ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.L1.integral_eq
theorem integral_eq (f : α →₁[μ] E) : integral f = integralCLM (E := E) f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.L1.integral f = MeasureTheory.L1.integralCLM f
-- MeasureTheory.L1.integral_eq_setToL1
theorem integral_eq_setToL1 (f : α →₁[μ] E) :
    integral f = setToL1 (E := E) (dominatedFinMeasAdditive_weightedSMul μ) f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.L1.integral f = (MeasureTheory.L1.setToL1 ⋯) f
-- MeasureTheory.L1.SimpleFunc.integral_L1_eq_integral
@[norm_cast]
theorem SimpleFunc.integral_L1_eq_integral (f : α →₁ₛ[μ] E) :
    L1.integral (f : α →₁[μ] E) = SimpleFunc.integral f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.integral ↑f = MeasureTheory.L1.SimpleFunc.integral f
-- MeasureTheory.L1.integral_zero
@[simp]
theorem integral_zero : integral (0 : α →₁[μ] E) = 0 
Arguments: (α : Type u_1) (E : Type u_2) [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E]
Goal: MeasureTheory.L1.integral 0 = 0
-- MeasureTheory.L1.integral_add
@[integral_simps]
theorem integral_add (f g : α →₁[μ] E) : integral (f + g) = integral f + integral g 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f g : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.L1.integral (f + g) = MeasureTheory.L1.integral f + MeasureTheory.L1.integral g
-- MeasureTheory.L1.integral_neg
@[integral_simps]
theorem integral_neg (f : α →₁[μ] E) : integral (-f) = -integral f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.L1.integral (-f) = -MeasureTheory.L1.integral f
-- MeasureTheory.L1.integral_sub
@[integral_simps]
theorem integral_sub (f g : α →₁[μ] E) : integral (f - g) = integral f - integral g 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f g : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.L1.integral (f - g) = MeasureTheory.L1.integral f - MeasureTheory.L1.integral g
-- MeasureTheory.L1.integral_smul
@[integral_simps]
theorem integral_smul (c : 𝕜) (f : α →₁[μ] E) : integral (c • f) = c • integral f 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℝ 𝕜 E] [CompleteSpace E] (c : 𝕜) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.L1.integral (c • f) = c • MeasureTheory.L1.integral f
-- MeasureTheory.L1.norm_Integral_le_one
theorem norm_Integral_le_one : ‖integralCLM (α := α) (E := E) (μ := μ)‖ ≤ 1 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E]
Goal: ‖MeasureTheory.L1.integralCLM‖ ≤ 1
-- MeasureTheory.L1.nnnorm_Integral_le_one
theorem nnnorm_Integral_le_one : ‖integralCLM (α := α) (E := E) (μ := μ)‖₊ ≤ 1 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E]
Goal: ‖MeasureTheory.L1.integralCLM‖₊ ≤ 1
-- MeasureTheory.L1.norm_integral_le
theorem norm_integral_le (f : α →₁[μ] E) : ‖integral f‖ ≤ ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: ‖MeasureTheory.L1.integral f‖ ≤ ‖f‖
-- MeasureTheory.L1.nnnorm_integral_le
theorem nnnorm_integral_le (f : α →₁[μ] E) : ‖integral f‖₊ ≤ ‖f‖₊ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E] (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: ‖MeasureTheory.L1.integral f‖₊ ≤ ‖f‖₊
-- MeasureTheory.L1.continuous_integral
@[continuity]
theorem continuous_integral : Continuous fun f : α →₁[μ] E => integral f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E]
Goal: Continuous fun f => MeasureTheory.L1.integral f
-- MeasureTheory.L1.integral_eq_norm_posPart_sub
theorem integral_eq_norm_posPart_sub (f : α →₁[μ] ℝ) :
    integral f = ‖Lp.posPart f‖ - ‖Lp.negPart f‖ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp ℝ 1 μ))
Goal: MeasureTheory.L1.integral f = ‖MeasureTheory.Lp.posPart f‖ - ‖MeasureTheory.Lp.negPart f‖
-- MeasureTheory.integral_def
irreducible_def integral {_ : MeasurableSpace α} (μ : Measure α) (f : α → G) : G 
Arguments: {α : Type u_6} {G : Type u_7} [NormedAddCommGroup G] [NormedSpace ℝ G]
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → G),
  MeasureTheory.integral μ f =
    if x_1 : CompleteSpace G then
      if hf : MeasureTheory.Integrable f μ then MeasureTheory.L1.integral (MeasureTheory.Integrable.toL1 f hf) else 0
    else 0
-- MeasureTheory.integral_eq
theorem integral_eq (f : α → E) (hf : Integrable f μ) : ∫ a, f a ∂μ = L1.integral (hf.toL1 f) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (a : α), f a ∂μ = MeasureTheory.L1.integral (MeasureTheory.Integrable.toL1 f hf)
-- MeasureTheory.integral_eq_setToFun
theorem integral_eq_setToFun (f : α → E) :
    ∫ a, f a ∂μ = setToFun μ (weightedSMul μ) (dominatedFinMeasAdditive_weightedSMul μ) f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → E)
