ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Integrable.to_average
theorem Integrable.to_average {f : α → β} (h : Integrable f μ) : Integrable f ((μ univ)⁻¹ • μ) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable f ((μ Set.univ)⁻¹ • μ)
-- MeasureTheory.integrable_average
theorem integrable_average [IsFiniteMeasure μ] {f : α → β} :
    Integrable f ((μ univ)⁻¹ • μ) ↔ Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [MeasureTheory.IsFiniteMeasure μ] {f : α → β}
Goal: MeasureTheory.Integrable f ((μ Set.univ)⁻¹ • μ) ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_map_measure
theorem integrable_map_measure {f : α → δ} {g : δ → β}
    (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
    Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace δ] [NormedAddCommGroup β] {f : α → δ} {g : δ → β} (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: MeasureTheory.Integrable g (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.Integrable (g ∘ f) μ
-- MeasureTheory.Integrable.comp_aemeasurable
theorem Integrable.comp_aemeasurable {f : α → δ} {g : δ → β} (hg : Integrable g (Measure.map f μ))
    (hf : AEMeasurable f μ) : Integrable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace δ] [NormedAddCommGroup β] {f : α → δ} {g : δ → β} (hg : MeasureTheory.Integrable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: MeasureTheory.Integrable (g ∘ f) μ
-- MeasureTheory.Integrable.comp_measurable
theorem Integrable.comp_measurable {f : α → δ} {g : δ → β} (hg : Integrable g (Measure.map f μ))
    (hf : Measurable f) : Integrable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace δ] [NormedAddCommGroup β] {f : α → δ} {g : δ → β} (hg : MeasureTheory.Integrable g (MeasureTheory.Measure.map f μ)) (hf : Measurable f)
Goal: MeasureTheory.Integrable (g ∘ f) μ
-- MeasurableEmbedding.integrable_map_iff
theorem _root_.MeasurableEmbedding.integrable_map_iff {f : α → δ} (hf : MeasurableEmbedding f)
    {g : δ → β} : Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace δ] [NormedAddCommGroup β] {f : α → δ} (hf : MeasurableEmbedding f) {g : δ → β}
Goal: MeasureTheory.Integrable g (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.Integrable (g ∘ f) μ
-- MeasureTheory.integrable_map_equiv
theorem integrable_map_equiv (f : α ≃ᵐ δ) (g : δ → β) :
    Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace δ] [NormedAddCommGroup β] (f : α ≃ᵐ δ) (g : δ → β)
Goal: MeasureTheory.Integrable g (MeasureTheory.Measure.map (⇑f) μ) ↔ MeasureTheory.Integrable (g ∘ ⇑f) μ
-- MeasureTheory.MeasurePreserving.integrable_comp
theorem MeasurePreserving.integrable_comp {ν : Measure δ} {g : δ → β} {f : α → δ}
    (hf : MeasurePreserving f μ ν) (hg : AEStronglyMeasurable g ν) :
    Integrable (g ∘ f) μ ↔ Integrable g ν 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace δ] [NormedAddCommGroup β] {ν : MeasureTheory.Measure δ} {g : δ → β} {f : α → δ} (hf : MeasureTheory.MeasurePreserving f μ ν) (hg : MeasureTheory.AEStronglyMeasurable g ν)
Goal: MeasureTheory.Integrable (g ∘ f) μ ↔ MeasureTheory.Integrable g ν
-- MeasureTheory.MeasurePreserving.integrable_comp_emb
theorem MeasurePreserving.integrable_comp_emb {f : α → δ} {ν} (h₁ : MeasurePreserving f μ ν)
    (h₂ : MeasurableEmbedding f) {g : δ → β} : Integrable (g ∘ f) μ ↔ Integrable g ν 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace δ] [NormedAddCommGroup β] {f : α → δ} {ν : MeasureTheory.Measure δ} (h₁ : MeasureTheory.MeasurePreserving f μ ν) (h₂ : MeasurableEmbedding f) {g : δ → β}
Goal: MeasureTheory.Integrable (g ∘ f) μ ↔ MeasureTheory.Integrable g ν
-- MeasureTheory.lintegral_edist_lt_top
theorem lintegral_edist_lt_top {f g : α → β} (hf : Integrable f μ) (hg : Integrable g μ) :
    (∫⁻ 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 : α → β} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: ∫⁻ (a : α), edist (f a) (g a) ∂μ < ⊤
-- MeasureTheory.integrable_zero
