ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoundedContinuousFunction.toLp_denseRange
theorem toLp_denseRange [μ.WeaklyRegular] [IsFiniteMeasure μ] :
    DenseRange (toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ) 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] (E : Type u_2) [NormedAddCommGroup E] (μ : MeasureTheory.Measure α) {p : ℝ≥0∞} [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (hp : p ≠ ⊤) (𝕜 : Type u_3) [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 E] [μ.WeaklyRegular] [MeasureTheory.IsFiniteMeasure μ]
Goal: DenseRange ⇑(BoundedContinuousFunction.toLp p μ 𝕜)
-- ContinuousMap.toLp_denseRange
theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] :
    DenseRange (toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] (E : Type u_2) [NormedAddCommGroup E] (μ : MeasureTheory.Measure α) {p : ℝ≥0∞} [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (hp : p ≠ ⊤) (𝕜 : Type u_3) [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 E] [CompactSpace α] [μ.WeaklyRegular] [MeasureTheory.IsFiniteMeasure μ]
Goal: DenseRange ⇑(ContinuousMap.toLp p μ 𝕜)
-- MeasureTheory.OuterMeasure.mono
structure OuterMeasure (α : Type*) where
    protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
Arguments: {α : Type u_2} (self : MeasureTheory.OuterMeasure α) {s₁ s₂ : Set α}
Goal: s₁ ⊆ s₂ → self.measureOf s₁ ≤ self.measureOf s₂
-- MeasureTheory.OuterMeasure.iUnion_nat
structure OuterMeasure (α : Type*) where
    protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
Arguments: {α : Type u_2} (self : MeasureTheory.OuterMeasure α) (s : ℕ → Set α)
Goal: Pairwise (Disjoint on s) → self.measureOf (⋃ i, s i) ≤ ∑' (i : ℕ), self.measureOf (s i)
-- MeasureTheory.OuterMeasure.empty
structure OuterMeasure (α : Type*) where
    protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
Arguments: {α : Type u_2} (self : MeasureTheory.OuterMeasure α)
Goal: self.measureOf ∅ = 0
-- MeasureTheory.OuterMeasureClass.measure_empty
class OuterMeasureClass (F : Type*) (α : outParam Type*) [FunLike F (Set α) ℝ≥0∞] : Prop where
  protected measure_empty (f : F) : f ∅ = 0
  protected measure_mono (f : F) {s t} : s ⊆ t → f s ≤ f t
  protected measure_iUnion_nat_le (f : F) (s : ℕ → Set α) : Pairwise (Disjoint on s) →
    f (⋃ i, s i) ≤ ∑' i, f (s i)
Arguments: {F : Type u_2} {α : outParam (Type u_3)} [FunLike F (Set α) ℝ≥0∞] [self : MeasureTheory.OuterMeasureClass F α] (f : F)
Goal: f ∅ = 0
-- MeasureTheory.OuterMeasureClass.measure_mono
class OuterMeasureClass (F : Type*) (α : outParam Type*) [FunLike F (Set α) ℝ≥0∞] : Prop where
  protected measure_empty (f : F) : f ∅ = 0
  protected measure_mono (f : F) {s t} : s ⊆ t → f s ≤ f t
  protected measure_iUnion_nat_le (f : F) (s : ℕ → Set α) : Pairwise (Disjoint on s) →
    f (⋃ i, s i) ≤ ∑' i, f (s i)
Arguments: {F : Type u_2} {α : outParam (Type u_3)} [FunLike F (Set α) ℝ≥0∞] [self : MeasureTheory.OuterMeasureClass F α] (f : F) {s t : Set α}
Goal: s ⊆ t → f s ≤ f t
-- MeasureTheory.OuterMeasureClass.measure_iUnion_nat_le
class OuterMeasureClass (F : Type*) (α : outParam Type*) [FunLike F (Set α) ℝ≥0∞] : Prop where
  protected measure_empty (f : F) : f ∅ = 0
  protected measure_mono (f : F) {s t} : s ⊆ t → f s ≤ f t
  protected measure_iUnion_nat_le (f : F) (s : ℕ → Set α) : Pairwise (Disjoint on s) →
    f (⋃ i, s i) ≤ ∑' i, f (s i)
Arguments: {F : Type u_2} {α : outParam (Type u_3)} [FunLike F (Set α) ℝ≥0∞] [self : MeasureTheory.OuterMeasureClass F α] (f : F) (s : ℕ → Set α)
Goal: Pairwise (Disjoint on s) → f (⋃ i, s i) ≤ ∑' (i : ℕ), f (s i)
-- MeasureTheory.OuterMeasure.measureOf_eq_coe
@[simp] theorem measureOf_eq_coe (m : OuterMeasure α) : m.measureOf = m 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α)
Goal: m.measureOf = ⇑m
-- MeasureTheory.unifTight_iff_ennreal
theorem unifTight_iff_ennreal {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) :
    UnifTight f p μ ↔ ∀ ⦃ε : ℝ≥0∞⦄, 0 < ε → ∃ s : Set α,
