ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.lpMeasSubgroupToLpTrim_sub
theorem lpMeasSubgroupToLpTrim_sub (hm : m ≤ m0) (f g : lpMeasSubgroup F m p μ) :
    lpMeasSubgroupToLpTrim F p μ hm (f - g) =
      lpMeasSubgroupToLpTrim F p μ hm f - lpMeasSubgroupToLpTrim F p μ hm g 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f g : ↥(MeasureTheory.lpMeasSubgroup F m p μ))
Goal: MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm (f - g) =
  MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm f - MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm g
-- MeasureTheory.lpMeasToLpTrim_smul
theorem lpMeasToLpTrim_smul (hm : m ≤ m0) (c : 𝕜) (f : lpMeas F 𝕜 m p μ) :
    lpMeasToLpTrim F 𝕜 p μ hm (c • f) = c • lpMeasToLpTrim F 𝕜 p μ hm f 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (c : 𝕜) (f : ↥(MeasureTheory.lpMeas F 𝕜 m p μ))
Goal: MeasureTheory.lpMeasToLpTrim F 𝕜 p μ hm (c • f) = c • MeasureTheory.lpMeasToLpTrim F 𝕜 p μ hm f
-- MeasureTheory.lpMeasSubgroupToLpTrim_norm_map
theorem lpMeasSubgroupToLpTrim_norm_map [hp : Fact (1 ≤ p)] (hm : m ≤ m0)
    (f : lpMeasSubgroup F m p μ) : ‖lpMeasSubgroupToLpTrim F p μ hm f‖ = ‖f‖ 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [hp : Fact (1 ≤ p)] (hm : m ≤ m0) (f : ↥(MeasureTheory.lpMeasSubgroup F m p μ))
Goal: ‖MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm f‖ = ‖f‖
-- MeasureTheory.isometry_lpMeasSubgroupToLpTrim
theorem isometry_lpMeasSubgroupToLpTrim [hp : Fact (1 ≤ p)] (hm : m ≤ m0) :
    Isometry (lpMeasSubgroupToLpTrim F p μ hm) 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [hp : Fact (1 ≤ p)] (hm : m ≤ m0)
Goal: Isometry (MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm)
-- MeasureTheory.isComplete_aeStronglyMeasurable'
theorem isComplete_aeStronglyMeasurable' [hp : Fact (1 ≤ p)] [CompleteSpace F] (hm : m ≤ m0) :
    IsComplete {f : Lp F p μ | AEStronglyMeasurable' m f μ} 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [hp : Fact (1 ≤ p)] [CompleteSpace F] (hm : m ≤ m0)
Goal: IsComplete {f | MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ}
-- MeasureTheory.isClosed_aeStronglyMeasurable'
theorem isClosed_aeStronglyMeasurable' [Fact (1 ≤ p)] [CompleteSpace F] (hm : m ≤ m0) :
    IsClosed {f : Lp F p μ | AEStronglyMeasurable' m f μ} 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] [CompleteSpace F] (hm : m ≤ m0)
Goal: IsClosed {f | MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ}
-- MeasureTheory.lpMeas.ae_fin_strongly_measurable'
theorem lpMeas.ae_fin_strongly_measurable' (hm : m ≤ m0) (f : lpMeas F 𝕜 m p μ) (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) :
    -- Porting note: changed `f` to `f.1` in the next line. Not certain this is okay.
