ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.integrable_withDensity_iff
theorem integrable_withDensity_iff {f : α → ℝ≥0∞} (hf : Measurable f) (hflt : ∀ᵐ x ∂μ, f x < ∞)
    {g : α → ℝ} : Integrable g (μ.withDensity f) ↔ Integrable (fun x => g x * (f x).toReal) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤) {g : α → ℝ}
Goal: MeasureTheory.Integrable g (μ.withDensity f) ↔ MeasureTheory.Integrable (fun x => g x * (f x).toReal) μ
-- MeasureTheory.memℒ1_smul_of_L1_withDensity
theorem memℒ1_smul_of_L1_withDensity {f : α → ℝ≥0} (f_meas : Measurable f)
    (u : Lp E 1 (μ.withDensity fun x => f x)) : Memℒp (fun x => f x • u x) 1 μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → ℝ≥0} (f_meas : Measurable f) (u : ↥(MeasureTheory.Lp E 1 (μ.withDensity fun x => ↑(f x))))
Goal: MeasureTheory.Memℒp (fun x => f x • ↑↑u x) 1 μ
-- MeasureTheory.withDensitySMulLI_apply
@[simp]
theorem withDensitySMulLI_apply {f : α → ℝ≥0} (f_meas : Measurable f)
    (u : Lp E 1 (μ.withDensity fun x => f x)) :
    withDensitySMulLI μ (E := E) f_meas u =
      (memℒ1_smul_of_L1_withDensity f_meas u).toLp fun x => f x • u x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {E : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → ℝ≥0} (f_meas : Measurable f) (u : ↥(MeasureTheory.Lp E 1 (μ.withDensity fun x => ↑(f x))))
Goal: (MeasureTheory.withDensitySMulLI μ f_meas) u = MeasureTheory.Memℒp.toLp (fun x => f x • ↑↑u x) ⋯
-- MeasureTheory.mem_ℒ1_toReal_of_lintegral_ne_top
theorem mem_ℒ1_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ)
    (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) : Memℒp (fun x => (f x).toReal) 1 μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: MeasureTheory.Memℒp (fun x => (f x).toReal) 1 μ
-- MeasureTheory.integrable_toReal_of_lintegral_ne_top
theorem integrable_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ)
    (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) : Integrable (fun x => (f x).toReal) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: MeasureTheory.Integrable (fun x => (f x).toReal) μ
-- MeasureTheory.Integrable.pos_part
theorem Integrable.pos_part {f : α → ℝ} (hf : Integrable f μ) :
    Integrable (fun a => max (f a) 0) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun a => max (f a) 0) μ
-- MeasureTheory.Integrable.neg_part
theorem Integrable.neg_part {f : α → ℝ} (hf : Integrable f μ) :
    Integrable (fun a => max (-f a) 0) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun a => max (-f a) 0) μ
-- MeasureTheory.Integrable.smul
theorem Integrable.smul [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜)
    {f : α → β} (hf : Integrable f μ) : Integrable (c • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜) {f : α → β} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (c • f) μ
-- IsUnit.integrable_smul_iff
theorem _root_.IsUnit.integrable_smul_iff [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜}
    (hc : IsUnit c) (f : α → β) : Integrable (c • f) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜} (hc : IsUnit c) (f : α → β)
Goal: MeasureTheory.Integrable (c • f) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_smul_iff
theorem integrable_smul_iff [NormedDivisionRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜}
    (hc : c ≠ 0) (f : α → β) : Integrable (c • f) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedDivisionRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜} (hc : c ≠ 0) (f : α → β)
Goal: MeasureTheory.Integrable (c • f) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.smul_of_top_right
theorem Integrable.smul_of_top_right {f : α → β} {φ : α → 𝕜} (hf : Integrable f μ)
    (hφ : Memℒp φ ∞ μ) : Integrable (φ • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {f : α → β} {φ : α → 𝕜} (hf : MeasureTheory.Integrable f μ) (hφ : MeasureTheory.Memℒp φ ⊤ μ)
