ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.setIntegral_compLp
theorem setIntegral_compLp (L : E →L[𝕜] F) (φ : Lp E p μ) {s : Set X} (hs : MeasurableSet s) :
    ∫ x in s, (L.compLp φ) x ∂μ = ∫ x in s, L (φ x) ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : ℝ≥0∞} [NormedSpace ℝ F] (L : E →L[𝕜] F) (φ : ↥(MeasureTheory.Lp E p μ)) {s : Set X} (hs : MeasurableSet s)
Goal: ∫ (x : X) in s, ↑↑(L.compLp φ) x ∂μ = ∫ (x : X) in s, L (↑↑φ x) ∂μ
-- ContinuousLinearMap.continuous_integral_comp_L1
theorem continuous_integral_comp_L1 (L : E →L[𝕜] F) :
    Continuous fun φ : X →₁[μ] E => ∫ x : X, L (φ x) ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] (L : E →L[𝕜] F)
Goal: Continuous fun φ => ∫ (x : X), L (↑↑φ x) ∂μ
-- ContinuousLinearMap.integral_comp_comm
theorem integral_comp_comm [CompleteSpace E] (L : E →L[𝕜] F) {φ : X → E} (φ_int : Integrable φ μ) :
    ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [CompleteSpace F] [NormedSpace ℝ E] [CompleteSpace E] (L : E →L[𝕜] F) {φ : X → E} (φ_int : MeasureTheory.Integrable φ μ)
Goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)
-- ContinuousLinearMap.integral_apply
theorem integral_apply {H : Type*} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : X → H →L[𝕜] E}
    (φ_int : Integrable φ μ) (v : H) : (∫ x, φ x ∂μ) v = ∫ x, φ x v ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace ℝ E] {H : Type u_6} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : X → H →L[𝕜] E} (φ_int : MeasureTheory.Integrable φ μ) (v : H)
Goal: (∫ (x : X), φ x ∂μ) v = ∫ (x : X), (φ x) v ∂μ
-- ContinuousMultilinearMap.integral_apply
theorem _root_.ContinuousMultilinearMap.integral_apply {ι : Type*} [Fintype ι] {M : ι → Type*}
    [∀ i, NormedAddCommGroup (M i)] [∀ i, NormedSpace 𝕜 (M i)]
    {φ : X → ContinuousMultilinearMap 𝕜 M E} (φ_int : Integrable φ μ) (m : ∀ i, M i) :
    (∫ x, φ x ∂μ) m = ∫ x, φ x m ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace ℝ E] {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {φ : X → ContinuousMultilinearMap 𝕜 M E} (φ_int : MeasureTheory.Integrable φ μ) (m : (i : ι) → M i)
Goal: (∫ (x : X), φ x ∂μ) m = ∫ (x : X), (φ x) m ∂μ
-- ContinuousLinearMap.integral_comp_comm'
theorem integral_comp_comm' (L : E →L[𝕜] F) {K} (hL : AntilipschitzWith K L) (φ : X → E) :
    ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [CompleteSpace F] [NormedSpace ℝ E] [CompleteSpace E] (L : E →L[𝕜] F) {K : ℝ≥0} (hL : AntilipschitzWith K ⇑L) (φ : X → E)
Goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)
-- ContinuousLinearMap.integral_comp_L1_comm
theorem integral_comp_L1_comm (L : E →L[𝕜] F) (φ : X →₁[μ] E) :
    ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [CompleteSpace F] [NormedSpace ℝ E] [CompleteSpace E] (L : E →L[𝕜] F) (φ : ↥(MeasureTheory.Lp E 1 μ))
Goal: ∫ (x : X), L (↑↑φ x) ∂μ = L (∫ (x : X), ↑↑φ x ∂μ)
-- LinearIsometry.integral_comp_comm
theorem integral_comp_comm (L : E →ₗᵢ[𝕜] F) (φ : X → E) : ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] [NormedSpace ℝ F] [CompleteSpace E] [NormedSpace ℝ E] (L : E →ₗᵢ[𝕜] F) (φ : X → E)
Goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)
-- ContinuousLinearEquiv.integral_comp_comm
