ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.eLpNorm_indicator_le_of_bound
theorem eLpNorm_indicator_le_of_bound {f : α → β} (hp_top : p ≠ ∞) {ε : ℝ} (hε : 0 < ε) {M : ℝ}
    (hf : ∀ x, ‖f x‖ < M) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s →
      μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : α → β} (hp_top : p ≠ ⊤) {ε : ℝ} (hε : 0 < ε) {M : ℝ} (hf : ∀ (x : α), ‖f x‖ < M)
Goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.Memℒp.eLpNorm_indicator_le'
theorem Memℒp.eLpNorm_indicator_le' (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (hf : Memℒp f p μ)
    (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s → μ s ≤ ENNReal.ofReal δ →
      eLpNorm (s.indicator f) p μ ≤ 2 * ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : α → β} (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) (hf : MeasureTheory.Memℒp f p μ) (hmeas : MeasureTheory.StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ 2 * ENNReal.ofReal ε
-- MeasureTheory.Memℒp.eLpNorm_indicator_le_of_meas
theorem Memℒp.eLpNorm_indicator_le_of_meas (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (hf : Memℒp f p μ)
    (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s → μ s ≤ ENNReal.ofReal δ →
      eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : α → β} (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) (hf : MeasureTheory.Memℒp f p μ) (hmeas : MeasureTheory.StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.Memℒp.eLpNorm_indicator_le
theorem Memℒp.eLpNorm_indicator_le (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (hf : Memℒp f p μ) {ε : ℝ}
    (hε : 0 < ε) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s → μ s ≤ ENNReal.ofReal δ →
      eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : α → β} (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) (hf : MeasureTheory.Memℒp f p μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.unifIntegrable_const
theorem unifIntegrable_const {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ∞) (hg : Memℒp g p μ) :
    UnifIntegrable (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 : ℝ≥0∞} {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ⊤) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.UnifIntegrable (fun x => g) p μ
-- MeasureTheory.unifIntegrable_subsingleton
theorem unifIntegrable_subsingleton [Subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
    {f : ι → α → β} (hf : ∀ i, Memℒp (f i) p μ) : UnifIntegrable 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 : ℝ≥0∞} [Subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) {f : ι → α → β} (hf : ∀ (i : ι), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.unifIntegrable_fin
theorem unifIntegrable_fin (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {n : ℕ} {f : Fin n → α → β}
    (hf : ∀ i, Memℒp (f i) p μ) : UnifIntegrable f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) {n : ℕ} {f : Fin n → α → β} (hf : ∀ (i : Fin n), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.unifIntegrable_finite
theorem unifIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {f : ι → α → β}
    (hf : ∀ i, Memℒp (f i) p μ) : UnifIntegrable 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 : ℝ≥0∞} [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) {f : ι → α → β} (hf : ∀ (i : ι), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.eLpNorm_sub_le_of_dist_bdd
theorem eLpNorm_sub_le_of_dist_bdd (μ : Measure α)
    {p : ℝ≥0∞} (hp' : p ≠ ∞) {s : Set α} (hs : MeasurableSet[m] s)
    {f g : α → β} {c : ℝ} (hc : 0 ≤ c) (hf : ∀ x ∈ s, dist (f x) (g x) ≤ c) :
    eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] (μ : MeasureTheory.Measure α) {p : ℝ≥0∞} (hp' : p ≠ ⊤) {s : Set α} (hs : MeasurableSet s) {f g : α → β} {c : ℝ} (hc : 0 ≤ c) (hf : ∀ x ∈ s, dist (f x) (g x) ≤ c)
Goal: MeasureTheory.eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal)
-- MeasureTheory.tendsto_Lp_finite_of_tendsto_ae_of_meas
theorem tendsto_Lp_finite_of_tendsto_ae_of_meas [IsFiniteMeasure μ] (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)
    (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} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} [MeasureTheory.IsFiniteMeasure μ] (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) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)
-- MeasureTheory.tendsto_Lp_finite_of_tendsto_ae
theorem tendsto_Lp_finite_of_tendsto_ae [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    {f : ℕ → α → β} {g : α → β} (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : Memℒp g p μ)