Goal: MeasureTheory.Integrable (φ • f) μ
-- MeasureTheory.Integrable.smul_of_top_left
theorem Integrable.smul_of_top_left {f : α → β} {φ : α → 𝕜} (hφ : Integrable φ μ)
    (hf : Memℒp f ∞ μ) : Integrable (φ • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {f : α → β} {φ : α → 𝕜} (hφ : MeasureTheory.Integrable φ μ) (hf : MeasureTheory.Memℒp f ⊤ μ)
Goal: MeasureTheory.Integrable (φ • f) μ
-- MeasureTheory.Integrable.smul_const
theorem Integrable.smul_const {f : α → 𝕜} (hf : Integrable f μ) (c : β) :
    Integrable (fun x => f x • c) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {f : α → 𝕜} (hf : MeasureTheory.Integrable f μ) (c : β)
Goal: MeasureTheory.Integrable (fun x => f x • c) μ
-- MeasureTheory.integrable_smul_const
theorem integrable_smul_const {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
    Integrable (fun x => f x • c) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : α → 𝕜} {c : E} (hc : c ≠ 0)
Goal: MeasureTheory.Integrable (fun x => f x • c) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.const_mul
theorem Integrable.const_mul {f : α → 𝕜} (h : Integrable f μ) (c : 𝕜) :
    Integrable (fun x => c * f x) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedRing 𝕜] {f : α → 𝕜} (h : MeasureTheory.Integrable f μ) (c : 𝕜)
Goal: MeasureTheory.Integrable (fun x => c * f x) μ
-- MeasureTheory.Integrable.const_mul'
theorem Integrable.const_mul' {f : α → 𝕜} (h : Integrable f μ) (c : 𝕜) :
    Integrable ((fun _ : α => c) * f) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedRing 𝕜] {f : α → 𝕜} (h : MeasureTheory.Integrable f μ) (c : 𝕜)
Goal: MeasureTheory.Integrable ((fun x => c) * f) μ
-- MeasureTheory.Integrable.mul_const
theorem Integrable.mul_const {f : α → 𝕜} (h : Integrable f μ) (c : 𝕜) :
    Integrable (fun x => f x * c) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedRing 𝕜] {f : α → 𝕜} (h : MeasureTheory.Integrable f μ) (c : 𝕜)
Goal: MeasureTheory.Integrable (fun x => f x * c) μ
-- MeasureTheory.Integrable.mul_const'
theorem Integrable.mul_const' {f : α → 𝕜} (h : Integrable f μ) (c : 𝕜) :
    Integrable (f * fun _ : α => c) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedRing 𝕜] {f : α → 𝕜} (h : MeasureTheory.Integrable f μ) (c : 𝕜)
Goal: MeasureTheory.Integrable (f * fun x => c) μ
-- MeasureTheory.integrable_const_mul_iff
theorem integrable_const_mul_iff {c : 𝕜} (hc : IsUnit c) (f : α → 𝕜) :
    Integrable (fun x => c * f x) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedRing 𝕜] {c : 𝕜} (hc : IsUnit c) (f : α → 𝕜)
Goal: MeasureTheory.Integrable (fun x => c * f x) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_mul_const_iff
theorem integrable_mul_const_iff {c : 𝕜} (hc : IsUnit c) (f : α → 𝕜) :
    Integrable (fun x => f x * c) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedRing 𝕜] {c : 𝕜} (hc : IsUnit c) (f : α → 𝕜)
Goal: MeasureTheory.Integrable (fun x => f x * c) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.bdd_mul'
theorem Integrable.bdd_mul' {f g : α → 𝕜} {c : ℝ} (hg : Integrable g μ)
    (hf : AEStronglyMeasurable f μ) (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
    Integrable (fun x => f x * g x) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedRing 𝕜] {f g : α → 𝕜} {c : ℝ} (hg : MeasureTheory.Integrable g μ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c)
Goal: MeasureTheory.Integrable (fun x => f x * g x) μ
-- MeasureTheory.Integrable.div_const
theorem Integrable.div_const {f : α → 𝕜} (h : Integrable f μ) (c : 𝕜) :
    Integrable (fun x => f x / c) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [NormedDivisionRing 𝕜] {f : α → 𝕜} (h : MeasureTheory.Integrable f μ) (c : 𝕜)
Goal: MeasureTheory.Integrable (fun x => f x / c) μ
-- MeasureTheory.Integrable.ofReal