      μ s ≠ ∞ ∧ ∀ i, eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} [NormedAddCommGroup β]
Goal: ∀ {x : MeasurableSpace α} (f : ι → α → β) (p : ENNReal) (μ : MeasureTheory.Measure α),
  MeasureTheory.UnifTight f p μ ↔
    ∀ ⦃ε : ENNReal⦄, 0 < ε → ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ι), MeasureTheory.eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε
-- MeasureTheory.unifTight_iff_real
theorem unifTight_iff_real {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) :
    UnifTight f p μ ↔ ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ s : Set α,
      μ s ≠ ∞ ∧ ∀ i, eLpNorm (sᶜ.indicator (f i)) p μ ≤ .ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} [NormedAddCommGroup β]
Goal: ∀ {x : MeasurableSpace α} (f : ι → α → β) (p : ENNReal) (μ : MeasureTheory.Measure α),
  MeasureTheory.UnifTight f p μ ↔
    ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ι), MeasureTheory.eLpNorm (sᶜ.indicator (f i)) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.UnifTight.eventually_cofinite_indicator
theorem eventually_cofinite_indicator (hf : UnifTight f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∀ᶠ s in μ.cofinite.smallSets, ∀ i, eLpNorm (s.indicator (f i)) p μ ≤ ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ENNReal} (hf : MeasureTheory.UnifTight f p μ) {ε : ENNReal} (hε : ε ≠ 0)
Goal: ∀ᶠ (s : Set α) in μ.cofinite.smallSets, ∀ (i : ι), MeasureTheory.eLpNorm (s.indicator (f i)) p μ ≤ ε
-- MeasureTheory.UnifTight.exists_measurableSet_indicator
protected theorem exists_measurableSet_indicator (hf : UnifTight f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ s, MeasurableSet s ∧ μ s < ∞ ∧ ∀ i, eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ENNReal} (hf : MeasureTheory.UnifTight f p μ) {ε : ENNReal} (hε : ε ≠ 0)
Goal: ∃ s, MeasurableSet s ∧ μ s < ⊤ ∧ ∀ (i : ι), MeasureTheory.eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε
-- MeasureTheory.UnifTight.add
protected theorem add (hf : UnifTight f p μ) (hg : UnifTight g p μ)
    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :
    UnifTight (f + g) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : ι → α → β} {p : ENNReal} (hf : MeasureTheory.UnifTight f p μ) (hg : MeasureTheory.UnifTight g p μ) (hf_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (hg_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (g i) μ)
Goal: MeasureTheory.UnifTight (f + g) p μ
-- MeasureTheory.UnifTight.neg
protected theorem neg (hf : UnifTight f p μ) : UnifTight (-f) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ENNReal} (hf : MeasureTheory.UnifTight f p μ)
Goal: MeasureTheory.UnifTight (-f) p μ
-- MeasureTheory.UnifTight.sub
protected theorem sub (hf : UnifTight f p μ) (hg : UnifTight g p μ)
    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :
    UnifTight (f - g) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : ι → α → β} {p : ENNReal} (hf : MeasureTheory.UnifTight f p μ) (hg : MeasureTheory.UnifTight g p μ) (hf_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (hg_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (g i) μ)
Goal: MeasureTheory.UnifTight (f - g) p μ
-- MeasureTheory.UnifTight.aeeq
protected theorem aeeq (hf : UnifTight f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UnifTight g p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : ι → α → β} {p : ENNReal} (hf : MeasureTheory.UnifTight f p μ) (hfg : ∀ (n : ι), f n =ᵐ[μ] g n)
Goal: MeasureTheory.UnifTight g p μ
-- MeasureTheory.unifTight_congr_ae
theorem unifTight_congr_ae {g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UnifTight f p μ ↔ UnifTight g p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ENNReal} {g : ι → α → β} (hfg : ∀ (n : ι), f n =ᵐ[μ] g n)
Goal: MeasureTheory.UnifTight f p μ ↔ MeasureTheory.UnifTight g p μ
-- MeasureTheory.unifTight_const
theorem unifTight_const {g : α → β} (hp_ne_top : p ≠ ∞) (hg : Memℒp g p μ) :
    UnifTight (fun _ : ι => g) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ENNReal} {g : α → β} (hp_ne_top : p ≠ ⊤) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.UnifTight (fun x => g) p μ
