_eq_coe_toFun (K : Compacts G) : μ K = μ.toFun K 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (K : TopologicalSpace.Compacts G)
Goal: (fun s => ↑(μ.toFun s)) K = ↑(μ.toFun K)
-- MeasureTheory.Content.mono
theorem mono (K₁ K₂ : Compacts G) (h : (K₁ : Set G) ⊆ K₂) : μ K₁ ≤ μ K₂ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (K₁ K₂ : TopologicalSpace.Compacts G) (h : ↑K₁ ⊆ ↑K₂)
Goal: (fun s => ↑(μ.toFun s)) K₁ ≤ (fun s => ↑(μ.toFun s)) K₂
-- MeasureTheory.Content.sup_disjoint
theorem sup_disjoint (K₁ K₂ : Compacts G) (h : Disjoint (K₁ : Set G) K₂)
    (h₁ : IsClosed (K₁ : Set G)) (h₂ : IsClosed (K₂ : Set G)) :
    μ (K₁ ⊔ K₂) = μ K₁ + μ K₂ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (K₁ K₂ : TopologicalSpace.Compacts G) (h : Disjoint ↑K₁ ↑K₂) (h₁ : IsClosed ↑K₁) (h₂ : IsClosed ↑K₂)
Goal: (fun s => ↑(μ.toFun s)) (K₁ ⊔ K₂) = (fun s => ↑(μ.toFun s)) K₁ + (fun s => ↑(μ.toFun s)) K₂
-- MeasureTheory.Content.sup_le
theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (K₁ K₂ : TopologicalSpace.Compacts G)
Goal: (fun s => ↑(μ.toFun s)) (K₁ ⊔ K₂) ≤ (fun s => ↑(μ.toFun s)) K₁ + (fun s => ↑(μ.toFun s)) K₂
-- MeasureTheory.Content.lt_top
theorem lt_top (K : Compacts G) : μ K < ∞ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (K : TopologicalSpace.Compacts G)
Goal: (fun s => ↑(μ.toFun s)) K < ⊤
-- MeasureTheory.Content.empty
theorem empty : μ ⊥ = 0 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G)
Goal: (fun s => ↑(μ.toFun s)) ⊥ = 0
-- MeasureTheory.Content.le_innerContent
theorem le_innerContent (K : Compacts G) (U : Opens G) (h2 : (K : Set G) ⊆ U) :
    μ K ≤ μ.innerContent U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (K : TopologicalSpace.Compacts G) (U : TopologicalSpace.Opens G) (h2 : ↑K ⊆ ↑U)
Goal: (fun s => ↑(μ.toFun s)) K ≤ μ.innerContent U
-- MeasureTheory.Content.innerContent_le
theorem innerContent_le (U : Opens G) (K : Compacts G) (h2 : (U : Set G) ⊆ K) :
    μ.innerContent U ≤ μ K 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (U : TopologicalSpace.Opens G) (K : TopologicalSpace.Compacts G) (h2 : ↑U ⊆ ↑K)
Goal: μ.innerContent U ≤ (fun s => ↑(μ.toFun s)) K
-- MeasureTheory.Content.innerContent_of_isCompact
theorem innerContent_of_isCompact {K : Set G} (h1K : IsCompact K) (h2K : IsOpen K) :
    μ.innerContent ⟨K, h2K⟩ = μ ⟨K, h1K⟩ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) {K : Set G} (h1K : IsCompact K) (h2K : IsOpen K)
Goal: μ.innerContent { carrier := K, is_open' := h2K } = (fun s => ↑(μ.toFun s)) { carrier := K, isCompact' := h1K }
-- MeasureTheory.Content.innerContent_bot
theorem innerContent_bot : μ.innerContent ⊥ = 0 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G)
Goal: μ.innerContent ⊥ = 0
-- MeasureTheory.Content.innerContent_mono
theorem innerContent_mono ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
    μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V)
Goal: μ.innerContent { carrier := U, is_open' := hU } ≤ μ.innerContent { carrier := V, is_open' := hV }
-- MeasureTheory.Content.innerContent_exists_compact
theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
    (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) {U : TopologicalSpace.Opens G} (hU : μ.innerContent U ≠ ⊤) {ε : ℝ≥0} (hε : ε ≠ 0)
Goal: ∃ K, ↑K ⊆ ↑U ∧ μ.innerContent U ≤ (fun s => ↑(μ.toFun s)) K + ↑ε
-- MeasureTheory.Content.innerContent_iSup_nat
theorem innerContent_iSup_nat [R1Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (U : ℕ → TopologicalSpace.Opens G)
Goal: μ.innerContent (⨆ i, U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
-- MeasureTheory.Content.innerContent_iUnion_nat
theorem innerContent_iUnion_nat [R1Space G] ⦃U : ℕ → Set G⦄
    (hU : ∀ i : ℕ, IsOpen (U i)) :
    μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ (i : ℕ), IsOpen (U i))
Goal: μ.innerContent { carrier := ⋃ i, U i, is_open' := ⋯ } ≤ ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }
-- MeasureTheory.Content.innerContent_comap
theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.continuous) = μ K)