    ∃ g, FinStronglyMeasurable g (μ.trim hm) ∧ f.1 =ᵐ[μ] g 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.lpMeas F 𝕜 m p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: ∃ g, MeasureTheory.FinStronglyMeasurable g (μ.trim hm) ∧ ↑↑↑f =ᵐ[μ] g
-- MeasureTheory.lpMeasToLpTrimLie_symm_indicator
theorem lpMeasToLpTrimLie_symm_indicator [one_le_p : Fact (1 ≤ p)] [NormedSpace ℝ F] {hm : m ≤ m0}
    {s : Set α} {μ : Measure α} (hs : MeasurableSet[m] s) (hμs : μ.trim hm s ≠ ∞) (c : F) :
    ((lpMeasToLpTrimLie F ℝ p μ hm).symm (indicatorConstLp p hs hμs c) : Lp F p μ) =
      indicatorConstLp p (hm s hs) ((le_trim hm).trans_lt hμs.lt_top).ne c 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} [one_le_p : Fact (1 ≤ p)] [NormedSpace ℝ F] {hm : m ≤ m0} {s : Set α} {μ : MeasureTheory.Measure α} (hs : MeasurableSet s) (hμs : (μ.trim hm) s ≠ ⊤) (c : F)
Goal: ↑((MeasureTheory.lpMeasToLpTrimLie F ℝ p μ hm).symm (MeasureTheory.indicatorConstLp p hs hμs c)) =
  MeasureTheory.indicatorConstLp p ⋯ ⋯ c
-- MeasureTheory.lpMeasToLpTrimLie_symm_toLp
theorem lpMeasToLpTrimLie_symm_toLp [one_le_p : Fact (1 ≤ p)] [NormedSpace ℝ F] (hm : m ≤ m0)
    (f : α → F) (hf : Memℒp f p (μ.trim hm)) :
    ((lpMeasToLpTrimLie F ℝ p μ hm).symm (hf.toLp f) : Lp F p μ) =
      (memℒp_of_memℒp_trim hm hf).toLp f 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [one_le_p : Fact (1 ≤ p)] [NormedSpace ℝ F] (hm : m ≤ m0) (f : α → F) (hf : MeasureTheory.Memℒp f p (μ.trim hm))
Goal: ↑((MeasureTheory.lpMeasToLpTrimLie F ℝ p μ hm).symm (MeasureTheory.Memℒp.toLp f hf)) = MeasureTheory.Memℒp.toLp f ⋯
-- MeasureTheory.Lp.induction_stronglyMeasurable_aux
@[elab_as_elim]
theorem Lp.induction_stronglyMeasurable_aux (hm : m ≤ m0) (hp_ne_top : p ≠ ∞) (P : Lp F p μ → Prop)
    (h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet[m] s) (hμs : μ s < ∞),
      P (Lp.simpleFunc.indicatorConst p (hm s hs) hμs.ne c))
    (h_add : ∀ ⦃f g⦄, ∀ hf : Memℒp f p μ, ∀ hg : Memℒp g p μ, AEStronglyMeasurable' m f μ →
      AEStronglyMeasurable' m g μ → Disjoint (Function.support f) (Function.support g) →
        P (hf.toLp f) → P (hg.toLp g) → P (hf.toLp f + hg.toLp g))
    (h_closed : IsClosed {f : lpMeas F ℝ m p μ | P f}) :
    ∀ f : Lp F p μ, AEStronglyMeasurable' m f μ → P f 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] [NormedSpace ℝ F] (hm : m ≤ m0) (hp_ne_top : p ≠ ⊤) (P : ↥(MeasureTheory.Lp F p μ) → Prop) (h_ind :
  ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(MeasureTheory.Lp.simpleFunc.indicatorConst p ⋯ ⋯ c)) (h_add :
  ∀ ⦃f g : α → F⦄ (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ),
    MeasureTheory.AEStronglyMeasurable' m f μ →
      MeasureTheory.AEStronglyMeasurable' m g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (MeasureTheory.Memℒp.toLp f hf) →
            P (MeasureTheory.Memℒp.toLp g hg) → P (MeasureTheory.Memℒp.toLp f hf + MeasureTheory.Memℒp.toLp g hg)) (h_closed : IsClosed {f | P ↑f}) (f : ↥(MeasureTheory.Lp F p μ))
Goal: MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ → P f