    (hui : UnifIntegrable 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} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} [MeasureTheory.IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ⊤) {f : ℕ → α → β} {g : α → β} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (hg : MeasureTheory.Memℒp g p μ) (hui : MeasureTheory.UnifIntegrable f p μ) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)
-- MeasureTheory.unifIntegrable_of_tendsto_Lp_zero
theorem unifIntegrable_of_tendsto_Lp_zero (hp : 1 ≤ p) (hp' : p ≠ ∞) (hf : ∀ n, Memℒp (f n) p μ)
    (hf_tendsto : Tendsto (fun n => eLpNorm (f n) p μ) atTop (𝓝 0)) : UnifIntegrable f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : ℕ → α → β} (hp : 1 ≤ p) (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 (𝓝 0))
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.unifIntegrable_of_tendsto_Lp
theorem unifIntegrable_of_tendsto_Lp (hp : 1 ≤ p) (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)) :
    UnifIntegrable f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {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) (hfg : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0))
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.tendsto_Lp_finite_of_tendstoInMeasure
theorem tendsto_Lp_finite_of_tendstoInMeasure [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : Memℒp g p μ) (hui : UnifIntegrable 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} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : ℕ → α → β} {g : α → β} [MeasureTheory.IsFiniteMeasure μ] (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) (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g)
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)
-- MeasureTheory.tendstoInMeasure_iff_tendsto_Lp_finite
theorem tendstoInMeasure_iff_tendsto_Lp_finite [IsFiniteMeasure μ] (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 ↔
      Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : ℕ → α → β} {g : α → β} [MeasureTheory.IsFiniteMeasure μ] (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 ↔
  Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)
-- MeasureTheory.unifIntegrable_of'
theorem unifIntegrable_of' (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
    (hf : ∀ i, StronglyMeasurable (f i))
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0, 0 < C ∧
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UnifIntegrable 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 : ℝ≥0∞} (hp : 1 ≤ p) (hp' : p ≠ ⊤) {f : ι → α → β} (hf : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (h :
  ∀ (ε : ℝ),
    0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε)
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.unifIntegrable_of
theorem unifIntegrable_of (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
    (hf : ∀ i, AEStronglyMeasurable (f i) μ)
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UnifIntegrable 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 : ℝ≥0∞} (hp : 1 ≤ p) (hp' : p ≠ ⊤) {f : ι → α → β} (hf : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (h :
  ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε)
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.uniformIntegrable_zero_meas
theorem uniformIntegrable_zero_meas [MeasurableSpace α] : UniformIntegrable f p (0 : Measure α) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : ι → α → β} [MeasurableSpace α]
Goal: MeasureTheory.UniformIntegrable f p 0
-- MeasureTheory.UniformIntegrable.ae_eq
theorem UniformIntegrable.ae_eq {g : ι → α → β} (hf : UniformIntegrable f p μ)
    (hfg : ∀ n, f n =ᵐ[μ] g n) : UniformIntegrable 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 : ℝ≥0∞} {f g : ι → α → β} (hf : MeasureTheory.UniformIntegrable f p μ) (hfg : ∀ (n : ι), f n =ᵐ[μ] g n)
Goal: MeasureTheory.UniformIntegrable g p μ
-- MeasureTheory.uniformIntegrable_congr_ae
theorem uniformIntegrable_congr_ae {g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UniformIntegrable f p μ ↔ UniformIntegrable 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 : ℝ≥0∞} {f g : ι → α → β} (hfg : ∀ (n : ι), f n =ᵐ[μ] g n)
Goal: MeasureTheory.UniformIntegrable f p μ ↔ MeasureTheory.UniformIntegrable g p μ
-- MeasureTheory.uniformIntegrable_finite
theorem uniformIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
    (hf : ∀ i, Memℒp (f i) p μ) : UniformIntegrable 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 : ℝ≥0∞} {f : ι → α → β} [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) (hf : ∀ (i : ι), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UniformIntegrable f p μ
-- MeasureTheory.uniformIntegrable_subsingleton
theorem uniformIntegrable_subsingleton [Subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
