ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.hasFiniteIntegral_congr
theorem hasFiniteIntegral_congr {f g : α → β} (h : f =ᵐ[μ] g) :
    HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : α → β} (h : f =ᵐ[μ] g)
Goal: MeasureTheory.HasFiniteIntegral f μ ↔ MeasureTheory.HasFiniteIntegral g μ
-- MeasureTheory.hasFiniteIntegral_const_iff
theorem hasFiniteIntegral_const_iff {c : β} :
    HasFiniteIntegral (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {c : β}
Goal: MeasureTheory.HasFiniteIntegral (fun x => c) μ ↔ c = 0 ∨ μ Set.univ < ⊤
-- MeasureTheory.hasFiniteIntegral_const
theorem hasFiniteIntegral_const [IsFiniteMeasure μ] (c : β) :
    HasFiniteIntegral (fun _ : α => c) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [MeasureTheory.IsFiniteMeasure μ] (c : β)
Goal: MeasureTheory.HasFiniteIntegral (fun x => c) μ
-- MeasureTheory.hasFiniteIntegral_of_bounded
theorem hasFiniteIntegral_of_bounded [IsFiniteMeasure μ] {f : α → β} {C : ℝ}
    (hC : ∀ᵐ a ∂μ, ‖f a‖ ≤ C) : HasFiniteIntegral 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 : ℝ} (hC : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ C)
Goal: MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.of_finite
theorem HasFiniteIntegral.of_finite [Finite α] [IsFiniteMeasure μ] {f : α → β} :
    HasFiniteIntegral f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [Finite α] [MeasureTheory.IsFiniteMeasure μ] {f : α → β}
Goal: MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.mono_measure
theorem HasFiniteIntegral.mono_measure {f : α → β} (h : HasFiniteIntegral f ν) (hμ : μ ≤ ν) :
    HasFiniteIntegral f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.HasFiniteIntegral f ν) (hμ : μ ≤ ν)
Goal: MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.add_measure
theorem HasFiniteIntegral.add_measure {f : α → β} (hμ : HasFiniteIntegral f μ)
    (hν : HasFiniteIntegral f ν) : HasFiniteIntegral f (μ + ν) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hμ : MeasureTheory.HasFiniteIntegral f μ) (hν : MeasureTheory.HasFiniteIntegral f ν)
Goal: MeasureTheory.HasFiniteIntegral f (μ + ν)
-- MeasureTheory.HasFiniteIntegral.left_of_add_measure
theorem HasFiniteIntegral.left_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) :
    HasFiniteIntegral f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.HasFiniteIntegral f (μ + ν))
Goal: MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.right_of_add_measure
theorem HasFiniteIntegral.right_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) :
    HasFiniteIntegral f ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.HasFiniteIntegral f (μ + ν))
Goal: MeasureTheory.HasFiniteIntegral f ν
-- MeasureTheory.hasFiniteIntegral_add_measure
@[simp]
theorem hasFiniteIntegral_add_measure {f : α → β} :
    HasFiniteIntegral f (μ + ν) ↔ HasFiniteIntegral f μ ∧ HasFiniteIntegral f ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β}
Goal: MeasureTheory.HasFiniteIntegral f (μ + ν) ↔ MeasureTheory.HasFiniteIntegral f μ ∧ MeasureTheory.HasFiniteIntegral f ν
-- MeasureTheory.HasFiniteIntegral.smul_measure
theorem HasFiniteIntegral.smul_measure {f : α → β} (h : HasFiniteIntegral f μ) {c : ℝ≥0∞}
    (hc : c ≠ ∞) : HasFiniteIntegral f (c • μ) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.HasFiniteIntegral f μ) {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: MeasureTheory.HasFiniteIntegral f (c • μ)
-- MeasureTheory.hasFiniteIntegral_zero_measure
@[simp]
theorem hasFiniteIntegral_zero_measure {m : MeasurableSpace α} (f : α → β) :
    HasFiniteIntegral f (0 : Measure α) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {m : MeasurableSpace α} (f : α → β)
Goal: MeasureTheory.HasFiniteIntegral f 0
-- MeasureTheory.hasFiniteIntegral_zero
@[simp]
theorem hasFiniteIntegral_zero : HasFiniteIntegral (fun _ : α => (0 : β)) μ 
Arguments: (α : Type u_1) (β : Type u_2) {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [NormedAddCommGroup β]
Goal: MeasureTheory.HasFiniteIntegral (fun x => 0) μ
-- MeasureTheory.HasFiniteIntegral.neg
theorem HasFiniteIntegral.neg {f : α → β} (hfi : HasFiniteIntegral f μ) :
    HasFiniteIntegral (-f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hfi : MeasureTheory.HasFiniteIntegral f μ)
Goal: MeasureTheory.HasFiniteIntegral (-f) μ
-- MeasureTheory.hasFiniteIntegral_neg_iff
@[simp]
theorem hasFiniteIntegral_neg_iff {f : α → β} : HasFiniteIntegral (-f) μ ↔ HasFiniteIntegral 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.HasFiniteIntegral (-f) μ ↔ MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.norm