-- MeasureTheory.Lp.induction_stronglyMeasurable
@[elab_as_elim]
theorem Lp.induction_stronglyMeasurable (hm : m ≤ m0) (hp_ne_top : p ≠ ∞) (P : Lp F p μ → Prop)
    (h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet[m] s) (hμs : μ s < ∞),
      P (Lp.simpleFunc.indicatorConst p (hm s hs) hμs.ne c))
    (h_add : ∀ ⦃f g⦄, ∀ hf : Memℒp f p μ, ∀ hg : Memℒp g p μ, StronglyMeasurable[m] f →
      StronglyMeasurable[m] g → Disjoint (Function.support f) (Function.support g) →
        P (hf.toLp f) → P (hg.toLp g) → P (hf.toLp f + hg.toLp g))
    (h_closed : IsClosed {f : lpMeas F ℝ m p μ | P f}) :
    ∀ f : Lp F p μ, AEStronglyMeasurable' m f μ → P f 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] [NormedSpace ℝ F] (hm : m ≤ m0) (hp_ne_top : p ≠ ⊤) (P : ↥(MeasureTheory.Lp F p μ) → Prop) (h_ind :
  ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(MeasureTheory.Lp.simpleFunc.indicatorConst p ⋯ ⋯ c)) (h_add :
  ∀ ⦃f g : α → F⦄ (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ),
    MeasureTheory.StronglyMeasurable f →
      MeasureTheory.StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (MeasureTheory.Memℒp.toLp f hf) →
            P (MeasureTheory.Memℒp.toLp g hg) → P (MeasureTheory.Memℒp.toLp f hf + MeasureTheory.Memℒp.toLp g hg)) (h_closed : IsClosed {f | P ↑f}) (f : ↥(MeasureTheory.Lp F p μ))
Goal: MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ → P f
-- MeasureTheory.Memℒp.induction_stronglyMeasurable
@[elab_as_elim]
theorem Memℒp.induction_stronglyMeasurable (hm : m ≤ m0) (hp_ne_top : p ≠ ∞) (P : (α → F) → Prop)
    (h_ind : ∀ (c : F) ⦃s⦄, MeasurableSet[m] s → μ s < ∞ → P (s.indicator fun _ => c))
    (h_add : ∀ ⦃f g : α → F⦄, Disjoint (Function.support f) (Function.support g) →
      Memℒp f p μ → Memℒp g p μ → StronglyMeasurable[m] f → StronglyMeasurable[m] g →
        P f → P g → P (f + g))
    (h_closed : IsClosed {f : lpMeas F ℝ m p μ | P f})
    (h_ae : ∀ ⦃f g⦄, f =ᵐ[μ] g → Memℒp f p μ → P f → P g) :
    ∀ ⦃f : α → F⦄, Memℒp f p μ → AEStronglyMeasurable' m f μ → P f 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] [NormedSpace ℝ F] (hm : m ≤ m0) (hp_ne_top : p ≠ ⊤) (P : (α → F) → Prop) (h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)) (h_add :
  ∀ ⦃f g : α → F⦄,
    Disjoint (Function.support f) (Function.support g) →
      MeasureTheory.Memℒp f p μ →
        MeasureTheory.Memℒp g p μ →
          MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable g → P f → P g → P (f + g)) (h_closed : IsClosed {f | P ↑↑↑f}) (h_ae : ∀ ⦃f g : α → F⦄, f =ᵐ[μ] g → MeasureTheory.Memℒp f p μ → P f → P g) ⦃f : α → F⦄
Goal: MeasureTheory.Memℒp f p μ → MeasureTheory.AEStronglyMeasurable' m f μ → P f
-- volume_regionBetween_eq_integral'
theorem volume_regionBetween_eq_integral' [SigmaFinite μ] (f_int : IntegrableOn f s μ)
    (g_int : IntegrableOn g s μ) (hs : MeasurableSet s) (hfg : f ≤ᵐ[μ.restrict s] g) :
    μ.prod volume (regionBetween f g s) = ENNReal.ofReal (∫ y in s, (g - f) y ∂μ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α → ℝ} {s : Set α} [MeasureTheory.SigmaFinite μ] (f_int : MeasureTheory.IntegrableOn f s μ) (g_int : MeasureTheory.IntegrableOn g s μ) (hs : MeasurableSet s) (hfg : f ≤ᵐ[μ.restrict s] g)