    (U : Opens G) : μ.innerContent (Opens.comap f.toContinuousMap U) = μ.innerContent U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (f : G ≃ₜ G) (h :
  ∀ ⦃K : TopologicalSpace.Compacts G⦄,
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map ⇑f ⋯ K) = (fun s => ↑(μ.toFun s)) K) (U : TopologicalSpace.Opens G)
Goal: μ.innerContent ((TopologicalSpace.Opens.comap f.toContinuousMap) U) = μ.innerContent U
-- MeasureTheory.Content.is_add_left_invariant_innerContent
@[to_additive]
theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
    (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
    (U : Opens G) :
    μ.innerContent (Opens.comap (Homeomorph.mulLeft g).toContinuousMap U) = μ.innerContent U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [AddGroup G] [TopologicalAddGroup G] (h :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (g : G) (U : TopologicalSpace.Opens G)
Goal: μ.innerContent ((TopologicalSpace.Opens.comap (Homeomorph.addLeft g).toContinuousMap) U) = μ.innerContent U
-- MeasureTheory.Content.is_mul_left_invariant_innerContent
@[to_additive]
theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
    (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
    (U : Opens G) :
    μ.innerContent (Opens.comap (Homeomorph.mulLeft g).toContinuousMap U) = μ.innerContent U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [Group G] [TopologicalGroup G] (h :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (g : G) (U : TopologicalSpace.Opens G)
Goal: μ.innerContent ((TopologicalSpace.Opens.comap (Homeomorph.mulLeft g).toContinuousMap) U) = μ.innerContent U
-- MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant
@[to_additive]
theorem innerContent_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
    (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
    (hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [Group G] [TopologicalGroup G] (h3 :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (K : TopologicalSpace.Compacts G) (hK : (fun s => ↑(μ.toFun s)) K ≠ 0) (U : TopologicalSpace.Opens G) (hU : (↑U).Nonempty)
Goal: 0 < μ.innerContent U
-- MeasureTheory.Content.innerContent_pos_of_is_add_left_invariant
@[to_additive]
theorem innerContent_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
    (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
    (hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [AddGroup G] [TopologicalAddGroup G] (h3 :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (K : TopologicalSpace.Compacts G) (hK : (fun s => ↑(μ.toFun s)) K ≠ 0) (U : TopologicalSpace.Opens G) (hU : (↑U).Nonempty)
Goal: 0 < μ.innerContent U
-- MeasureTheory.Content.innerContent_mono'
theorem innerContent_mono' ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
    μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V)
Goal: μ.innerContent { carrier := U, is_open' := hU } ≤ μ.innerContent { carrier := V, is_open' := hV }
-- MeasureTheory.Content.outerMeasure_opens
theorem outerMeasure_opens (U : Opens G) : μ.outerMeasure U = μ.innerContent U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (U : TopologicalSpace.Opens G)
Goal: μ.outerMeasure ↑U = μ.innerContent U
-- MeasureTheory.Content.outerMeasure_of_isOpen
theorem outerMeasure_of_isOpen (U : Set G) (hU : IsOpen U) :
    μ.outerMeasure U = μ.innerContent ⟨U, hU⟩ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (U : Set G) (hU : IsOpen U)
Goal: μ.outerMeasure U = μ.innerContent { carrier := U, is_open' := hU }
-- MeasureTheory.Content.outerMeasure_le
theorem outerMeasure_le (U : Opens G) (K : Compacts G) (hUK : (U : Set G) ⊆ K) :
    μ.outerMeasure U ≤ μ K 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (U : TopologicalSpace.Opens G) (K : TopologicalSpace.Compacts G) (hUK : ↑U ⊆ ↑K)
Goal: μ.outerMeasure ↑U ≤ (fun s => ↑(μ.toFun s)) K
-- MeasureTheory.Content.le_outerMeasure_compacts
theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.outerMeasure K 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (K : TopologicalSpace.Compacts G)
Goal: (fun s => ↑(μ.toFun s)) K ≤ μ.outerMeasure ↑K