theorem HasFiniteIntegral.norm {f : α → β} (hfi : HasFiniteIntegral f μ) :
    HasFiniteIntegral (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} (hfi : MeasureTheory.HasFiniteIntegral f μ)
Goal: MeasureTheory.HasFiniteIntegral (fun a => ‖f a‖) μ
-- MeasureTheory.hasFiniteIntegral_norm_iff
theorem hasFiniteIntegral_norm_iff (f : α → β) :
    HasFiniteIntegral (fun a => ‖f a‖) μ ↔ HasFiniteIntegral 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.HasFiniteIntegral (fun a => ‖f a‖) μ ↔ MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top
theorem hasFiniteIntegral_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hf : (∫⁻ x, f x ∂μ) ≠ ∞) :
    HasFiniteIntegral (fun x => (f x).toReal) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: MeasureTheory.HasFiniteIntegral (fun x => (f x).toReal) μ
-- MeasureTheory.isFiniteMeasure_withDensity_ofReal
theorem isFiniteMeasure_withDensity_ofReal {f : α → ℝ} (hfi : HasFiniteIntegral f μ) :
    IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal <| f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hfi : MeasureTheory.HasFiniteIntegral f μ)
Goal: MeasureTheory.IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal (f x))
-- MeasureTheory.all_ae_ofReal_F_le_bound
theorem all_ae_ofReal_F_le_bound (h : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) :
    ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {F : ℕ → α → β} {bound : α → ℝ} (h : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) (n : ℕ)
Goal: ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
-- MeasureTheory.all_ae_tendsto_ofReal_norm
theorem all_ae_tendsto_ofReal_norm (h : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop <| 𝓝 <| f a) :
    ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a‖) atTop <| 𝓝 <| ENNReal.ofReal ‖f a‖ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {F : ℕ → α → β} {f : α → β} (h : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a)))
Goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => ENNReal.ofReal ‖F n a‖) Filter.atTop (𝓝 (ENNReal.ofReal ‖f a‖))
-- MeasureTheory.all_ae_ofReal_f_le_bound
theorem all_ae_ofReal_f_le_bound (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    ∀ᵐ a ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} (h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a)))
Goal: ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)
-- MeasureTheory.hasFiniteIntegral_of_dominated_convergence
theorem hasFiniteIntegral_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : HasFiniteIntegral f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} (bound_hasFiniteIntegral : MeasureTheory.HasFiniteIntegral bound μ) (h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a)))
Goal: MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence
theorem tendsto_lintegral_norm_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ)
    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} (F_measurable : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (F n) μ) (bound_hasFiniteIntegral : MeasureTheory.HasFiniteIntegral bound μ) (h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a)))
Goal: Filter.Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) Filter.atTop (𝓝 0)
-- MeasureTheory.HasFiniteIntegral.max_zero
theorem HasFiniteIntegral.max_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) :
    HasFiniteIntegral (fun a => max (f a) 0) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.HasFiniteIntegral f μ)
Goal: MeasureTheory.HasFiniteIntegral (fun a => max (f a) 0) μ
-- MeasureTheory.HasFiniteIntegral.min_zero
theorem HasFiniteIntegral.min_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) :
    HasFiniteIntegral (fun a => min (f a) 0) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.HasFiniteIntegral f μ)
Goal: MeasureTheory.HasFiniteIntegral (fun a => min (f a) 0) μ
-- MeasureTheory.HasFiniteIntegral.smul
theorem HasFiniteIntegral.smul [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜)
    {f : α → β} : HasFiniteIntegral f μ → HasFiniteIntegral (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}
Goal: MeasureTheory.HasFiniteIntegral f μ → MeasureTheory.HasFiniteIntegral (c • f) μ
-- MeasureTheory.hasFiniteIntegral_smul_iff
theorem hasFiniteIntegral_smul_iff [NormedRing 𝕜] [MulActionWithZero 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜}