Goal: (μ.prod MeasureTheory.volume) (regionBetween f g s) = ENNReal.ofReal (∫ (y : α) in s, (g - f) y ∂μ)
-- volume_regionBetween_eq_integral
theorem volume_regionBetween_eq_integral [SigmaFinite μ] (f_int : IntegrableOn f s μ)
    (g_int : IntegrableOn g s μ) (hs : MeasurableSet s) (hfg : ∀ x ∈ s, f x ≤ g x) :
    μ.prod volume (regionBetween f g s) = ENNReal.ofReal (∫ y in s, (g - f) y ∂μ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α → ℝ} {s : Set α} [MeasureTheory.SigmaFinite μ] (f_int : MeasureTheory.IntegrableOn f s μ) (g_int : MeasureTheory.IntegrableOn g s μ) (hs : MeasurableSet s) (hfg : ∀ x ∈ s, f x ≤ g x)
Goal: (μ.prod MeasureTheory.volume) (regionBetween f g s) = ENNReal.ofReal (∫ (y : α) in s, (g - f) y ∂μ)
-- Real.integrable_of_summable_norm_Icc
theorem Real.integrable_of_summable_norm_Icc {E : Type*} [NormedAddCommGroup E] {f : C(ℝ, E)}
    (hf : Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)‖) :
    Integrable f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : C(ℝ, E)} (hf : Summable fun n => ‖ContinuousMap.restrict (Set.Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖)
Goal: MeasureTheory.Integrable (⇑f) MeasureTheory.volume
-- integral_comp_neg_Iic
theorem integral_comp_neg_Iic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (c : ℝ) (f : ℝ → E) : (∫ x in Iic c, f (-x)) = ∫ x in Ioi (-c), f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (c : ℝ) (f : ℝ → E)
Goal: ∫ (x : ℝ) in Set.Iic c, f (-x) = ∫ (x : ℝ) in Set.Ioi (-c), f x
-- integral_comp_neg_Ioi
theorem integral_comp_neg_Ioi {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (c : ℝ) (f : ℝ → E) : (∫ x in Ioi c, f (-x)) = ∫ x in Iic (-c), f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (c : ℝ) (f : ℝ → E)
Goal: ∫ (x : ℝ) in Set.Ioi c, f (-x) = ∫ (x : ℝ) in Set.Iic (-c), f x
-- integral_comp_abs
theorem integral_comp_abs {f : ℝ → ℝ} :
    ∫ x, f |x| = 2 * ∫ x in Ioi (0 : ℝ), f x 
Arguments: {f : ℝ → ℝ}
Goal: ∫ (x : ℝ), f |x| = 2 * ∫ (x : ℝ) in Set.Ioi 0, f x
-- MeasureTheory.le_measure_compl_liminf_of_limsup_measure_le
theorem le_measure_compl_liminf_of_limsup_measure_le {ι : Type*} {L : Filter ι} {μ : Measure Ω}
    {μs : ι → Measure Ω} [IsProbabilityMeasure μ] [∀ i, IsProbabilityMeasure (μs i)] {E : Set Ω}
    (E_mble : MeasurableSet E) (h : (L.limsup fun i => μs i E) ≤ μ E) :
    μ Eᶜ ≤ L.liminf fun i => μs i Eᶜ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure Ω} {μs : ι → MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] {E : Set Ω} (E_mble : MeasurableSet E) (h : Filter.limsup (fun i => (μs i) E) L ≤ μ E)
Goal: μ Eᶜ ≤ Filter.liminf (fun i => (μs i) Eᶜ) L
-- MeasureTheory.le_measure_liminf_of_limsup_measure_compl_le
theorem le_measure_liminf_of_limsup_measure_compl_le {ι : Type*} {L : Filter ι} {μ : Measure Ω}
    {μs : ι → Measure Ω} [IsProbabilityMeasure μ] [∀ i, IsProbabilityMeasure (μs i)] {E : Set Ω}
    (E_mble : MeasurableSet E) (h : (L.limsup fun i => μs i Eᶜ) ≤ μ Eᶜ) :