@[simp]
theorem integrable_zero : Integrable (fun _ => (0 : β)) μ 
Arguments: (α : Type u_1) (β : Type u_2) {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [NormedAddCommGroup β]
Goal: MeasureTheory.Integrable (fun x => 0) μ
-- MeasureTheory.Integrable.add'
theorem Integrable.add' {f g : α → β} (hf : Integrable f μ) (hg : Integrable g μ) :
    HasFiniteIntegral (f + g) μ 
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.HasFiniteIntegral (f + g) μ
-- MeasureTheory.Integrable.add
theorem Integrable.add {f g : α → β} (hf : Integrable f μ) (hg : Integrable g μ) :
    Integrable (f + g) μ 
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 (f + g) μ
-- MeasureTheory.integrable_finset_sum'
theorem integrable_finset_sum' {ι} (s : Finset ι) {f : ι → α → β}
    (hf : ∀ i ∈ s, Integrable (f i) μ) : Integrable (∑ i ∈ s, f i) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {ι : Type u_5} (s : Finset ι) {f : ι → α → β} (hf : ∀ i ∈ s, MeasureTheory.Integrable (f i) μ)
Goal: MeasureTheory.Integrable (∑ i ∈ s, f i) μ
-- MeasureTheory.integrable_finset_sum
theorem integrable_finset_sum {ι} (s : Finset ι) {f : ι → α → β}
    (hf : ∀ i ∈ s, Integrable (f i) μ) : Integrable (fun a => ∑ i ∈ s, f i a) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {ι : Type u_5} (s : Finset ι) {f : ι → α → β} (hf : ∀ i ∈ s, MeasureTheory.Integrable (f i) μ)
Goal: MeasureTheory.Integrable (fun a => ∑ i ∈ s, f i a) μ
-- MeasureTheory.Integrable.neg
theorem Integrable.neg {f : α → β} (hf : Integrable f μ) : 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.Integrable f μ)
Goal: MeasureTheory.Integrable (-f) μ
-- MeasureTheory.integrable_neg_iff
@[simp]
theorem integrable_neg_iff {f : α → β} : Integrable (-f) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β}
Goal: MeasureTheory.Integrable (-f) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_add_iff_integrable_right
@[simp]
lemma integrable_add_iff_integrable_right {f g : α → β} (hf : Integrable f μ) :
    Integrable (f + g) μ ↔ Integrable g μ 
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)
Goal: MeasureTheory.Integrable (f + g) μ ↔ MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_add_iff_integrable_left
@[simp]
lemma integrable_add_iff_integrable_left {f g : α → β} (hf : Integrable f μ) :
    Integrable (g + f) μ ↔ Integrable g μ 
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)
Goal: MeasureTheory.Integrable (g + f) μ ↔ MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_left_of_integrable_add_of_nonneg
lemma integrable_left_of_integrable_add_of_nonneg {f g : α → ℝ}
    (h_meas : AEStronglyMeasurable f μ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g)
    (h_int : Integrable (f + g) μ) : Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (h_meas : MeasureTheory.AEStronglyMeasurable f μ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g) (h_int : MeasureTheory.Integrable (f + g) μ)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_right_of_integrable_add_of_nonneg
lemma integrable_right_of_integrable_add_of_nonneg {f g : α → ℝ}
    (h_meas : AEStronglyMeasurable f μ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g)
    (h_int : Integrable (f + g) μ) : Integrable g μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (h_meas : MeasureTheory.AEStronglyMeasurable f μ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g) (h_int : MeasureTheory.Integrable (f + g) μ)
Goal: MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_add_iff_of_nonneg
lemma integrable_add_iff_of_nonneg {f g : α → ℝ} (h_meas : AEStronglyMeasurable f μ)
    (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g) :
    Integrable (f + g) μ ↔ Integrable f μ ∧ Integrable g μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (h_meas : MeasureTheory.AEStronglyMeasurable f μ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g)