theorem integral_comp_comm (L : E ≃L[𝕜] F) (φ : X → E) : ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [NormedSpace ℝ E] (L : E ≃L[𝕜] F) (φ : X → E)
Goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)
-- integral_ofReal
@[norm_cast]
theorem integral_ofReal {f : X → ℝ} : ∫ x, (f x : 𝕜) ∂μ = ↑(∫ x, f x ∂μ) 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] {f : X → ℝ}
Goal: ∫ (x : X), ↑(f x) ∂μ = ↑(∫ (x : X), f x ∂μ)
-- integral_re
theorem integral_re {f : X → 𝕜} (hf : Integrable f μ) :
    ∫ x, RCLike.re (f x) ∂μ = RCLike.re (∫ x, f x ∂μ) 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] {f : X → 𝕜} (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X), RCLike.re (f x) ∂μ = RCLike.re (∫ (x : X), f x ∂μ)
-- integral_im
theorem integral_im {f : X → 𝕜} (hf : Integrable f μ) :
    ∫ x, RCLike.im (f x) ∂μ = RCLike.im (∫ x, f x ∂μ) 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] {f : X → 𝕜} (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X), RCLike.im (f x) ∂μ = RCLike.im (∫ (x : X), f x ∂μ)
-- integral_conj
theorem integral_conj {f : X → 𝕜} : ∫ x, conj (f x) ∂μ = conj (∫ x, f x ∂μ) 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] {f : X → 𝕜}
Goal: ∫ (x : X), (starRingEnd 𝕜) (f x) ∂μ = (starRingEnd 𝕜) (∫ (x : X), f x ∂μ)
-- integral_coe_re_add_coe_im
theorem integral_coe_re_add_coe_im {f : X → 𝕜} (hf : Integrable f μ) :
    ∫ x, (re (f x) : 𝕜) ∂μ + (∫ x, (im (f x) : 𝕜) ∂μ) * RCLike.I = ∫ x, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] {f : X → 𝕜} (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X), ↑(RCLike.re (f x)) ∂μ + (∫ (x : X), ↑(RCLike.im (f x)) ∂μ) * RCLike.I = ∫ (x : X), f x ∂μ
-- integral_re_add_im
theorem integral_re_add_im {f : X → 𝕜} (hf : Integrable f μ) :
    ((∫ x, RCLike.re (f x) ∂μ : ℝ) : 𝕜) + (∫ x, RCLike.im (f x) ∂μ : ℝ) * RCLike.I =
      ∫ x, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] {f : X → 𝕜} (hf : MeasureTheory.Integrable f μ)
Goal: ↑(∫ (x : X), RCLike.re (f x) ∂μ) + ↑(∫ (x : X), RCLike.im (f x) ∂μ) * RCLike.I = ∫ (x : X), f x ∂μ
-- setIntegral_re_add_im
theorem setIntegral_re_add_im {f : X → 𝕜} {i : Set X} (hf : IntegrableOn f i μ) :
    ((∫ x in i, RCLike.re (f x) ∂μ : ℝ) : 𝕜) + (∫ x in i, RCLike.im (f x) ∂μ : ℝ) * RCLike.I =
      ∫ x in i, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_5} [RCLike 𝕜] {f : X → 𝕜} {i : Set X} (hf : MeasureTheory.IntegrableOn f i μ)
Goal: ↑(∫ (x : X) in i, RCLike.re (f x) ∂μ) + ↑(∫ (x : X) in i, RCLike.im (f x) ∂μ) * RCLike.I = ∫ (x : X) in i, f x ∂μ
-- swap_integral
lemma swap_integral (f : X → E × F) : (∫ x, f x ∂μ).swap = ∫ x, (f x).swap ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] (f : X → E × F)
Goal: (∫ (x : X), f x ∂μ).swap = ∫ (x : X), (f x).swap ∂μ
-- fst_integral
theorem fst_integral [CompleteSpace F] {f : X → E × F} (hf : Integrable f μ) :
    (∫ x, f x ∂μ).1 = ∫ x, (f x).1 ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [CompleteSpace F] {f : X → E × F} (hf : MeasureTheory.Integrable f μ)
Goal: (∫ (x : X), f x ∂μ).1 = ∫ (x : X), (f x).1 ∂μ
-- snd_integral
theorem snd_integral [CompleteSpace E] {f : X → E × F} (hf : Integrable f μ) :