    μ E ≤ L.liminf fun i => μs i E 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure Ω} {μs : ι → MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] {E : Set Ω} (E_mble : MeasurableSet E) (h : Filter.limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ)
Goal: μ E ≤ Filter.liminf (fun i => (μs i) E) L
-- MeasureTheory.limsup_measure_compl_le_of_le_liminf_measure
theorem limsup_measure_compl_le_of_le_liminf_measure {ι : Type*} {L : Filter ι} {μ : Measure Ω}
    {μs : ι → Measure Ω} [IsProbabilityMeasure μ] [∀ i, IsProbabilityMeasure (μs i)] {E : Set Ω}
    (E_mble : MeasurableSet E) (h : μ E ≤ L.liminf fun i => μs i E) :
    (L.limsup fun i => μs i Eᶜ) ≤ μ Eᶜ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure Ω} {μs : ι → MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] {E : Set Ω} (E_mble : MeasurableSet E) (h : μ E ≤ Filter.liminf (fun i => (μs i) E) L)
Goal: Filter.limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ
-- MeasureTheory.limsup_measure_le_of_le_liminf_measure_compl
theorem limsup_measure_le_of_le_liminf_measure_compl {ι : Type*} {L : Filter ι} {μ : Measure Ω}
    {μs : ι → Measure Ω} [IsProbabilityMeasure μ] [∀ i, IsProbabilityMeasure (μs i)] {E : Set Ω}
    (E_mble : MeasurableSet E) (h : μ Eᶜ ≤ L.liminf fun i => μs i Eᶜ) :
    (L.limsup fun i => μs i E) ≤ μ E 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure Ω} {μs : ι → MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] {E : Set Ω} (E_mble : MeasurableSet E) (h : μ Eᶜ ≤ Filter.liminf (fun i => (μs i) Eᶜ) L)
Goal: Filter.limsup (fun i => (μs i) E) L ≤ μ E
-- MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge
theorem limsup_measure_closed_le_iff_liminf_measure_open_ge {ι : Type*} {L : Filter ι}
    {μ : Measure Ω} {μs : ι → Measure Ω} [IsProbabilityMeasure μ]
    [∀ i, IsProbabilityMeasure (μs i)] :
    (∀ F, IsClosed F → (L.limsup fun i => μs i F) ≤ μ F) ↔
      ∀ G, IsOpen G → μ G ≤ L.liminf fun i => μs i G 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure Ω} {μs : ι → MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)]
Goal: (∀ (F : Set Ω), IsClosed F → Filter.limsup (fun i => (μs i) F) L ≤ μ F) ↔
  ∀ (G : Set Ω), IsOpen G → μ G ≤ Filter.liminf (fun i => (μs i) G) L
-- MeasureTheory.tendsto_measure_of_le_liminf_measure_of_limsup_measure_le
theorem tendsto_measure_of_le_liminf_measure_of_limsup_measure_le {ι : Type*} {L : Filter ι}
    {μ : Measure Ω} {μs : ι → Measure Ω} {E₀ E E₁ : Set Ω} (E₀_subset : E₀ ⊆ E) (subset_E₁ : E ⊆ E₁)
    (nulldiff : μ (E₁ \ E₀) = 0) (h_E₀ : μ E₀ ≤ L.liminf fun i => μs i E₀)
    (h_E₁ : (L.limsup fun i => μs i E₁) ≤ μ E₁) : L.Tendsto (fun i => μs i E) (𝓝 (μ E)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure Ω} {μs : ι → MeasureTheory.Measure Ω} {E₀ E E₁ : Set Ω} (E₀_subset : E₀ ⊆ E) (subset_E₁ : E ⊆ E₁) (nulldiff : μ (E₁ \ E₀) = 0) (h_E₀ : μ E₀ ≤ Filter.liminf (fun i => (μs i) E₀) L) (h_E₁ : Filter.limsup (fun i => (μs i) E₁) L ≤ μ E₁)
Goal: Filter.Tendsto (fun i => (μs i) E) L (𝓝 (μ E))
-- MeasureTheory.tendsto_measure_of_null_frontier