Goal: MeasureTheory.Integrable (f + g) μ ↔ MeasureTheory.Integrable f μ ∧ MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_add_iff_of_nonpos
lemma integrable_add_iff_of_nonpos {f g : α → ℝ} (h_meas : AEStronglyMeasurable f μ)
    (hf : f ≤ᵐ[μ] 0) (hg : g ≤ᵐ[μ] 0) :
    Integrable (f + g) μ ↔ Integrable f μ ∧ Integrable g μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (h_meas : MeasureTheory.AEStronglyMeasurable f μ) (hf : f ≤ᵐ[μ] 0) (hg : g ≤ᵐ[μ] 0)
Goal: MeasureTheory.Integrable (f + g) μ ↔ MeasureTheory.Integrable f μ ∧ MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_add_const_iff
@[simp]
lemma integrable_add_const_iff [IsFiniteMeasure μ] {f : α → β} {c : β} :
    Integrable (fun x ↦ f x + c) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [MeasureTheory.IsFiniteMeasure μ] {f : α → β} {c : β}
Goal: MeasureTheory.Integrable (fun x => f x + c) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_const_add_iff
@[simp]
lemma integrable_const_add_iff [IsFiniteMeasure μ] {f : α → β} {c : β} :
    Integrable (fun x ↦ c + f x) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [MeasureTheory.IsFiniteMeasure μ] {f : α → β} {c : β}
Goal: MeasureTheory.Integrable (fun x => c + f x) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.sub
theorem Integrable.sub {f g : α → β} (hf : Integrable f μ) (hg : Integrable g μ) :
    Integrable (f - g) μ 
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 (f - g) μ
-- MeasureTheory.Integrable.norm
theorem Integrable.norm {f : α → β} (hf : Integrable f μ) : Integrable (fun 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 (fun a => ‖f a‖) μ
-- MeasureTheory.Integrable.inf
theorem Integrable.inf {β} [NormedLatticeAddCommGroup β] {f g : α → β} (hf : Integrable f μ)
    (hg : Integrable g μ) : Integrable (f ⊓ g) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [NormedLatticeAddCommGroup β] {f g : α → β} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable (f ⊓ g) μ
-- MeasureTheory.Integrable.sup
theorem Integrable.sup {β} [NormedLatticeAddCommGroup β] {f g : α → β} (hf : Integrable f μ)
    (hg : Integrable g μ) : Integrable (f ⊔ g) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [NormedLatticeAddCommGroup β] {f g : α → β} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable (f ⊔ g) μ
-- MeasureTheory.Integrable.abs
theorem Integrable.abs {β} [NormedLatticeAddCommGroup β] {f : α → β} (hf : Integrable f μ) :
    Integrable (fun a => |f a|) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [NormedLatticeAddCommGroup β] {f : α → β} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun a => |f a|) μ
-- MeasureTheory.Integrable.bdd_mul
theorem Integrable.bdd_mul {F : Type*} [NormedDivisionRing F] {f g : α → F} (hint : Integrable g μ)
    (hm : AEStronglyMeasurable f μ) (hfbdd : ∃ C, ∀ x, ‖f x‖ ≤ C) :
    Integrable (fun x => f x * g x) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {F : Type u_5} [NormedDivisionRing F] {f g : α → F} (hint : MeasureTheory.Integrable g μ) (hm : MeasureTheory.AEStronglyMeasurable f μ) (hfbdd : ∃ C, ∀ (x : α), ‖f x‖ ≤ C)
Goal: MeasureTheory.Integrable (fun x => f x * g x) μ
-- MeasureTheory.Integrable.essSup_smul
theorem Integrable.essSup_smul {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 β] {f : α → β}
    (hf : Integrable f μ) {g : α → 𝕜} (g_aestronglyMeasurable : AEStronglyMeasurable g μ)
    (ess_sup_g : essSup (fun x => (‖g x‖₊ : ℝ≥0∞)) μ ≠ ∞) :
    Integrable (fun x : α => g x • f x) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 β] {f : α → β} (hf : MeasureTheory.Integrable f μ) {g : α → 𝕜} (g_aestronglyMeasurable : MeasureTheory.AEStronglyMeasurable g μ) (ess_sup_g : essSup (fun x => ↑‖g x‖₊) μ ≠ ⊤)
Goal: MeasureTheory.Integrable (fun x => g x • f x) μ
-- MeasureTheory.Integrable.smul_essSup
theorem Integrable.smul_essSup {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
    {f : α → 𝕜} (hf : Integrable f μ) {g : α → β}