    (∫ x, f x ∂μ).2 = ∫ x, (f x).2 ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [CompleteSpace E] {f : X → E × F} (hf : MeasureTheory.Integrable f μ)
Goal: (∫ (x : X), f x ∂μ).2 = ∫ (x : X), (f x).2 ∂μ
-- integral_pair
theorem integral_pair [CompleteSpace E] [CompleteSpace F] {f : X → E} {g : X → F}
    (hf : Integrable f μ) (hg : Integrable g μ) :
    ∫ x, (f x, g x) ∂μ = (∫ x, f x ∂μ, ∫ x, g x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [CompleteSpace E] [CompleteSpace F] {f : X → E} {g : X → F} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: ∫ (x : X), (f x, g x) ∂μ = (∫ (x : X), f x ∂μ, ∫ (x : X), g x ∂μ)
-- integral_smul_const
theorem integral_smul_const {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [CompleteSpace E]
    (f : X → 𝕜) (c : E) :
    ∫ x, f x • c ∂μ = (∫ x, f x ∂μ) • c 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_6} [RCLike 𝕜] [NormedSpace 𝕜 E] [CompleteSpace E] (f : X → 𝕜) (c : E)
Goal: ∫ (x : X), f x • c ∂μ = (∫ (x : X), f x ∂μ) • c
-- integral_withDensity_eq_integral_smul
theorem integral_withDensity_eq_integral_smul {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E) :
    ∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E)
Goal: (∫ (x : X), g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • g x ∂μ
-- integral_withDensity_eq_integral_smul₀
theorem integral_withDensity_eq_integral_smul₀ {f : X → ℝ≥0} (hf : AEMeasurable f μ) (g : X → E) :
    ∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → ℝ≥0} (hf : AEMeasurable f μ) (g : X → E)
Goal: (∫ (x : X), g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • g x ∂μ
-- setIntegral_withDensity_eq_setIntegral_smul
theorem setIntegral_withDensity_eq_setIntegral_smul {f : X → ℝ≥0} (f_meas : Measurable f)
    (g : X → E) {s : Set X} (hs : MeasurableSet s) :
    ∫ x in s, g x ∂μ.withDensity (fun x => f x) = ∫ x in s, f x • g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E) {s : Set X} (hs : MeasurableSet s)
Goal: (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ
-- setIntegral_withDensity_eq_setIntegral_smul₀
theorem setIntegral_withDensity_eq_setIntegral_smul₀ {f : X → ℝ≥0} {s : Set X}
    (hf : AEMeasurable f (μ.restrict s)) (g : X → E) (hs : MeasurableSet s) :
    ∫ x in s, g x ∂μ.withDensity (fun x => f x) = ∫ x in s, f x • g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → ℝ≥0} {s : Set X} (hf : AEMeasurable f (μ.restrict s)) (g : X → E) (hs : MeasurableSet s)
Goal: (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ
-- setIntegral_withDensity_eq_setIntegral_smul₀'
theorem setIntegral_withDensity_eq_setIntegral_smul₀' [SFinite μ] {f : X → ℝ≥0} (s : Set X)
    (hf : AEMeasurable f (μ.restrict s)) (g : X → E)  :
    ∫ x in s, g x ∂μ.withDensity (fun x => f x) = ∫ x in s, f x • g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] {f : X → ℝ≥0} (s : Set X) (hf : AEMeasurable f (μ.restrict s)) (g : X → E)
Goal: (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ
-- measure_le_lintegral_thickenedIndicatorAux
theorem measure_le_lintegral_thickenedIndicatorAux (μ : Measure X) {E : Set X}
    (E_mble : MeasurableSet E) (δ : ℝ) : μ E ≤ ∫⁻ x, (thickenedIndicatorAux δ E x : ℝ≥0∞) ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] [PseudoEMetricSpace X] (μ : MeasureTheory.Measure X) {E : Set X} (E_mble : MeasurableSet E) (δ : ℝ)