-- MeasureTheory.unifTight_of_subsingleton
theorem unifTight_of_subsingleton [Subsingleton ι] (hp_top : p ≠ ∞)
    {f : ι → α → β} (hf : ∀ i, Memℒp (f i) p μ) : UnifTight f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ENNReal} [Subsingleton ι] (hp_top : p ≠ ⊤) {f : ι → α → β} (hf : ∀ (i : ι), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UnifTight f p μ
-- _private.Mathlib.MeasureTheory.Function.UnifTight.0.MeasureTheory.unifTight_fin
private theorem unifTight_fin (hp_top : p ≠ ∞) {n : ℕ} {f : Fin n → α → β}
    (hf : ∀ i, Memℒp (f i) p μ) : UnifTight f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ENNReal} (hp_top : p ≠ ⊤) {n : ℕ} {f : Fin n → α → β} (hf : ∀ (i : Fin n), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UnifTight f p μ
-- MeasureTheory.unifTight_finite
theorem unifTight_finite [Finite ι] (hp_top : p ≠ ∞) {f : ι → α → β}
    (hf : ∀ i, Memℒp (f i) p μ) : UnifTight f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ENNReal} [Finite ι] (hp_top : p ≠ ⊤) {f : ι → α → β} (hf : ∀ (i : ι), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UnifTight f p μ
-- _private.Mathlib.MeasureTheory.Function.UnifTight.0.MeasureTheory.unifTight_of_tendsto_Lp_zero
private theorem unifTight_of_tendsto_Lp_zero (hp' : p ≠ ∞) (hf : ∀ n, Memℒp (f n) p μ)
    (hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)) : UnifTight f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] {μ : MeasureTheory.Measure α} {p : ENNReal} {f : ℕ → α → β} (hp' : p ≠ ⊤) (hf : ∀ (n : ℕ), MeasureTheory.Memℒp (f n) p μ) (hf_tendsto : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n) p μ) Filter.atTop (nhds 0))
Goal: MeasureTheory.UnifTight f p μ
-- _private.Mathlib.MeasureTheory.Function.UnifTight.0.MeasureTheory.unifTight_of_tendsto_Lp
private theorem unifTight_of_tendsto_Lp (hp' : p ≠ ∞) (hf : ∀ n, Memℒp (f n) p μ)
    (hg : Memℒp g p μ) (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)) :
    UnifTight f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] {μ : MeasureTheory.Measure α} {p : ENNReal} {f : ℕ → α → β} {g : α → β} (hp' : p ≠ ⊤) (hf : ∀ (n : ℕ), MeasureTheory.Memℒp (f n) p μ) (hg : MeasureTheory.Memℒp g p μ) (hfg : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0))
Goal: MeasureTheory.UnifTight f p μ
-- _private.Mathlib.MeasureTheory.Function.UnifTight.0.MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas
private theorem tendsto_Lp_of_tendsto_ae_of_meas (hp : 1 ≤ p) (hp' : p ≠ ∞)
    {f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    (hg' : Memℒp g p μ) (hui : UnifIntegrable f p μ) (hut : UnifTight f p μ)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] {μ : MeasureTheory.Measure α} {p : ENNReal} (hp : 1 ≤ p) (hp' : p ≠ ⊤) {f : ℕ → α → β} {g : α → β} (hf : ∀ (n : ℕ), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hg' : MeasureTheory.Memℒp g p μ) (hui : MeasureTheory.UnifIntegrable f p μ) (hut : MeasureTheory.UnifTight f p μ) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (nhds (g x)))
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0)
-- _private.Mathlib.MeasureTheory.Function.UnifTight.0.MeasureTheory.ae_tendsto_ae_congr
private theorem ae_tendsto_ae_congr {f f' : ℕ → α → β} {g g' : α → β}
    (hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n) (hgg' : g =ᵐ[μ] g')
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    ∀ᵐ x ∂μ, Tendsto (fun n => f' n x) atTop (𝓝 (g' x)) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] {μ : MeasureTheory.Measure α} {f f' : ℕ → α → β} {g g' : α → β} (hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n) (hgg' : g =ᵐ[μ] g') (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (nhds (g x)))
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f' n x) Filter.atTop (nhds (g' x))
-- MeasureTheory.tendsto_Lp_of_tendsto_ae
theorem tendsto_Lp_of_tendsto_ae (hp : 1 ≤ p) (hp' : p ≠ ∞)
    {f : ℕ → α → β} {g : α → β} (haef : ∀ n, AEStronglyMeasurable (f n) μ)