theorem Integrable.ofReal {f : α → ℝ} (hf : Integrable f μ) :
    Integrable (fun x => (f x : 𝕜)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [RCLike 𝕜] {f : α → ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => ↑(f x)) μ
-- MeasureTheory.Integrable.re_im_iff
theorem Integrable.re_im_iff :
    Integrable (fun x => RCLike.re (f x)) μ ∧ Integrable (fun x => RCLike.im (f x)) μ ↔
      Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [RCLike 𝕜] {f : α → 𝕜}
Goal: MeasureTheory.Integrable (fun x => RCLike.re (f x)) μ ∧ MeasureTheory.Integrable (fun x => RCLike.im (f x)) μ ↔
  MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.re
theorem Integrable.re (hf : Integrable f μ) : Integrable (fun x => RCLike.re (f x)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [RCLike 𝕜] {f : α → 𝕜} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => RCLike.re (f x)) μ
-- MeasureTheory.Integrable.im
theorem Integrable.im (hf : Integrable f μ) : Integrable (fun x => RCLike.im (f x)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [RCLike 𝕜] {f : α → 𝕜} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => RCLike.im (f x)) μ
-- MeasureTheory.Integrable.trim
theorem Integrable.trim (hm : m ≤ m0) (hf_int : Integrable f μ') (hf : StronglyMeasurable[m] f) :
    Integrable f (μ'.trim hm) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {H : Type u_5} [NormedAddCommGroup H] {m0 : MeasurableSpace α} {μ' : MeasureTheory.Measure α} {f : α → H} (hm : m ≤ m0) (hf_int : MeasureTheory.Integrable f μ') (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.Integrable f (μ'.trim hm)
-- MeasureTheory.integrable_of_integrable_trim
theorem integrable_of_integrable_trim (hm : m ≤ m0) (hf_int : Integrable f (μ'.trim hm)) :
    Integrable f μ' 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {H : Type u_5} [NormedAddCommGroup H] {m0 : MeasurableSpace α} {μ' : MeasureTheory.Measure α} {f : α → H} (hm : m ≤ m0) (hf_int : MeasureTheory.Integrable f (μ'.trim hm))
Goal: MeasureTheory.Integrable f μ'
-- MeasureTheory.integrable_of_forall_fin_meas_le'
theorem integrable_of_forall_fin_meas_le' {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) (hC : C < ∞) {f : α → E} (hf_meas : AEStronglyMeasurable f μ)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → (∫⁻ x in s, ‖f x‖₊ ∂μ) ≤ C) : Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {E : Type u_5} {m0 : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) (hC : C < ⊤) {f : α → E} (hf_meas : MeasureTheory.AEStronglyMeasurable f μ) (hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, ↑‖f x‖₊ ∂μ ≤ C)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_of_forall_fin_meas_le
theorem integrable_of_forall_fin_meas_le [SigmaFinite μ] (C : ℝ≥0∞) (hC : C < ∞) {f : α → E}
    (hf_meas : AEStronglyMeasurable f μ)
    (hf : ∀ s : Set α, MeasurableSet[m] s → μ s ≠ ∞ → (∫⁻ x in s, ‖f x‖₊ ∂μ) ≤ C) :
    Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite μ] (C : ℝ≥0∞) (hC : C < ⊤) {f : α → E} (hf_meas : MeasureTheory.AEStronglyMeasurable f μ) (hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, ↑‖f x‖₊ ∂μ ≤ C)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AEEqFun.integrable_mk
theorem integrable_mk {f : α → β} (hf : AEStronglyMeasurable f μ) :
    Integrable (mk f hf : α →ₘ[μ] β) ↔ MeasureTheory.Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: (MeasureTheory.AEEqFun.mk f hf).Integrable ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.AEEqFun.integrable_coeFn
theorem integrable_coeFn {f : α →ₘ[μ] β} : MeasureTheory.Integrable f μ ↔ Integrable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α →ₘ[μ] β}
Goal: MeasureTheory.Integrable (↑f) μ ↔ f.Integrable
-- MeasureTheory.AEEqFun.integrable_zero
theorem integrable_zero : Integrable (0 : α →ₘ[μ] β) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β]