theorem tendsto_measure_of_null_frontier {ι : Type*} {L : Filter ι} {μ : Measure Ω}
    {μs : ι → Measure Ω} [IsProbabilityMeasure μ] [∀ i, IsProbabilityMeasure (μs i)]
    (h_opens : ∀ G, IsOpen G → μ G ≤ L.liminf fun i => μs i G) {E : Set Ω}
    (E_nullbdry : μ (frontier E) = 0) : L.Tendsto (fun i => μs i E) (𝓝 (μ E)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure Ω} {μs : ι → MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] (h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ Filter.liminf (fun i => (μs i) G) L) {E : Set Ω} (E_nullbdry : μ (frontier E) = 0)
Goal: Filter.Tendsto (fun i => (μs i) E) L (𝓝 (μ E))
-- MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto
theorem FiniteMeasure.limsup_measure_closed_le_of_tendsto {Ω ι : Type*} {L : Filter ι}
    [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω]
    [OpensMeasurableSpace Ω] {μ : FiniteMeasure Ω}
    {μs : ι → FiniteMeasure Ω} (μs_lim : Tendsto μs L (𝓝 μ)) {F : Set Ω} (F_closed : IsClosed F) :
    (L.limsup fun i => (μs i : Measure Ω) F) ≤ (μ : Measure Ω) F 
Arguments: {Ω : Type u_1} {ι : Type u_2} {L : Filter ι} [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω] [OpensMeasurableSpace Ω] {μ : MeasureTheory.FiniteMeasure Ω} {μs : ι → MeasureTheory.FiniteMeasure Ω} (μs_lim : Filter.Tendsto μs L (𝓝 μ)) {F : Set Ω} (F_closed : IsClosed F)
Goal: Filter.limsup (fun i => ↑(μs i) F) L ≤ ↑μ F
-- MeasureTheory.ProbabilityMeasure.limsup_measure_closed_le_of_tendsto
theorem ProbabilityMeasure.limsup_measure_closed_le_of_tendsto {Ω ι : Type*} {L : Filter ι}
    [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω]
    {μ : ProbabilityMeasure Ω} {μs : ι → ProbabilityMeasure Ω} (μs_lim : Tendsto μs L (𝓝 μ))
    {F : Set Ω} (F_closed : IsClosed F) :
    (L.limsup fun i => (μs i : Measure Ω) F) ≤ (μ : Measure Ω) F 
Arguments: {Ω : Type u_1} {ι : Type u_2} {L : Filter ι} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω] {μ : MeasureTheory.ProbabilityMeasure Ω} {μs : ι → MeasureTheory.ProbabilityMeasure Ω} (μs_lim : Filter.Tendsto μs L (𝓝 μ)) {F : Set Ω} (F_closed : IsClosed F)
Goal: Filter.limsup (fun i => ↑(μs i) F) L ≤ ↑μ F
-- MeasureTheory.ProbabilityMeasure.le_liminf_measure_open_of_tendsto
theorem ProbabilityMeasure.le_liminf_measure_open_of_tendsto {Ω ι : Type*} {L : Filter ι}
    [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω]
    {μ : ProbabilityMeasure Ω} {μs : ι → ProbabilityMeasure Ω} (μs_lim : Tendsto μs L (𝓝 μ))
    {G : Set Ω} (G_open : IsOpen G) :
    (μ : Measure Ω) G ≤ L.liminf fun i => (μs i : Measure Ω) G 
Arguments: {Ω : Type u_1} {ι : Type u_2} {L : Filter ι} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω] {μ : MeasureTheory.ProbabilityMeasure Ω} {μs : ι → MeasureTheory.ProbabilityMeasure Ω} (μs_lim : Filter.Tendsto μs L (𝓝 μ)) {G : Set Ω} (G_open : IsOpen G)
Goal: ↑μ G ≤ Filter.liminf (fun i => ↑(μs i) G) L
-- MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto'
theorem ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto' {Ω ι : Type*}