    (hg' : Memℒp g p μ) (hui : UnifIntegrable f p μ) (hut : UnifTight f p μ)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] {μ : MeasureTheory.Measure α} {p : ENNReal} (hp : 1 ≤ p) (hp' : p ≠ ⊤) {f : ℕ → α → β} {g : α → β} (haef : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (hg' : MeasureTheory.Memℒp g p μ) (hui : MeasureTheory.UnifIntegrable f p μ) (hut : MeasureTheory.UnifTight f p μ) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (nhds (g x)))
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0)
-- MeasureTheory.tendsto_Lp_of_tendstoInMeasure
theorem tendsto_Lp_of_tendstoInMeasure (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : Memℒp g p μ)
    (hui : UnifIntegrable f p μ) (hut : UnifTight f p μ)
    (hfg : TendstoInMeasure μ f atTop g) : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] {μ : MeasureTheory.Measure α} {p : ENNReal} {f : ℕ → α → β} {g : α → β} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (hg : MeasureTheory.Memℒp g p μ) (hui : MeasureTheory.UnifIntegrable f p μ) (hut : MeasureTheory.UnifTight f p μ) (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g)
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0)
-- MeasureTheory.tendstoInMeasure_iff_tendsto_Lp
theorem tendstoInMeasure_iff_tendsto_Lp (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ n, Memℒp (f n) p μ) (hg : Memℒp g p μ) :
    TendstoInMeasure μ f atTop g ∧ UnifIntegrable f p μ ∧ UnifTight f p μ
      ↔ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] {μ : MeasureTheory.Measure α} {p : ENNReal} {f : ℕ → α → β} {g : α → β} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hf : ∀ (n : ℕ), MeasureTheory.Memℒp (f n) p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.TendstoInMeasure μ f Filter.atTop g ∧ MeasureTheory.UnifIntegrable f p μ ∧ MeasureTheory.UnifTight f p μ ↔
  Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0)
-- MeasureTheory.eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ
theorem eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q)
    (hf : AEStronglyMeasurable f μ) :
    eLpNorm' f p μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q) 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : α → E} {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.eLpNorm' f p μ ≤ MeasureTheory.eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)
-- MeasureTheory.eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ
theorem eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ {q : ℝ} (hq_pos : 0 < q) :
    eLpNorm' f q μ ≤ eLpNormEssSup f μ * μ Set.univ ^ (1 / q) 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : α → E} {q : ℝ} (hq_pos : 0 < q)
Goal: MeasureTheory.eLpNorm' f q μ ≤ MeasureTheory.eLpNormEssSup f μ * μ Set.univ ^ (1 / q)
-- MeasureTheory.eLpNorm_le_eLpNorm_mul_rpow_measure_univ
theorem eLpNorm_le_eLpNorm_mul_rpow_measure_univ {p q : ℝ≥0∞} (hpq : p ≤ q)
    (hf : AEStronglyMeasurable f μ) :
    eLpNorm f p μ ≤ eLpNorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal) 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : α → E} {p q : ℝ≥0∞} (hpq : p ≤ q) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal)
-- MeasureTheory.eLpNorm'_le_eLpNorm'_of_exponent_le
theorem eLpNorm'_le_eLpNorm'_of_exponent_le {p q : ℝ} (hp0_lt : 0 < p)
    (hpq : p ≤ q) (μ : Measure α) [IsProbabilityMeasure μ] (hf : AEStronglyMeasurable f μ) :
    eLpNorm' f p μ ≤ eLpNorm' f q μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {f : α → E} {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q) (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.eLpNorm' f p μ ≤ MeasureTheory.eLpNorm' f q μ
-- MeasureTheory.eLpNorm'_le_eLpNormEssSup
theorem eLpNorm'_le_eLpNormEssSup {q : ℝ} (hq_pos : 0 < q) [IsProbabilityMeasure μ] :
    eLpNorm' f q μ ≤ eLpNormEssSup f μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : α → E} {q : ℝ} (hq_pos : 0 < q) [MeasureTheory.IsProbabilityMeasure μ]