Goal: MeasureTheory.AEEqFun.Integrable 0
-- MeasureTheory.AEEqFun.Integrable.neg
theorem Integrable.neg {f : α →ₘ[μ] β} : Integrable f → Integrable (-f) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α →ₘ[μ] β}
Goal: f.Integrable → (-f).Integrable
-- MeasureTheory.AEEqFun.integrable_iff_mem_L1
theorem integrable_iff_mem_L1 {f : α →ₘ[μ] β} : Integrable f ↔ f ∈ (α →₁[μ] β) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α →ₘ[μ] β}
Goal: f.Integrable ↔ f ∈ MeasureTheory.Lp β 1 μ
-- MeasureTheory.AEEqFun.Integrable.add
theorem Integrable.add {f g : α →ₘ[μ] β} : Integrable f → Integrable g → Integrable (f + g) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : α →ₘ[μ] β}
Goal: f.Integrable → g.Integrable → (f + g).Integrable
-- MeasureTheory.AEEqFun.Integrable.sub
theorem Integrable.sub {f g : α →ₘ[μ] β} (hf : Integrable f) (hg : Integrable g) :
    Integrable (f - g) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : α →ₘ[μ] β} (hf : f.Integrable) (hg : g.Integrable)
Goal: (f - g).Integrable
-- MeasureTheory.AEEqFun.Integrable.smul
theorem Integrable.smul {c : 𝕜} {f : α →ₘ[μ] β} : Integrable f → Integrable (c • f) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜} {f : α →ₘ[μ] β}
Goal: f.Integrable → (c • f).Integrable
-- MeasureTheory.L1.integrable_coeFn
theorem integrable_coeFn (f : α →₁[μ] β) : Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: MeasureTheory.Integrable (↑↑f) μ
-- MeasureTheory.L1.hasFiniteIntegral_coeFn
theorem hasFiniteIntegral_coeFn (f : α →₁[μ] β) : HasFiniteIntegral f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: MeasureTheory.HasFiniteIntegral (↑↑f) μ
-- MeasureTheory.L1.stronglyMeasurable_coeFn
theorem stronglyMeasurable_coeFn (f : α →₁[μ] β) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: MeasureTheory.StronglyMeasurable ↑↑f
-- MeasureTheory.L1.measurable_coeFn
theorem measurable_coeFn [MeasurableSpace β] [BorelSpace β] (f : α →₁[μ] β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [MeasurableSpace β] [BorelSpace β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: Measurable ↑↑f
-- MeasureTheory.L1.aestronglyMeasurable_coeFn
theorem aestronglyMeasurable_coeFn (f : α →₁[μ] β) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: MeasureTheory.AEStronglyMeasurable (↑↑f) μ
-- MeasureTheory.L1.aemeasurable_coeFn
theorem aemeasurable_coeFn [MeasurableSpace β] [BorelSpace β] (f : α →₁[μ] β) : AEMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [MeasurableSpace β] [BorelSpace β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: AEMeasurable (↑↑f) μ
-- MeasureTheory.L1.edist_def
theorem edist_def (f g : α →₁[μ] β) : edist f g = ∫⁻ a, edist (f a) (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : ↥(MeasureTheory.Lp β 1 μ))
Goal: edist f g = ∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ
-- MeasureTheory.L1.dist_def
theorem dist_def (f g : α →₁[μ] β) : dist f g = (∫⁻ a, edist (f a) (g a) ∂μ).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : ↥(MeasureTheory.Lp β 1 μ))
Goal: dist f g = (∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ).toReal
-- MeasureTheory.L1.norm_def
theorem norm_def (f : α →₁[μ] β) : ‖f‖ = (∫⁻ a, ‖f a‖₊ ∂μ).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: ‖f‖ = (∫⁻ (a : α), ↑‖↑↑f a‖₊ ∂μ).toReal
-- MeasureTheory.L1.norm_sub_eq_lintegral
theorem norm_sub_eq_lintegral (f g : α →₁[μ] β) :
    ‖f - g‖ = (∫⁻ x, (‖f x - g x‖₊ : ℝ≥0∞) ∂μ).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : ↥(MeasureTheory.Lp β 1 μ))
Goal: ‖f - g‖ = (∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ).toReal
-- MeasureTheory.L1.ofReal_norm_eq_lintegral
theorem ofReal_norm_eq_lintegral (f : α →₁[μ] β) :
    ENNReal.ofReal ‖f‖ = ∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : ↥(MeasureTheory.Lp β 1 μ))
Goal: ENNReal.ofReal ‖f‖ = ∫⁻ (x : α), ↑‖↑↑f x‖₊ ∂μ