Goal: μ E ≤ ∫⁻ (x : X), thickenedIndicatorAux δ E x ∂μ
-- measure_le_lintegral_thickenedIndicator
theorem measure_le_lintegral_thickenedIndicator (μ : Measure X) {E : Set X}
    (E_mble : MeasurableSet E) {δ : ℝ} (δ_pos : 0 < δ) :
    μ E ≤ ∫⁻ x, (thickenedIndicator δ_pos E x : ℝ≥0∞) ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] [PseudoEMetricSpace X] (μ : MeasureTheory.Measure X) {E : Set X} (E_mble : MeasurableSet E) {δ : ℝ} (δ_pos : 0 < δ)
Goal: μ E ≤ ∫⁻ (x : X), ↑((thickenedIndicator δ_pos E) x) ∂μ
-- MeasureTheory.Integrable.simpleFunc_mul
theorem Integrable.simpleFunc_mul (g : SimpleFunc X ℝ) (hf : Integrable f μ) :
    Integrable (⇑g * f) μ 
Arguments: {X : Type u_1} {f : X → ℝ} {m0 : MeasurableSpace X} {μ : MeasureTheory.Measure X} (g : MeasureTheory.SimpleFunc X ℝ) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (↑g * f) μ
-- MeasureTheory.Integrable.simpleFunc_mul'
theorem Integrable.simpleFunc_mul' (hm : m ≤ m0) (g : @SimpleFunc X m ℝ) (hf : Integrable f μ) :
    Integrable (⇑g * f) μ 
Arguments: {X : Type u_1} {f : X → ℝ} {m m0 : MeasurableSpace X} {μ : MeasureTheory.Measure X} (hm : m ≤ m0) (g : MeasureTheory.SimpleFunc X ℝ) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (↑g * f) μ
-- continuous_parametric_integral_of_continuous
theorem continuous_parametric_integral_of_continuous
    [FirstCountableTopology X] [LocallyCompactSpace X]
    [OpensMeasurableSpace Y] [SecondCountableTopologyEither Y E] [IsLocallyFiniteMeasure μ]
    {f : X → Y → E} (hf : Continuous f.uncurry) {s : Set Y} (hs : IsCompact s) :
    Continuous (∫ y in s, f · y ∂μ) 
Arguments: {X : Type u_1} {Y : Type u_2} {E : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace Y] {μ : MeasureTheory.Measure Y} [NormedAddCommGroup E] [NormedSpace ℝ E] [FirstCountableTopology X] [LocallyCompactSpace X] [OpensMeasurableSpace Y] [SecondCountableTopologyEither Y E] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : X → Y → E} (hf : Continuous (Function.uncurry f)) {s : Set Y} (hs : IsCompact s)
Goal: Continuous fun x => ∫ (y : Y) in s, f x y ∂μ
-- continuousOn_integral_bilinear_of_locally_integrable_of_compact_support
lemma continuousOn_integral_bilinear_of_locally_integrable_of_compact_support
    [NormedSpace 𝕜 E] (L : F →L[𝕜] G →L[𝕜] E)
    {f : X → Y → G} {s : Set X} {k : Set Y} {g : Y → F}
    (hk : IsCompact k) (hf : ContinuousOn f.uncurry (s ×ˢ univ))
    (hfs : ∀ p, ∀ x, p ∈ s → x ∉ k → f p x = 0) (hg : IntegrableOn g k μ) :
    ContinuousOn (fun x ↦ ∫ y, L (g y) (f x y) ∂μ) s 
Arguments: {X : Type u_1} {Y : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {𝕜 : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] {μ : MeasureTheory.Measure Y} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedSpace 𝕜 E] (L : F →L[𝕜] G →L[𝕜] E) {f : X → Y → G} {s : Set X} {k : Set Y} {g : Y → F} (hk : IsCompact k) (hf : ContinuousOn (Function.uncurry f) (s ×ˢ Set.univ)) (hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0) (hg : MeasureTheory.IntegrableOn g k μ)
Goal: ContinuousOn (fun x => ∫ (y : Y), (L (g y)) (f x y) ∂μ) s
-- continuousOn_integral_of_compact_support