    {L : Filter ι} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
    [HasOuterApproxClosed Ω] {μ : ProbabilityMeasure Ω} {μs : ι → ProbabilityMeasure Ω}
    (μs_lim : Tendsto μs L (𝓝 μ)) {E : Set Ω} (E_nullbdry : (μ : Measure Ω) (frontier E) = 0) :
    Tendsto (fun i => (μs i : Measure Ω) E) L (𝓝 ((μ : Measure Ω) E)) 
Arguments: {Ω : Type u_1} {ι : Type u_2} {L : Filter ι} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω] {μ : MeasureTheory.ProbabilityMeasure Ω} {μs : ι → MeasureTheory.ProbabilityMeasure Ω} (μs_lim : Filter.Tendsto μs L (𝓝 μ)) {E : Set Ω} (E_nullbdry : ↑μ (frontier E) = 0)
Goal: Filter.Tendsto (fun i => ↑(μs i) E) L (𝓝 (↑μ E))
-- MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto
theorem ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto {Ω ι : Type*} {L : Filter ι}
    [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω]
    {μ : ProbabilityMeasure Ω} {μs : ι → ProbabilityMeasure Ω} (μs_lim : Tendsto μs L (𝓝 μ))
    {E : Set Ω} (E_nullbdry : μ (frontier E) = 0) : Tendsto (fun i => μs i E) L (𝓝 (μ E)) 
Arguments: {Ω : Type u_1} {ι : Type u_2} {L : Filter ι} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω] {μ : MeasureTheory.ProbabilityMeasure Ω} {μs : ι → MeasureTheory.ProbabilityMeasure Ω} (μs_lim : Filter.Tendsto μs L (𝓝 μ)) {E : Set Ω} (E_nullbdry : μ (frontier E) = 0)
Goal: Filter.Tendsto (fun i => (μs i) E) L (𝓝 (μ E))
-- MeasureTheory.exists_null_frontier_thickening
theorem exists_null_frontier_thickening (μ : Measure Ω) [SFinite μ] (s : Set Ω) {a b : ℝ}
    (hab : a < b) : ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0 
Arguments: {Ω : Type u_1} [PseudoEMetricSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.SFinite μ] (s : Set Ω) {a b : ℝ} (hab : a < b)
Goal: ∃ r ∈ Set.Ioo a b, μ (frontier (Metric.thickening r s)) = 0
-- MeasureTheory.exists_null_frontiers_thickening
theorem exists_null_frontiers_thickening (μ : Measure Ω) [SFinite μ] (s : Set Ω) :
    ∃ rs : ℕ → ℝ,
      Tendsto rs atTop (𝓝 0) ∧ ∀ n, 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0 
Arguments: {Ω : Type u_1} [PseudoEMetricSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.SFinite μ] (s : Set Ω)
Goal: ∃ rs, Filter.Tendsto rs Filter.atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0
-- MeasureTheory.limsup_measure_closed_le_of_forall_tendsto_measure
lemma limsup_measure_closed_le_of_forall_tendsto_measure
    {Ω ι : Type*} {L : Filter ι} [NeBot L]
    [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
    {μ : Measure Ω} [IsFiniteMeasure μ] {μs : ι → Measure Ω}
    (h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 →
            Tendsto (fun i ↦ μs i E) L (𝓝 (μ E)))
    (F : Set Ω) (F_closed : IsClosed F) :
    L.limsup (fun i ↦ μs i F) ≤ μ F 
Arguments: {Ω : Type u_2} {ι : Type u_3} {L : Filter ι} [L.NeBot] [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {μs : ι → MeasureTheory.Measure Ω} (h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Filter.Tendsto (fun i => (μs i) E) L (𝓝 (μ E))) (F : Set Ω) (F_closed : IsClosed F)
Goal: Filter.limsup (fun i => (μs i) F) L ≤ μ F