    (hc : IsUnit c) (f : α → β) : HasFiniteIntegral (c • f) μ ↔ HasFiniteIntegral 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] [MulActionWithZero 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜} (hc : IsUnit c) (f : α → β)
Goal: MeasureTheory.HasFiniteIntegral (c • f) μ ↔ MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.const_mul
theorem HasFiniteIntegral.const_mul [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) :
    HasFiniteIntegral (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.HasFiniteIntegral f μ) (c : 𝕜)
Goal: MeasureTheory.HasFiniteIntegral (fun x => c * f x) μ
-- MeasureTheory.HasFiniteIntegral.mul_const
theorem HasFiniteIntegral.mul_const [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) :
    HasFiniteIntegral (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.HasFiniteIntegral f μ) (c : 𝕜)
Goal: MeasureTheory.HasFiniteIntegral (fun x => f x * c) μ
-- MeasureTheory.memℒp_one_iff_integrable
theorem memℒp_one_iff_integrable {f : α → β} : Memℒp f 1 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.Memℒp f 1 μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.aestronglyMeasurable
theorem Integrable.aestronglyMeasurable {f : α → β} (hf : Integrable f μ) :
    AEStronglyMeasurable 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.AEStronglyMeasurable f μ
-- MeasureTheory.Integrable.aemeasurable
theorem Integrable.aemeasurable [MeasurableSpace β] [BorelSpace β] {f : α → β}
    (hf : Integrable f μ) : 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 : α → β} (hf : MeasureTheory.Integrable f μ)
Goal: AEMeasurable f μ
-- MeasureTheory.Integrable.hasFiniteIntegral
theorem Integrable.hasFiniteIntegral {f : α → β} (hf : Integrable f μ) : HasFiniteIntegral 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.HasFiniteIntegral f μ
-- MeasureTheory.Integrable.mono
theorem Integrable.mono {f : α → β} {g : α → γ} (hg : Integrable g μ)
    (hf : AEStronglyMeasurable f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : 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] {f : α → β} {g : α → γ} (hg : MeasureTheory.Integrable g μ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ ‖g a‖)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.mono'
theorem Integrable.mono' {f : α → β} {g : α → ℝ} (hg : Integrable g μ)
    (hf : AEStronglyMeasurable f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} {g : α → ℝ} (hg : MeasureTheory.Integrable g μ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ g a)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.congr'
theorem Integrable.congr' {f : α → β} {g : α → γ} (hf : Integrable f μ)
    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : Integrable g 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.AEStronglyMeasurable g μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
Goal: MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_congr'
theorem integrable_congr' {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) :
    Integrable f μ ↔ Integrable g 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.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
Goal: MeasureTheory.Integrable f μ ↔ MeasureTheory.Integrable g μ
-- MeasureTheory.Integrable.congr
theorem Integrable.congr {f g : α → β} (hf : Integrable f μ) (h : f =ᵐ[μ] g) : 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) (h : f =ᵐ[μ] g)
Goal: MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_congr
theorem integrable_congr {f g : α → β} (h : f =ᵐ[μ] g) : Integrable 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} (h : f =ᵐ[μ] g)
Goal: MeasureTheory.Integrable f μ ↔ MeasureTheory.Integrable g μ
-- MeasureTheory.integrable_const_iff
theorem integrable_const_iff {c : β} : Integrable (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {c : β}
Goal: MeasureTheory.Integrable (fun x => c) μ ↔ c = 0 ∨ μ Set.univ < ⊤
-- MeasureTheory.integrable_const
@[simp]
theorem integrable_const [IsFiniteMeasure μ] (c : β) : Integrable (fun _ : α => c) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [MeasureTheory.IsFiniteMeasure μ] (c : β)
Goal: MeasureTheory.Integrable (fun x => c) μ
-- MeasureTheory.Integrable.of_finite
@[simp]
theorem Integrable.of_finite [Finite α] [MeasurableSpace α] [MeasurableSingletonClass α]
    (μ : Measure α) [IsFiniteMeasure μ] (f : α → β) : Integrable (fun a ↦ f a) μ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] [Finite α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → β)