    (g_aestronglyMeasurable : AEStronglyMeasurable g μ)
    (ess_sup_g : essSup (fun x => (‖g x‖₊ : ℝ≥0∞)) μ ≠ ∞) :
    Integrable (fun x : α => f x • g x) 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) {g : α → β} (g_aestronglyMeasurable : MeasureTheory.AEStronglyMeasurable g μ) (ess_sup_g : essSup (fun x => ↑‖g x‖₊) μ ≠ ⊤)
Goal: MeasureTheory.Integrable (fun x => f x • g x) μ
-- MeasureTheory.integrable_norm_iff
theorem integrable_norm_iff {f : α → β} (hf : AEStronglyMeasurable f μ) :
    Integrable (fun a => ‖f a‖) μ ↔ 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.Integrable (fun a => ‖f a‖) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_of_norm_sub_le
theorem integrable_of_norm_sub_le {f₀ f₁ : α → β} {g : α → ℝ} (hf₁_m : AEStronglyMeasurable f₁ μ)
    (hf₀_i : Integrable f₀ μ) (hg_i : Integrable g μ) (h : ∀ᵐ a ∂μ, ‖f₀ a - f₁ a‖ ≤ g a) :
    Integrable f₁ μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f₀ f₁ : α → β} {g : α → ℝ} (hf₁_m : MeasureTheory.AEStronglyMeasurable f₁ μ) (hf₀_i : MeasureTheory.Integrable f₀ μ) (hg_i : MeasureTheory.Integrable g μ) (h : ∀ᵐ (a : α) ∂μ, ‖f₀ a - f₁ a‖ ≤ g a)
Goal: MeasureTheory.Integrable f₁ μ
-- MeasureTheory.Integrable.prod_mk
theorem Integrable.prod_mk {f : α → β} {g : α → γ} (hf : Integrable f μ) (hg : Integrable g μ) :
    Integrable (fun x => (f x, g x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [NormedAddCommGroup γ] {f : α → β} {g : α → γ} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable (fun x => (f x, g x)) μ
-- MeasureTheory.Memℒp.integrable
theorem Memℒp.integrable {q : ℝ≥0∞} (hq1 : 1 ≤ q) {f : α → β} [IsFiniteMeasure μ]
    (hfq : Memℒp f q μ) : Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {q : ℝ≥0∞} (hq1 : 1 ≤ q) {f : α → β} [MeasureTheory.IsFiniteMeasure μ] (hfq : MeasureTheory.Memℒp f q μ)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.measure_norm_ge_lt_top
theorem Integrable.measure_norm_ge_lt_top {f : α → β} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
    μ { x | ε ≤ ‖f x‖ } < ∞ 
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) {ε : ℝ} (hε : 0 < ε)
Goal: μ {x | ε ≤ ‖f x‖} < ⊤
-- MeasureTheory.Integrable.measure_norm_gt_lt_top
lemma Integrable.measure_norm_gt_lt_top {f : α → β} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
    μ {x | ε < ‖f x‖} < ∞ 
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) {ε : ℝ} (hε : 0 < ε)
Goal: μ {x | ε < ‖f x‖} < ⊤
-- MeasureTheory.Integrable.measure_ge_lt_top
lemma Integrable.measure_ge_lt_top {f : α → ℝ} (hf : Integrable f μ) {ε : ℝ} (ε_pos : 0 < ε) :
    μ {a : α | ε ≤ f a} < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) {ε : ℝ} (ε_pos : 0 < ε)
Goal: μ {a | ε ≤ f a} < ⊤
-- MeasureTheory.Integrable.measure_le_lt_top
lemma Integrable.measure_le_lt_top {f : α → ℝ} (hf : Integrable f μ) {c : ℝ} (c_neg : c < 0) :
    μ {a : α | f a ≤ c} < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) {c : ℝ} (c_neg : c < 0)
Goal: μ {a | f a ≤ c} < ⊤
-- MeasureTheory.Integrable.measure_gt_lt_top
lemma Integrable.measure_gt_lt_top {f : α → ℝ} (hf : Integrable f μ) {ε : ℝ} (ε_pos : 0 < ε) :
    μ {a : α | ε < f a} < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) {ε : ℝ} (ε_pos : 0 < ε)
Goal: μ {a | ε < f a} < ⊤
-- MeasureTheory.Integrable.measure_lt_lt_top
lemma Integrable.measure_lt_lt_top {f : α → ℝ} (hf : Integrable f μ) {c : ℝ} (c_neg : c < 0) :
    μ {a : α | f a < c} < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) {c : ℝ} (c_neg : c < 0)
Goal: μ {a | f a < c} < ⊤
-- MeasureTheory.LipschitzWith.integrable_comp_iff_of_antilipschitz
theorem LipschitzWith.integrable_comp_iff_of_antilipschitz {K K'} {f : α → β} {g : β → γ}
    (hg : LipschitzWith K g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0) :
    Integrable (g ∘ f) μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [NormedAddCommGroup γ] {K K' : ℝ≥0} {f : α → β} {g : β → γ} (hg : LipschitzWith K g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0)
Goal: MeasureTheory.Integrable (g ∘ f) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.real_toNNReal
theorem Integrable.real_toNNReal {f : α → ℝ} (hf : Integrable f μ) :
    Integrable (fun x => ((f x).toNNReal : ℝ)) 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 x => ↑(f x).toNNReal) μ
-- MeasureTheory.ofReal_toReal_ae_eq
theorem ofReal_toReal_ae_eq {f : α → ℝ≥0∞} (hf : ∀ᵐ x ∂μ, f x < ∞) :
    (fun x => ENNReal.ofReal (f x).toReal) =ᵐ[μ] f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : ∀ᵐ (x : α) ∂μ, f x < ⊤)
Goal: (fun x => ENNReal.ofReal (f x).toReal) =ᵐ[μ] f
-- MeasureTheory.coe_toNNReal_ae_eq
theorem coe_toNNReal_ae_eq {f : α → ℝ≥0∞} (hf : ∀ᵐ x ∂μ, f x < ∞) :
    (fun x => ((f x).toNNReal : ℝ≥0∞)) =ᵐ[μ] f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : ∀ᵐ (x : α) ∂μ, f x < ⊤)
Goal: (fun x => ↑(f x).toNNReal) =ᵐ[μ] f
-- MeasureTheory.hasFiniteIntegral_count_iff
lemma hasFiniteIntegral_count_iff :
    HasFiniteIntegral f Measure.count ↔ Summable (‖f ·‖) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] [MeasurableSingletonClass α] {f : α → β}
Goal: MeasureTheory.HasFiniteIntegral f MeasureTheory.Measure.count ↔ Summable fun x => ‖f x‖
-- MeasureTheory.integrable_count_iff
lemma integrable_count_iff :
    Integrable f Measure.count ↔ Summable (‖f ·‖) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup β] [MeasurableSingletonClass α] {f : α → β}
Goal: MeasureTheory.Integrable f MeasureTheory.Measure.count ↔ Summable fun x => ‖f x‖
-- MeasureTheory.integrable_withDensity_iff_integrable_coe_smul
theorem integrable_withDensity_iff_integrable_coe_smul {f : α → ℝ≥0} (hf : Measurable f)
    {g : α → E} :
    Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => (f x : ℝ) • g x) 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} (hf : Measurable f) {g : α → E}
Goal: MeasureTheory.Integrable g (μ.withDensity fun x => ↑(f x)) ↔ MeasureTheory.Integrable (fun x => ↑(f x) • g x) μ
-- MeasureTheory.integrable_withDensity_iff_integrable_smul
theorem integrable_withDensity_iff_integrable_smul {f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
    Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => f x • g x) 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} (hf : Measurable f) {g : α → E}
Goal: MeasureTheory.Integrable g (μ.withDensity fun x => ↑(f x)) ↔ MeasureTheory.Integrable (fun x => f x • g x) μ
-- MeasureTheory.integrable_withDensity_iff_integrable_smul'
theorem integrable_withDensity_iff_integrable_smul' {f : α → ℝ≥0∞} (hf : Measurable f)
    (hflt : ∀ᵐ x ∂μ, f x < ∞) {g : α → E} :
    Integrable g (μ.withDensity f) ↔ Integrable (fun x => (f x).toReal • g x) 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∞} (hf : Measurable f) (hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤) {g : α → E}
Goal: MeasureTheory.Integrable g (μ.withDensity f) ↔ MeasureTheory.Integrable (fun x => (f x).toReal • g x) μ
-- MeasureTheory.integrable_withDensity_iff_integrable_coe_smul₀
theorem integrable_withDensity_iff_integrable_coe_smul₀ {f : α → ℝ≥0} (hf : AEMeasurable f μ)
    {g : α → E} :
    Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => (f x : ℝ) • g x) 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} (hf : AEMeasurable f μ) {g : α → E}
Goal: MeasureTheory.Integrable g (μ.withDensity fun x => ↑(f x)) ↔ MeasureTheory.Integrable (fun x => ↑(f x) • g x) μ
-- MeasureTheory.integrable_withDensity_iff_integrable_smul₀
theorem integrable_withDensity_iff_integrable_smul₀ {f : α → ℝ≥0} (hf : AEMeasurable f μ)
    {g : α → E} : Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => f x • g x) 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} (hf : AEMeasurable f μ) {g : α → E}
Goal: MeasureTheory.Integrable g (μ.withDensity fun x => ↑(f x)) ↔ MeasureTheory.Integrable (fun x => f x • g x) μ