-- MeasureTheory.L1.ofReal_norm_sub_eq_lintegral
theorem ofReal_norm_sub_eq_lintegral (f g : α →₁[μ] β) :
    ENNReal.ofReal ‖f - g‖ = ∫⁻ x, (‖f x - g x‖₊ : ℝ≥0∞) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : ↥(MeasureTheory.Lp β 1 μ))
Goal: ENNReal.ofReal ‖f - g‖ = ∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ
-- MeasureTheory.Integrable.toL1_coeFn
@[simp]
theorem toL1_coeFn (f : α →₁[μ] β) (hf : Integrable f μ) : hf.toL1 f = f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : ↥(MeasureTheory.Lp β 1 μ)) (hf : MeasureTheory.Integrable (↑↑f) μ)
Goal: MeasureTheory.Integrable.toL1 (↑↑f) hf = f
-- MeasureTheory.Integrable.coeFn_toL1
theorem coeFn_toL1 {f : α → β} (hf : Integrable f μ) : hf.toL1 f =ᵐ[μ] f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.Integrable f μ)
Goal: ↑↑(MeasureTheory.Integrable.toL1 f hf) =ᵐ[μ] f
-- MeasureTheory.Integrable.toL1_zero
@[simp]
theorem toL1_zero (h : Integrable (0 : α → β) μ) : h.toL1 0 = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (h : MeasureTheory.Integrable 0 μ)
Goal: MeasureTheory.Integrable.toL1 0 h = 0
-- MeasureTheory.Integrable.toL1_eq_mk
@[simp]
theorem toL1_eq_mk (f : α → β) (hf : Integrable f μ) :
    (hf.toL1 f : α →ₘ[μ] β) = AEEqFun.mk f hf.aestronglyMeasurable 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : α → β) (hf : MeasureTheory.Integrable f μ)
Goal: ↑(MeasureTheory.Integrable.toL1 f hf) = MeasureTheory.AEEqFun.mk f ⋯
-- MeasureTheory.Integrable.toL1_eq_toL1_iff
@[simp]
theorem toL1_eq_toL1_iff (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
    toL1 f hf = toL1 g hg ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : α → β) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable.toL1 f hf = MeasureTheory.Integrable.toL1 g hg ↔ f =ᵐ[μ] g
-- MeasureTheory.Integrable.toL1_add
theorem toL1_add (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
    toL1 (f + g) (hf.add hg) = toL1 f hf + toL1 g hg 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : α → β) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable.toL1 (f + g) ⋯ = MeasureTheory.Integrable.toL1 f hf + MeasureTheory.Integrable.toL1 g hg
-- MeasureTheory.Integrable.toL1_neg
theorem toL1_neg (f : α → β) (hf : Integrable f μ) : toL1 (-f) (Integrable.neg hf) = -toL1 f hf 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : α → β) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable.toL1 (-f) ⋯ = -MeasureTheory.Integrable.toL1 f hf
-- MeasureTheory.Integrable.toL1_sub
theorem toL1_sub (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
    toL1 (f - g) (hf.sub hg) = toL1 f hf - toL1 g hg 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : α → β) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable.toL1 (f - g) ⋯ = MeasureTheory.Integrable.toL1 f hf - MeasureTheory.Integrable.toL1 g hg
-- MeasureTheory.Integrable.norm_toL1
theorem norm_toL1 (f : α → β) (hf : Integrable f μ) :
    ‖hf.toL1 f‖ = ENNReal.toReal (∫⁻ a, edist (f a) 0 ∂μ) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : α → β) (hf : MeasureTheory.Integrable f μ)
Goal: ‖MeasureTheory.Integrable.toL1 f hf‖ = (∫⁻ (a : α), edist (f a) 0 ∂μ).toReal
-- MeasureTheory.Integrable.nnnorm_toL1
theorem nnnorm_toL1 {f : α → β} (hf : Integrable f μ) :
    (‖hf.toL1 f‖₊ : ℝ≥0∞) = ∫⁻ a, ‖f a‖₊ ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.Integrable f μ)
Goal: ↑‖MeasureTheory.Integrable.toL1 f hf‖₊ = ∫⁻ (a : α), ↑‖f a‖₊ ∂μ
-- MeasureTheory.Integrable.norm_toL1_eq_lintegral_norm
theorem norm_toL1_eq_lintegral_norm (f : α → β) (hf : Integrable f μ) :
    ‖hf.toL1 f‖ = ENNReal.toReal (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : α → β) (hf : MeasureTheory.Integrable f μ)
Goal: ‖MeasureTheory.Integrable.toL1 f hf‖ = (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal