ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.volume_preserving_transvectionStruct
theorem volume_preserving_transvectionStruct [DecidableEq ι] (t : TransvectionStruct ι ℝ) :
    MeasurePreserving (toLin' t.toMatrix) 
Arguments: {ι : Type u_1} [Fintype ι] [DecidableEq ι] (t : Matrix.TransvectionStruct ι ℝ)
Goal: MeasureTheory.MeasurePreserving (⇑(Matrix.toLin' t.toMatrix)) MeasureTheory.volume MeasureTheory.volume
-- Real.map_matrix_volume_pi_eq_smul_volume_pi
theorem map_matrix_volume_pi_eq_smul_volume_pi [DecidableEq ι] {M : Matrix ι ι ℝ} (hM : det M ≠ 0) :
    Measure.map (toLin' M) volume = ENNReal.ofReal (abs (det M)⁻¹) • volume 
Arguments: {ι : Type u_1} [Fintype ι] [DecidableEq ι] {M : Matrix ι ι ℝ} (hM : M.det ≠ 0)
Goal: MeasureTheory.Measure.map (⇑(Matrix.toLin' M)) MeasureTheory.volume = ENNReal.ofReal |M.det⁻¹| • MeasureTheory.volume
-- Real.map_linearMap_volume_pi_eq_smul_volume_pi
theorem map_linearMap_volume_pi_eq_smul_volume_pi {f : (ι → ℝ) →ₗ[ℝ] ι → ℝ}
    (hf : LinearMap.det f ≠ 0) : Measure.map f volume =
      ENNReal.ofReal (abs (LinearMap.det f)⁻¹) • volume 
Arguments: {ι : Type u_1} [Fintype ι] {f : (ι → ℝ) →ₗ[ℝ] ι → ℝ} (hf : LinearMap.det f ≠ 0)
Goal: MeasureTheory.Measure.map (⇑f) MeasureTheory.volume = ENNReal.ofReal |(LinearMap.det f)⁻¹| • MeasureTheory.volume
-- regionBetween_subset
theorem regionBetween_subset (f g : α → ℝ) (s : Set α) : regionBetween f g s ⊆ s ×ˢ univ 
Arguments: {α : Type u_1} (f g : α → ℝ) (s : Set α)
Goal: regionBetween f g s ⊆ s ×ˢ Set.univ
-- measurableSet_regionBetween
theorem measurableSet_regionBetween (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s) :
    MeasurableSet (regionBetween f g s) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f g : α → ℝ} {s : Set α} (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s)
Goal: MeasurableSet (regionBetween f g s)
-- measurableSet_region_between_oc
theorem measurableSet_region_between_oc (hf : Measurable f) (hg : Measurable g)
    (hs : MeasurableSet s) :
    MeasurableSet { p : α × ℝ | p.fst ∈ s ∧ p.snd ∈ Ioc (f p.fst) (g p.fst) } 
Arguments: {α : Type u_1} [MeasurableSpace α] {f g : α → ℝ} {s : Set α} (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s)
Goal: MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Set.Ioc (f p.1) (g p.1)}
-- measurableSet_region_between_co
theorem measurableSet_region_between_co (hf : Measurable f) (hg : Measurable g)
    (hs : MeasurableSet s) :
    MeasurableSet { p : α × ℝ | p.fst ∈ s ∧ p.snd ∈ Ico (f p.fst) (g p.fst) } 
Arguments: {α : Type u_1} [MeasurableSpace α] {f g : α → ℝ} {s : Set α} (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s)
Goal: MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Set.Ico (f p.1) (g p.1)}
-- measurableSet_region_between_cc
theorem measurableSet_region_between_cc (hf : Measurable f) (hg : Measurable g)
    (hs : MeasurableSet s) :
    MeasurableSet { p : α × ℝ | p.fst ∈ s ∧ p.snd ∈ Icc (f p.fst) (g p.fst) } 
Arguments: {α : Type u_1} [MeasurableSpace α] {f g : α → ℝ} {s : Set α} (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s)
Goal: MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Set.Icc (f p.1) (g p.1)}
-- measurableSet_graph
theorem measurableSet_graph (hf : Measurable f) :
    MeasurableSet { p : α × ℝ | p.snd = f p.fst } 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (hf : Measurable f)
Goal: MeasurableSet {p | p.2 = f p.1}
-- volume_regionBetween_eq_lintegral'
theorem volume_regionBetween_eq_lintegral' (hf : Measurable f) (hg : Measurable g)
    (hs : MeasurableSet s) :
    μ.prod volume (regionBetween f g s) = ∫⁻ y in s, ENNReal.ofReal ((g - f) y) ∂μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α → ℝ} {s : Set α} (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s)
Goal: (μ.prod MeasureTheory.volume) (regionBetween f g s) = ∫⁻ (y : α) in s, ENNReal.ofReal ((g - f) y) ∂μ
-- volume_regionBetween_eq_lintegral
theorem volume_regionBetween_eq_lintegral [SFinite μ] (hf : AEMeasurable f (μ.restrict s))
    (hg : AEMeasurable g (μ.restrict s)) (hs : MeasurableSet s) :
    μ.prod volume (regionBetween f g s) = ∫⁻ y in s, ENNReal.ofReal ((g - f) y) ∂μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α → ℝ} {s : Set α} [MeasureTheory.SFinite μ] (hf : AEMeasurable f (μ.restrict s)) (hg : AEMeasurable g (μ.restrict s)) (hs : MeasurableSet s)
Goal: (μ.prod MeasureTheory.volume) (regionBetween f g s) = ∫⁻ (y : α) in s, ENNReal.ofReal ((g - f) y) ∂μ
-- nullMeasurableSet_regionBetween
lemma nullMeasurableSet_regionBetween (μ : Measure α)
    {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ)
    {s : Set α} (s_mble : NullMeasurableSet s μ) :
    NullMeasurableSet {p : α × ℝ | p.1 ∈ s ∧ p.snd ∈ Ioo (f p.fst) (g p.fst)} (μ.prod volume) 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ) {s : Set α} (s_mble : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Set.Ioo (f p.1) (g p.1)} (μ.prod MeasureTheory.volume)
-- nullMeasurableSet_region_between_oc
lemma nullMeasurableSet_region_between_oc (μ : Measure α)
    {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ)
    {s : Set α} (s_mble : NullMeasurableSet s μ) :
    NullMeasurableSet {p : α × ℝ | p.1 ∈ s ∧ p.snd ∈ Ioc (f p.fst) (g p.fst)} (μ.prod volume) 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ) {s : Set α} (s_mble : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Set.Ioc (f p.1) (g p.1)} (μ.prod MeasureTheory.volume)
-- nullMeasurableSet_region_between_co
lemma nullMeasurableSet_region_between_co (μ : Measure α)
    {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ)
    {s : Set α} (s_mble : NullMeasurableSet s μ) :
    NullMeasurableSet {p : α × ℝ | p.1 ∈ s ∧ p.snd ∈ Ico (f p.fst) (g p.fst)} (μ.prod volume) 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ) {s : Set α} (s_mble : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Set.Ico (f p.1) (g p.1)} (μ.prod MeasureTheory.volume)
-- nullMeasurableSet_region_between_cc
lemma nullMeasurableSet_region_between_cc (μ : Measure α)
    {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ)
    {s : Set α} (s_mble : NullMeasurableSet s μ) :
    NullMeasurableSet {p : α × ℝ | p.1 ∈ s ∧ p.snd ∈ Icc (f p.fst) (g p.fst)} (μ.prod volume) 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ) {s : Set α} (s_mble : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Set.Icc (f p.1) (g p.1)} (μ.prod MeasureTheory.volume)
-- ae_restrict_of_ae_restrict_inter_Ioo
theorem ae_restrict_of_ae_restrict_inter_Ioo {μ : Measure ℝ} [NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop}
    (h : ∀ a b, a ∈ s → b ∈ s → a < b → ∀ᵐ x ∂μ.restrict (s ∩ Ioo a b), p x) :
    ∀ᵐ x ∂μ.restrict s, p x 
Arguments: {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop} (h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Set.Ioo a b), p x)
Goal: ∀ᵐ (x : ℝ) ∂μ.restrict s, p x
-- ae_of_mem_of_ae_of_mem_inter_Ioo
theorem ae_of_mem_of_ae_of_mem_inter_Ioo {μ : Measure ℝ} [NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop}
    (h : ∀ a b, a ∈ s → b ∈ s → a < b → ∀ᵐ x ∂μ, x ∈ s ∩ Ioo a b → p x) :
    ∀ᵐ x ∂μ, x ∈ s → p x 
Arguments: {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop} (h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Set.Ioo a b → p x)
Goal: ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x
-- MeasureTheory.SimpleFunc.nnnorm_approxOn_le
theorem nnnorm_approxOn_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
    {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
    ‖approxOn f hf s y₀ h₀ n x - f x‖₊ ≤ ‖f x - y₀‖₊ 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (x : β) (n : ℕ)
Goal: ‖↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊
-- MeasureTheory.SimpleFunc.norm_approxOn_y₀_le
theorem norm_approxOn_y₀_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
    {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
    ‖approxOn f hf s y₀ h₀ n x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (x : β) (n : ℕ)
Goal: ‖↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖
-- MeasureTheory.SimpleFunc.norm_approxOn_zero_le
theorem norm_approxOn_zero_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
    (h₀ : (0 : E) ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
    ‖approxOn f hf s 0 h₀ n x‖ ≤ ‖f x‖ + ‖f x‖ 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E} (h₀ : 0 ∈ s) [TopologicalSpace.SeparableSpace ↑s] (x : β) (n : ℕ)
Goal: ‖↑(MeasureTheory.SimpleFunc.approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖
-- MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_eLpNorm
theorem tendsto_approxOn_Lp_eLpNorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
    {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
    (hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : eLpNorm (fun x => f x - y₀) p μ < ∞) :
    Tendsto (fun n => eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0) 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] {p : ℝ≥0∞} [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (hp_ne_top : p ≠ ⊤) {μ : MeasureTheory.Measure β} (hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s) (hi : MeasureTheory.eLpNorm (fun x => f x - y₀) p μ < ⊤)
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) - f) p μ)
  Filter.atTop (𝓝 0)
-- MeasureTheory.SimpleFunc.memℒp_approxOn
theorem memℒp_approxOn [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
    (hf : Memℒp f p μ) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s]
    (hi₀ : Memℒp (fun _ => y₀) p μ) (n : ℕ) : Memℒp (approxOn f fmeas s y₀ h₀ n) p μ 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] {p : ℝ≥0∞} [BorelSpace E] {f : β → E} {μ : MeasureTheory.Measure β} (fmeas : Measurable f) (hf : MeasureTheory.Memℒp f p μ) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (hi₀ : MeasureTheory.Memℒp (fun x => y₀) p μ) (n : ℕ)
Goal: MeasureTheory.Memℒp (↑(MeasureTheory.SimpleFunc.approxOn f fmeas s y₀ h₀ n)) p μ
-- MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_eLpNorm
theorem tendsto_approxOn_range_Lp_eLpNorm [BorelSpace E] {f : β → E} (hp_ne_top : p ≠ ∞)
    {μ : Measure β} (fmeas : Measurable f) [SeparableSpace (range f ∪ {0} : Set E)]
    (hf : eLpNorm f p μ < ∞) :
    Tendsto (fun n => eLpNorm (⇑(approxOn f fmeas (range f ∪ {0}) 0 (by simp) n) - f) p μ)
      atTop (𝓝 0) 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] {p : ℝ≥0∞} [BorelSpace E] {f : β → E} (hp_ne_top : p ≠ ⊤) {μ : MeasureTheory.Measure β} (fmeas : Measurable f) [TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})] (hf : MeasureTheory.eLpNorm f p μ < ⊤)
Goal: Filter.Tendsto
  (fun n => MeasureTheory.eLpNorm (↑(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) - f) p μ)
  Filter.atTop (𝓝 0)
-- MeasureTheory.SimpleFunc.memℒp_approxOn_range
theorem memℒp_approxOn_range [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
    [SeparableSpace (range f ∪ {0} : Set E)] (hf : Memℒp f p μ) (n : ℕ) :
    Memℒp (approxOn f fmeas (range f ∪ {0}) 0 (by simp) n) p μ 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] {p : ℝ≥0∞} [BorelSpace E] {f : β → E} {μ : MeasureTheory.Measure β} (fmeas : Measurable f) [TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})] (hf : MeasureTheory.Memℒp f p μ) (n : ℕ)
Goal: MeasureTheory.Memℒp (↑(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n)) p μ
-- MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp
theorem tendsto_approxOn_range_Lp [BorelSpace E] {f : β → E} [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞)
    {μ : Measure β} (fmeas : Measurable f) [SeparableSpace (range f ∪ {0} : Set E)]
    (hf : Memℒp f p μ) :
    Tendsto
      (fun n =>
        (memℒp_approxOn_range fmeas hf n).toLp (approxOn f fmeas (range f ∪ {0}) 0 (by simp) n))
      atTop (𝓝 (hf.toLp f)) 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] {p : ℝ≥0∞} [BorelSpace E] {f : β → E} [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤) {μ : MeasureTheory.Measure β} (fmeas : Measurable f) [TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})] (hf : MeasureTheory.Memℒp f p μ)
Goal: Filter.Tendsto
  (fun n => MeasureTheory.Memℒp.toLp ↑(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) ⋯)
  Filter.atTop (𝓝 (MeasureTheory.Memℒp.toLp f hf))
-- MeasureTheory.Memℒp.exists_simpleFunc_eLpNorm_sub_lt
theorem _root_.MeasureTheory.Memℒp.exists_simpleFunc_eLpNorm_sub_lt {E : Type*}
    [NormedAddCommGroup E] {f : β → E} {μ : Measure β} (hf : Memℒp f p μ) (hp_ne_top : p ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ g : β →ₛ E, eLpNorm (f - ⇑g) p μ < ε ∧ Memℒp g p μ 
Arguments: {β : Type u_2} [MeasurableSpace β] {p : ℝ≥0∞} {E : Type u_7} [NormedAddCommGroup E] {f : β → E} {μ : MeasureTheory.Measure β} (hf : MeasureTheory.Memℒp f p μ) (hp_ne_top : p ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ g, MeasureTheory.eLpNorm (f - ↑g) p μ < ε ∧ MeasureTheory.Memℒp (↑g) p μ
-- MeasureTheory.SimpleFunc.tendsto_approxOn_L1_nnnorm
theorem tendsto_approxOn_L1_nnnorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
    {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] {μ : Measure β}
    (hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : HasFiniteIntegral (fun x => f x - y₀) μ) :
    Tendsto (fun n => ∫⁻ x, ‖approxOn f hf s y₀ h₀ n x - f x‖₊ ∂μ) atTop (𝓝 0) 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] {μ : MeasureTheory.Measure β} (hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s) (hi : MeasureTheory.HasFiniteIntegral (fun x => f x - y₀) μ)
Goal: Filter.Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x - f x‖₊ ∂μ) Filter.atTop
  (𝓝 0)
-- MeasureTheory.SimpleFunc.integrable_approxOn
theorem integrable_approxOn [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
    (hf : Integrable f μ) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s]
    (hi₀ : Integrable (fun _ => y₀) μ) (n : ℕ) : Integrable (approxOn f fmeas s y₀ h₀ n) μ 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] [BorelSpace E] {f : β → E} {μ : MeasureTheory.Measure β} (fmeas : Measurable f) (hf : MeasureTheory.Integrable f μ) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (hi₀ : MeasureTheory.Integrable (fun x => y₀) μ) (n : ℕ)