    (hf : ∀ i, Memℒp (f i) p μ) : UniformIntegrable 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 : ℝ≥0∞} {f : ι → α → β} [Subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ⊤) (hf : ∀ (i : ι), MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.UniformIntegrable f p μ
-- MeasureTheory.uniformIntegrable_const
theorem uniformIntegrable_const {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ∞) (hg : Memℒp g p μ) :
    UniformIntegrable (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 : ℝ≥0∞} {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ⊤) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.UniformIntegrable (fun x => g) p μ
-- MeasureTheory.uniformIntegrable_of'
theorem uniformIntegrable_of' [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ i, StronglyMeasurable (f i))
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UniformIntegrable 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 : ℝ≥0∞} {f : ι → α → β} [MeasureTheory.IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hf : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (h :
  ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε)
Goal: MeasureTheory.UniformIntegrable f p μ
-- MeasureTheory.uniformIntegrable_of
theorem uniformIntegrable_of [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ i, AEStronglyMeasurable (f i) μ)
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UniformIntegrable 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 : ℝ≥0∞} {f : ι → α → β} [MeasureTheory.IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hf : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (h :
  ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε)
Goal: MeasureTheory.UniformIntegrable f p μ
-- MeasureTheory.UniformIntegrable.spec'
theorem UniformIntegrable.spec' (hp : p ≠ 0) (hp' : p ≠ ∞) (hf : ∀ i, StronglyMeasurable (f i))
    (hfu : UniformIntegrable f p μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ≥0, ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : ι → α → β} (hp : p ≠ 0) (hp' : p ≠ ⊤) (hf : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (hfu : MeasureTheory.UniformIntegrable f p μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.UniformIntegrable.spec
theorem UniformIntegrable.spec (hp : p ≠ 0) (hp' : p ≠ ∞) (hfu : UniformIntegrable f p μ) {ε : ℝ}
    (hε : 0 < ε) :
    ∃ C : ℝ≥0, ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : ι → α → β} (hp : p ≠ 0) (hp' : p ≠ ⊤) (hfu : MeasureTheory.UniformIntegrable f p μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.uniformIntegrable_iff
theorem uniformIntegrable_iff [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞) :
    UniformIntegrable f p μ ↔
      (∀ i, AEStronglyMeasurable (f i) μ) ∧
        ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
          ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : ι → α → β} [MeasureTheory.IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ⊤)
Goal: MeasureTheory.UniformIntegrable f p μ ↔
  (∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) ∧
    ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.uniformIntegrable_average
theorem uniformIntegrable_average
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (hp : 1 ≤ p) {f : ℕ → α → E} (hf : UniformIntegrable f p μ) :
    UniformIntegrable (fun (n : ℕ) => (n : ℝ)⁻¹ • (∑ i ∈ Finset.range n, f i)) p μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] (hp : 1 ≤ p) {f : ℕ → α → E} (hf : MeasureTheory.UniformIntegrable f p μ)
Goal: MeasureTheory.UniformIntegrable (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, f i) p μ
-- MeasureTheory.uniformIntegrable_average_real
theorem uniformIntegrable_average_real (hp : 1 ≤ p) {f : ℕ → α → ℝ} (hf : UniformIntegrable f p μ) :
    UniformIntegrable (fun n => (∑ i ∈ Finset.range n, f i) / (n : α → ℝ)) p μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} (hp : 1 ≤ p) {f : ℕ → α → ℝ} (hf : MeasureTheory.UniformIntegrable f p μ)
Goal: MeasureTheory.UniformIntegrable (fun n => (∑ i ∈ Finset.range n, f i) / ↑n) p μ
-- MeasureTheory.lintegral_add_left_eq_self
@[to_additive
      "Translating a function by left-addition does not change its Lebesgue integral with
      respect to a left-invariant measure."]
theorem lintegral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → ℝ≥0∞) (g : G) :
    (∫⁻ x, f (g * x) ∂μ) = ∫⁻ x, f x ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [μ.IsAddLeftInvariant] (f : G → ENNReal) (g : G)
Goal: ∫⁻ (x : G), f (g + x) ∂μ = ∫⁻ (x : G), f x ∂μ
-- MeasureTheory.lintegral_mul_left_eq_self
@[to_additive
      "Translating a function by left-addition does not change its Lebesgue integral with
      respect to a left-invariant measure."]
theorem lintegral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → ℝ≥0∞) (g : G) :
    (∫⁻ x, f (g * x) ∂μ) = ∫⁻ x, f x ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [μ.IsMulLeftInvariant] (f : G → ENNReal) (g : G)
Goal: ∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ
-- MeasureTheory.lintegral_mul_right_eq_self
@[to_additive
      "Translating a function by right-addition does not change its Lebesgue integral with
      respect to a right-invariant measure."]
theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
    (∫⁻ x, f (x * g) ∂μ) = ∫⁻ x, f x ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [μ.IsMulRightInvariant] (f : G → ENNReal) (g : G)
Goal: ∫⁻ (x : G), f (x * g) ∂μ = ∫⁻ (x : G), f x ∂μ
-- MeasureTheory.lintegral_add_right_eq_self
@[to_additive
      "Translating a function by right-addition does not change its Lebesgue integral with
      respect to a right-invariant measure."]
theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
    (∫⁻ x, f (x * g) ∂μ) = ∫⁻ x, f x ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [μ.IsAddRightInvariant] (f : G → ENNReal) (g : G)
Goal: ∫⁻ (x : G), f (x + g) ∂μ = ∫⁻ (x : G), f x ∂μ
-- MeasureTheory.lintegral_div_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem lintegral_div_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
    (∫⁻ x, f (x / g) ∂μ) = ∫⁻ x, f x ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [μ.IsMulRightInvariant] (f : G → ENNReal) (g : G)
Goal: ∫⁻ (x : G), f (x / g) ∂μ = ∫⁻ (x : G), f x ∂μ
-- MeasureTheory.lintegral_sub_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem lintegral_div_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
    (∫⁻ x, f (x / g) ∂μ) = ∫⁻ x, f x ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [μ.IsAddRightInvariant] (f : G → ENNReal) (g : G)
Goal: ∫⁻ (x : G), f (x - g) ∂μ = ∫⁻ (x : G), f x ∂μ
-- MeasureTheory.lintegral_eq_zero_of_isMulLeftInvariant
@[to_additive
      "For nonzero regular left invariant measures, the integral of a continuous nonnegative
      function `f` is 0 iff `f` is 0."]
theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
    (hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [TopologicalSpace G] [Group G] [TopologicalGroup G] [BorelSpace G] [μ.IsMulLeftInvariant] [μ.Regular] [NeZero μ] {f : G → ENNReal} (hf : Continuous f)
Goal: ∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0
-- MeasureTheory.lintegral_eq_zero_of_isAddLeftInvariant
@[to_additive
      "For nonzero regular left invariant measures, the integral of a continuous nonnegative
      function `f` is 0 iff `f` is 0."]
theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
    (hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 
Arguments: {G : Type u_1} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [BorelSpace G] [μ.IsAddLeftInvariant] [μ.Regular] [NeZero μ] {f : G → ENNReal} (hf : Continuous f)
Goal: ∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0
-- MeasureTheory.toFiniteAux_apply
lemma toFiniteAux_apply (μ : Measure α) [SFinite μ] (s : Set α) :
    μ.toFiniteAux s = ∑' n, (2 ^ (n + 1) * sFiniteSeq μ n Set.univ)⁻¹ * sFiniteSeq μ n s 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (s : Set α)
Goal: μ.toFiniteAux s =
  ∑' (n : ℕ), (2 ^ (n + 1) * (MeasureTheory.sFiniteSeq μ n) Set.univ)⁻¹ * (MeasureTheory.sFiniteSeq μ n) s
-- MeasureTheory.toFinite_apply
lemma toFinite_apply (μ : Measure α) [SFinite μ] (s : Set α) :
    μ.toFinite s = (μ.toFiniteAux Set.univ)⁻¹ * μ.toFiniteAux s 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (s : Set α)
Goal: μ.toFinite s = (μ.toFiniteAux Set.univ)⁻¹ * μ.toFiniteAux s
-- MeasureTheory.toFiniteAux_zero
lemma toFiniteAux_zero : Measure.toFiniteAux (0 : Measure α) = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α}
Goal: MeasureTheory.Measure.toFiniteAux 0 = 0
-- MeasureTheory.toFinite_zero
@[simp]
lemma toFinite_zero : Measure.toFinite (0 : Measure α) = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α}
Goal: MeasureTheory.Measure.toFinite 0 = 0