lemma continuousOn_integral_of_compact_support
    {f : X → Y → E} {s : Set X} {k : Set Y} [IsFiniteMeasureOnCompacts μ]
    (hk : IsCompact k) (hf : ContinuousOn f.uncurry (s ×ˢ univ))
    (hfs : ∀ p, ∀ x, p ∈ s → x ∉ k → f p x = 0) :
    ContinuousOn (fun x ↦ ∫ y, f x y ∂μ) s 
Arguments: {X : Type u_1} {Y : Type u_2} {E : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] {μ : MeasureTheory.Measure Y} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → Y → E} {s : Set X} {k : Set Y} [MeasureTheory.IsFiniteMeasureOnCompacts μ] (hk : IsCompact k) (hf : ContinuousOn (Function.uncurry f) (s ×ˢ Set.univ)) (hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0)
Goal: ContinuousOn (fun x => ∫ (y : Y), f x y ∂μ) s
-- MeasureTheory.lintegral_nnnorm_eq_lintegral_edist
theorem lintegral_nnnorm_eq_lintegral_edist (f : α → β) :
    ∫⁻ a, ‖f a‖₊ ∂μ = ∫⁻ 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)
Goal: ∫⁻ (a : α), ↑‖f a‖₊ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ
-- MeasureTheory.lintegral_norm_eq_lintegral_edist
theorem lintegral_norm_eq_lintegral_edist (f : α → β) :
    ∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ 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)
Goal: ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ
-- MeasureTheory.lintegral_edist_triangle
theorem lintegral_edist_triangle {f g h : α → β} (hf : AEStronglyMeasurable f μ)
    (hh : AEStronglyMeasurable h μ) :
    (∫⁻ a, edist (f a) (g a) ∂μ) ≤ (∫⁻ a, edist (f a) (h a) ∂μ) + ∫⁻ a, edist (g a) (h a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g h : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hh : MeasureTheory.AEStronglyMeasurable h μ)
Goal: ∫⁻ (a : α), edist (f a) (g a) ∂μ ≤ ∫⁻ (a : α), edist (f a) (h a) ∂μ + ∫⁻ (a : α), edist (g a) (h a) ∂μ
-- MeasureTheory.lintegral_nnnorm_zero
theorem lintegral_nnnorm_zero : (∫⁻ _ : α, ‖(0 : β)‖₊ ∂μ) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β]
Goal: ∫⁻ (x : α), ↑‖0‖₊ ∂μ = 0
-- MeasureTheory.lintegral_nnnorm_add_left
theorem lintegral_nnnorm_add_left {f : α → β} (hf : AEStronglyMeasurable f μ) (g : α → γ) :
    ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂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} (hf : MeasureTheory.AEStronglyMeasurable f μ) (g : α → γ)
Goal: ∫⁻ (a : α), ↑‖f a‖₊ + ↑‖g a‖₊ ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ∂μ + ∫⁻ (a : α), ↑‖g a‖₊ ∂μ
-- MeasureTheory.lintegral_nnnorm_add_right
theorem lintegral_nnnorm_add_right (f : α → β) {g : α → γ} (hg : AEStronglyMeasurable g μ) :
    ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂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.AEStronglyMeasurable g μ)
Goal: ∫⁻ (a : α), ↑‖f a‖₊ + ↑‖g a‖₊ ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ∂μ + ∫⁻ (a : α), ↑‖g a‖₊ ∂μ
-- MeasureTheory.lintegral_nnnorm_neg
theorem lintegral_nnnorm_neg {f : α → β} : (∫⁻ a, ‖(-f) a‖₊ ∂μ) = ∫⁻ 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}
Goal: ∫⁻ (a : α), ↑‖(-f) a‖₊ ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ∂μ
-- MeasureTheory.hasFiniteIntegral_def
theorem hasFiniteIntegral_def {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) :