Goal: MeasureTheory.Integrable (↑(MeasureTheory.SimpleFunc.approxOn f fmeas s y₀ h₀ n)) μ
-- MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_nnnorm
theorem tendsto_approxOn_range_L1_nnnorm [OpensMeasurableSpace E] {f : β → E} {μ : Measure β}
    [SeparableSpace (range f ∪ {0} : Set E)] (fmeas : Measurable f) (hf : Integrable f μ) :
    Tendsto (fun n => ∫⁻ x, ‖approxOn f fmeas (range f ∪ {0}) 0 (by simp) n x - f x‖₊ ∂μ) atTop
      (𝓝 0) 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] [OpensMeasurableSpace E] {f : β → E} {μ : MeasureTheory.Measure β} [TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})] (fmeas : Measurable f) (hf : MeasureTheory.Integrable f μ)
Goal: Filter.Tendsto
  (fun n => ∫⁻ (x : β), ↑‖↑(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) x - f x‖₊ ∂μ)
  Filter.atTop (𝓝 0)
-- MeasureTheory.SimpleFunc.integrable_approxOn_range
theorem integrable_approxOn_range [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
    [SeparableSpace (range f ∪ {0} : Set E)] (hf : Integrable f μ) (n : ℕ) :
    Integrable (approxOn f fmeas (range f ∪ {0}) 0 (by simp) n) μ 
Arguments: {β : Type u_2} {E : Type u_4} [MeasurableSpace β] [MeasurableSpace E] [NormedAddCommGroup E] [BorelSpace E] {f : β → E} {μ : MeasureTheory.Measure β} (fmeas : Measurable f) [TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})] (hf : MeasureTheory.Integrable f μ) (n : ℕ)
Goal: MeasureTheory.Integrable (↑(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n)) μ
-- MeasureTheory.SimpleFunc.exists_forall_norm_le
theorem exists_forall_norm_le (f : α →ₛ F) : ∃ C, ∀ x, ‖f x‖ ≤ C 
Arguments: {α : Type u_1} {F : Type u_5} [MeasurableSpace α] [NormedAddCommGroup F] (f : MeasureTheory.SimpleFunc α F)
Goal: ∃ C, ∀ (x : α), ‖↑f x‖ ≤ C
-- MeasureTheory.SimpleFunc.memℒp_zero
theorem memℒp_zero (f : α →ₛ E) (μ : Measure α) : Memℒp f 0 μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] (f : MeasureTheory.SimpleFunc α E) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Memℒp (↑f) 0 μ
-- MeasureTheory.SimpleFunc.memℒp_top
theorem memℒp_top (f : α →ₛ E) (μ : Measure α) : Memℒp f ∞ μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] (f : MeasureTheory.SimpleFunc α E) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Memℒp ↑f ⊤ μ
-- MeasureTheory.SimpleFunc.eLpNorm'_eq
protected theorem eLpNorm'_eq {p : ℝ} (f : α →ₛ F) (μ : Measure α) :
    eLpNorm' f p μ = (∑ y ∈ f.range, (‖y‖₊ : ℝ≥0∞) ^ p * μ (f ⁻¹' {y})) ^ (1 / p) 
Arguments: {α : Type u_1} {F : Type u_5} [MeasurableSpace α] [NormedAddCommGroup F] {p : ℝ} (f : MeasureTheory.SimpleFunc α F) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.eLpNorm' (↑f) p μ = (∑ y ∈ f.range, ↑‖y‖₊ ^ p * μ (↑f ⁻¹' {y})) ^ (1 / p)
-- MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_memℒp
theorem measure_preimage_lt_top_of_memℒp (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) (f : α →ₛ E)
    (hf : Memℒp f p μ) (y : E) (hy_ne : y ≠ 0) : μ (f ⁻¹' {y}) < ∞ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ⊤) (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ) (y : E) (hy_ne : y ≠ 0)
Goal: μ (↑f ⁻¹' {y}) < ⊤
-- MeasureTheory.SimpleFunc.memℒp_of_finite_measure_preimage
theorem memℒp_of_finite_measure_preimage (p : ℝ≥0∞) {f : α →ₛ E}
    (hf : ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞) : Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} (p : ℝ≥0∞) {f : MeasureTheory.SimpleFunc α E} (hf : ∀ (y : E), y ≠ 0 → μ (↑f ⁻¹' {y}) < ⊤)