Goal: MeasureTheory.Integrable (fun a => f a) μ
-- MeasureTheory.Memℒp.integrable_norm_rpow
theorem Memℒp.integrable_norm_rpow {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) : Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} {p : ℝ≥0∞} (hf : MeasureTheory.Memℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.Integrable (fun x => ‖f x‖ ^ p.toReal) μ
-- MeasureTheory.Memℒp.integrable_norm_rpow'
theorem Memℒp.integrable_norm_rpow' [IsFiniteMeasure μ] {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) :
    Integrable (fun x : α => ‖f x‖ ^ p.toReal) 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} {p : ℝ≥0∞} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Integrable (fun x => ‖f x‖ ^ p.toReal) μ
-- MeasureTheory.Integrable.mono_measure
theorem Integrable.mono_measure {f : α → β} (h : Integrable f ν) (hμ : μ ≤ ν) : Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.Integrable f ν) (hμ : μ ≤ ν)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.of_measure_le_smul
theorem Integrable.of_measure_le_smul {μ' : Measure α} (c : ℝ≥0∞) (hc : c ≠ ∞) (hμ'_le : μ' ≤ c • μ)
    {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] {μ' : MeasureTheory.Measure α} (c : ℝ≥0∞) (hc : c ≠ ⊤) (hμ'_le : μ' ≤ c • μ) {f : α → β} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable f μ'
-- MeasureTheory.Integrable.add_measure
theorem Integrable.add_measure {f : α → β} (hμ : Integrable f μ) (hν : Integrable f ν) :
    Integrable f (μ + ν) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hμ : MeasureTheory.Integrable f μ) (hν : MeasureTheory.Integrable f ν)
Goal: MeasureTheory.Integrable f (μ + ν)
-- MeasureTheory.Integrable.left_of_add_measure
theorem Integrable.left_of_add_measure {f : α → β} (h : Integrable f (μ + ν)) : Integrable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.Integrable f (μ + ν))
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.right_of_add_measure
theorem Integrable.right_of_add_measure {f : α → β} (h : Integrable f (μ + ν)) :
    Integrable f ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (h : MeasureTheory.Integrable f (μ + ν))
Goal: MeasureTheory.Integrable f ν
-- MeasureTheory.integrable_add_measure
@[simp]
theorem integrable_add_measure {f : α → β} :
    Integrable f (μ + ν) ↔ Integrable f μ ∧ Integrable f ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β}
Goal: MeasureTheory.Integrable f (μ + ν) ↔ MeasureTheory.Integrable f μ ∧ MeasureTheory.Integrable f ν
-- MeasureTheory.integrable_zero_measure
@[simp]
theorem integrable_zero_measure {_ : MeasurableSpace α} {f : α → β} :
    Integrable f (0 : Measure α) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β]
Goal: ∀ {x : MeasurableSpace α} {f : α → β}, MeasureTheory.Integrable f 0
-- MeasureTheory.integrable_finset_sum_measure
theorem integrable_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → β} {μ : ι → Measure α}
    {s : Finset ι} : Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {ι : Type u_5} {m : MeasurableSpace α} {f : α → β} {μ : ι → MeasureTheory.Measure α} {s : Finset ι}
Goal: MeasureTheory.Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, MeasureTheory.Integrable f (μ i)
-- MeasureTheory.Integrable.smul_measure
theorem Integrable.smul_measure {f : α → β} (h : Integrable f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) :
    Integrable f (c • 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) {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: MeasureTheory.Integrable f (c • μ)
-- MeasureTheory.Integrable.smul_measure_nnreal
theorem Integrable.smul_measure_nnreal {f : α → β} (h : Integrable f μ) {c : ℝ≥0} :
    Integrable f (c • 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) {c : ℝ≥0}
Goal: MeasureTheory.Integrable f (c • μ)
-- MeasureTheory.integrable_smul_measure
theorem integrable_smul_measure {f : α → β} {c : ℝ≥0∞} (h₁ : c ≠ 0) (h₂ : c ≠ ∞) :
    Integrable f (c • 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} {c : ℝ≥0∞} (h₁ : c ≠ 0) (h₂ : c ≠ ⊤)
Goal: MeasureTheory.Integrable f (c • μ) ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_inv_smul_measure
theorem integrable_inv_smul_measure {f : α → β} {c : ℝ≥0∞} (h₁ : c ≠ 0) (h₂ : c ≠ ∞) :
    Integrable f (c⁻¹ • 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} {c : ℝ≥0∞} (h₁ : c ≠ 0) (h₂ : c ≠ ⊤)
Goal: MeasureTheory.Integrable f (c⁻¹ • μ) ↔ MeasureTheory.Integrable f μ