    HasFiniteIntegral f μ ↔ ((∫⁻ a, ‖f a‖₊ ∂μ) < ∞) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β]
Goal: ∀ {x : MeasurableSpace α} (f : α → β) (μ : MeasureTheory.Measure α),
  MeasureTheory.HasFiniteIntegral f μ ↔ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ < ⊤
-- MeasureTheory.hasFiniteIntegral_iff_norm
theorem hasFiniteIntegral_iff_norm (f : α → β) :
    HasFiniteIntegral f μ ↔ (∫⁻ 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)
Goal: MeasureTheory.HasFiniteIntegral f μ ↔ ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ < ⊤
-- MeasureTheory.hasFiniteIntegral_iff_edist
theorem hasFiniteIntegral_iff_edist (f : α → β) :
    HasFiniteIntegral f μ ↔ (∫⁻ 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)
Goal: MeasureTheory.HasFiniteIntegral f μ ↔ ∫⁻ (a : α), edist (f a) 0 ∂μ < ⊤
-- MeasureTheory.hasFiniteIntegral_iff_ofReal
theorem hasFiniteIntegral_iff_ofReal {f : α → ℝ} (h : 0 ≤ᵐ[μ] f) :
    HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal (f a) ∂μ) < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (h : 0 ≤ᵐ[μ] f)
Goal: MeasureTheory.HasFiniteIntegral f μ ↔ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ < ⊤
-- MeasureTheory.hasFiniteIntegral_iff_ofNNReal
theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} :
    HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0}
Goal: MeasureTheory.HasFiniteIntegral (fun x => ↑(f x)) μ ↔ ∫⁻ (a : α), ↑(f a) ∂μ < ⊤
-- MeasureTheory.HasFiniteIntegral.mono
theorem HasFiniteIntegral.mono {f : α → β} {g : α → γ} (hg : HasFiniteIntegral g μ)
    (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : HasFiniteIntegral 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.HasFiniteIntegral g μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ ‖g a‖)
Goal: MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.mono'
theorem HasFiniteIntegral.mono' {f : α → β} {g : α → ℝ} (hg : HasFiniteIntegral g μ)
    (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : HasFiniteIntegral 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.HasFiniteIntegral g μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ g a)
Goal: MeasureTheory.HasFiniteIntegral f μ
-- MeasureTheory.HasFiniteIntegral.congr'
theorem HasFiniteIntegral.congr' {f : α → β} {g : α → γ} (hf : HasFiniteIntegral f μ)
    (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : HasFiniteIntegral 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.HasFiniteIntegral f μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
Goal: MeasureTheory.HasFiniteIntegral g μ
-- MeasureTheory.hasFiniteIntegral_congr'
theorem hasFiniteIntegral_congr' {f : α → β} {g : α → γ} (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) :
    HasFiniteIntegral f μ ↔ HasFiniteIntegral 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} (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
Goal: MeasureTheory.HasFiniteIntegral f μ ↔ MeasureTheory.HasFiniteIntegral g μ
-- MeasureTheory.HasFiniteIntegral.congr
theorem HasFiniteIntegral.congr {f g : α → β} (hf : HasFiniteIntegral f μ) (h : f =ᵐ[μ] g) :
    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} (hf : MeasureTheory.HasFiniteIntegral f μ) (h : f =ᵐ[μ] g)
Goal: MeasureTheory.HasFiniteIntegral g μ