Goal: MeasureTheory.Memℒp (↑f) p μ
-- MeasureTheory.SimpleFunc.memℒp_iff
theorem memℒp_iff {f : α →ₛ E} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    Memℒp f p μ ↔ ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} {f : MeasureTheory.SimpleFunc α E} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.Memℒp (↑f) p μ ↔ ∀ (y : E), y ≠ 0 → μ (↑f ⁻¹' {y}) < ⊤
-- MeasureTheory.SimpleFunc.integrable_iff
theorem integrable_iff {f : α →ₛ E} : Integrable f μ ↔ ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α E}
Goal: MeasureTheory.Integrable (↑f) μ ↔ ∀ (y : E), y ≠ 0 → μ (↑f ⁻¹' {y}) < ⊤
-- MeasureTheory.SimpleFunc.memℒp_iff_integrable
theorem memℒp_iff_integrable {f : α →ₛ E} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    Memℒp f p μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} {f : MeasureTheory.SimpleFunc α E} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.Memℒp (↑f) p μ ↔ MeasureTheory.Integrable (↑f) μ
-- MeasureTheory.SimpleFunc.memℒp_iff_finMeasSupp
theorem memℒp_iff_finMeasSupp {f : α →ₛ E} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    Memℒp f p μ ↔ f.FinMeasSupp μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} {f : MeasureTheory.SimpleFunc α E} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.Memℒp (↑f) p μ ↔ f.FinMeasSupp μ
-- MeasureTheory.SimpleFunc.integrable_iff_finMeasSupp
theorem integrable_iff_finMeasSupp {f : α →ₛ E} : Integrable f μ ↔ f.FinMeasSupp μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α E}
Goal: MeasureTheory.Integrable (↑f) μ ↔ f.FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.integrable
theorem FinMeasSupp.integrable {f : α →ₛ E} (h : f.FinMeasSupp μ) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α E} (h : f.FinMeasSupp μ)
Goal: MeasureTheory.Integrable (↑f) μ
-- MeasureTheory.SimpleFunc.integrable_pair
theorem integrable_pair {f : α →ₛ E} {g : α →ₛ F} :
    Integrable f μ → Integrable g μ → Integrable (pair f g) μ 
Arguments: {α : Type u_1} {E : Type u_4} {F : Type u_5} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α E} {g : MeasureTheory.SimpleFunc α F}
Goal: MeasureTheory.Integrable (↑f) μ → MeasureTheory.Integrable (↑g) μ → MeasureTheory.Integrable (↑(f.pair g)) μ
-- MeasureTheory.SimpleFunc.memℒp_of_isFiniteMeasure
theorem memℒp_of_isFiniteMeasure (f : α →ₛ E) (p : ℝ≥0∞) (μ : Measure α) [IsFiniteMeasure μ] :
    Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] (f : MeasureTheory.SimpleFunc α E) (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ]
Goal: MeasureTheory.Memℒp (↑f) p μ
-- MeasureTheory.SimpleFunc.integrable_of_isFiniteMeasure
theorem integrable_of_isFiniteMeasure [IsFiniteMeasure μ] (f : α →ₛ E) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (f : MeasureTheory.SimpleFunc α E)
Goal: MeasureTheory.Integrable (↑f) μ
-- MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_integrable
theorem measure_preimage_lt_top_of_integrable (f : α →ₛ E) (hf : Integrable f μ) {x : E}
    (hx : x ≠ 0) : μ (f ⁻¹' {x}) < ∞ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ) {x : E} (hx : x ≠ 0)
Goal: μ (↑f ⁻¹' {x}) < ⊤
-- MeasureTheory.SimpleFunc.measure_support_lt_top
theorem measure_support_lt_top [Zero β] (f : α →ₛ β) (hf : ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞) :
    μ (support f) < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Zero β] (f : MeasureTheory.SimpleFunc α β) (hf : ∀ (y : β), y ≠ 0 → μ (↑f ⁻¹' {y}) < ⊤)
Goal: μ (Function.support ↑f) < ⊤