Goal: MeasureTheory.eLpNorm' f q μ ≤ MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.eLpNorm_le_eLpNorm_of_exponent_le
theorem eLpNorm_le_eLpNorm_of_exponent_le {p q : ℝ≥0∞} (hpq : p ≤ q) [IsProbabilityMeasure μ]
    (hf : AEStronglyMeasurable f μ) : eLpNorm f p μ ≤ eLpNorm f q μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : α → E} {p q : ℝ≥0∞} (hpq : p ≤ q) [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm f q μ
-- MeasureTheory.eLpNorm'_lt_top_of_eLpNorm'_lt_top_of_exponent_le
theorem eLpNorm'_lt_top_of_eLpNorm'_lt_top_of_exponent_le {p q : ℝ} [IsFiniteMeasure μ]
    (hf : AEStronglyMeasurable f μ) (hfq_lt_top : eLpNorm' f q μ < ∞) (hp_nonneg : 0 ≤ p)
    (hpq : p ≤ q) : eLpNorm' f p μ < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f : α → E} {p q : ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hfq_lt_top : MeasureTheory.eLpNorm' f q μ < ⊤) (hp_nonneg : 0 ≤ p) (hpq : p ≤ q)
Goal: MeasureTheory.eLpNorm' f p μ < ⊤
-- MeasureTheory.Memℒp.memℒp_of_exponent_le
theorem Memℒp.memℒp_of_exponent_le {p q : ℝ≥0∞} [IsFiniteMeasure μ] {f : α → E} (hfq : Memℒp f q μ)
    (hpq : p ≤ q) : Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p q : ℝ≥0∞} [MeasureTheory.IsFiniteMeasure μ] {f : α → E} (hfq : MeasureTheory.Memℒp f q μ) (hpq : p ≤ q)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.eLpNorm_le_eLpNorm_top_mul_eLpNorm
theorem eLpNorm_le_eLpNorm_top_mul_eLpNorm (p : ℝ≥0∞) (f : α → E) {g : α → F}
    (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f ∞ μ * eLpNorm g p μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {m : MeasurableSpace α} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] {μ : MeasureTheory.Measure α} (p : ℝ≥0∞) (f : α → E) {g : α → F} (hg : MeasureTheory.AEStronglyMeasurable g μ) (b : E → F → G) (h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊)
Goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f ⊤ μ * MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_top
theorem eLpNorm_le_eLpNorm_mul_eLpNorm_top (p : ℝ≥0∞) {f : α → E} (hf : AEStronglyMeasurable f μ)
    (g : α → F) (b : E → F → G) (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f p μ * eLpNorm g ∞ μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {m : MeasurableSpace α} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] {μ : MeasureTheory.Measure α} (p : ℝ≥0∞) {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (g : α → F) (b : E → F → G) (h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊)
Goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f p μ * MeasureTheory.eLpNorm g ⊤ μ
-- MeasureTheory.eLpNorm'_le_eLpNorm'_mul_eLpNorm'
theorem eLpNorm'_le_eLpNorm'_mul_eLpNorm' {p q r : ℝ} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hp0_lt : 0 < p) (hpq : p < q)
    (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm' (fun x => b (f x) (g x)) p μ ≤ eLpNorm' f q μ * eLpNorm' g r μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {m : MeasurableSpace α} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] {μ : MeasureTheory.Measure α} {f : α → E} {g : α → F} {p q r : ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (b : E → F → G) (h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hp0_lt : 0 < p) (hpq : p < q) (hpqr : 1 / p = 1 / q + 1 / r)
Goal: MeasureTheory.eLpNorm' (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm' f q μ * MeasureTheory.eLpNorm' g r μ
-- MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_nnnorm
theorem eLpNorm_le_eLpNorm_mul_eLpNorm_of_nnnorm {p q r : ℝ≥0∞}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f q μ * eLpNorm g r μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {m : MeasurableSpace α} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] {μ : MeasureTheory.Measure α} {f : α → E} {g : α → F} {p q r : ℝ≥0∞} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (b : E → F → G) (h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hpqr : 1 / p = 1 / q + 1 / r)
Goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f q μ * MeasureTheory.eLpNorm g r μ