Goal: ∫ (a : α), f a ∂μ = MeasureTheory.setToFun μ (MeasureTheory.weightedSMul μ) ⋯ f
-- MeasureTheory.L1.integral_eq_integral
theorem L1.integral_eq_integral (f : α →₁[μ] E) : L1.integral f = ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.L1.integral f = ∫ (a : α), ↑↑f a ∂μ
-- MeasureTheory.integral_undef
theorem integral_undef {f : α → G} (h : ¬Integrable f μ) : ∫ a, f a ∂μ = 0 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → G} (h : ¬MeasureTheory.Integrable f μ)
Goal: ∫ (a : α), f a ∂μ = 0
-- MeasureTheory.Integrable.of_integral_ne_zero
theorem Integrable.of_integral_ne_zero {f : α → G} (h : ∫ a, f a ∂μ ≠ 0) : Integrable f μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → G} (h : ∫ (a : α), f a ∂μ ≠ 0)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.integral_non_aestronglyMeasurable
theorem integral_non_aestronglyMeasurable {f : α → G} (h : ¬AEStronglyMeasurable f μ) :
    ∫ a, f a ∂μ = 0 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → G} (h : ¬MeasureTheory.AEStronglyMeasurable f μ)
Goal: ∫ (a : α), f a ∂μ = 0
-- MeasureTheory.integral_zero
@[simp]
theorem integral_zero : ∫ _ : α, (0 : G) ∂μ = 0 
Arguments: (α : Type u_1) (G : Type u_5) [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: ∫ (x : α), 0 ∂μ = 0
-- MeasureTheory.integral_zero'
@[simp]
theorem integral_zero' : integral μ (0 : α → G) = 0 
Arguments: (α : Type u_1) (G : Type u_5) [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.integral μ 0 = 0
-- MeasureTheory.integrable_of_integral_eq_one
theorem integrable_of_integral_eq_one {f : α → ℝ} (h : ∫ x, f x ∂μ = 1) : Integrable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (h : ∫ (x : α), f x ∂μ = 1)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.integral_add
theorem integral_add {f g : α → G} (hf : Integrable f μ) (hg : Integrable g μ) :
    ∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → G} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: ∫ (a : α), f a + g a ∂μ = ∫ (a : α), f a ∂μ + ∫ (a : α), g a ∂μ
-- MeasureTheory.integral_add'
theorem integral_add' {f g : α → G} (hf : Integrable f μ) (hg : Integrable g μ) :
    ∫ a, (f + g) a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → G} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: ∫ (a : α), (f + g) a ∂μ = ∫ (a : α), f a ∂μ + ∫ (a : α), g a ∂μ
-- MeasureTheory.integral_finset_sum
theorem integral_finset_sum {ι} (s : Finset ι) {f : ι → α → G} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    ∫ a, ∑ i ∈ s, f i a ∂μ = ∑ i ∈ s, ∫ a, f i a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_6} (s : Finset ι) {f : ι → α → G} (hf : ∀ i ∈ s, MeasureTheory.Integrable (f i) μ)
Goal: ∫ (a : α), ∑ i ∈ s, f i a ∂μ = ∑ i ∈ s, ∫ (a : α), f i a ∂μ
-- MeasureTheory.integral_neg
@[integral_simps]
theorem integral_neg (f : α → G) : ∫ a, -f a ∂μ = -∫ a, f a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → G)
Goal: ∫ (a : α), -f a ∂μ = -∫ (a : α), f a ∂μ
-- MeasureTheory.integral_neg'
theorem integral_neg' (f : α → G) : ∫ a, (-f) a ∂μ = -∫ a, f a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → G)
Goal: ∫ (a : α), (-f) a ∂μ = -∫ (a : α), f a ∂μ
-- MeasureTheory.integral_sub
theorem integral_sub {f g : α → G} (hf : Integrable f μ) (hg : Integrable g μ) :
    ∫ a, f a - g a ∂μ = ∫ a, f a ∂μ - ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → G} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: ∫ (a : α), f a - g a ∂μ = ∫ (a : α), f a ∂μ - ∫ (a : α), g a ∂μ
-- MeasureTheory.integral_sub'
theorem integral_sub' {f g : α → G} (hf : Integrable f μ) (hg : Integrable g μ) :
    ∫ a, (f - g) a ∂μ = ∫ a, f a ∂μ - ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → G} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: ∫ (a : α), (f - g) a ∂μ = ∫ (a : α), f a ∂μ - ∫ (a : α), g a ∂μ
-- MeasureTheory.integral_smul
@[integral_simps]
theorem integral_smul [NormedSpace 𝕜 G] [SMulCommClass ℝ 𝕜 G] (c : 𝕜) (f : α → G) :
    ∫ a, c • f a ∂μ = c • ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace 𝕜 G] [SMulCommClass ℝ 𝕜 G] (c : 𝕜) (f : α → G)
Goal: ∫ (a : α), c • f a ∂μ = c • ∫ (a : α), f a ∂μ
-- MeasureTheory.integral_mul_left
theorem integral_mul_left {L : Type*} [RCLike L] (r : L) (f : α → L) :
    ∫ a, r * f a ∂μ = r * ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {L : Type u_6} [RCLike L] (r : L) (f : α → L)
Goal: ∫ (a : α), r * f a ∂μ = r * ∫ (a : α), f a ∂μ
-- MeasureTheory.integral_mul_right
theorem integral_mul_right {L : Type*} [RCLike L] (r : L) (f : α → L) :
    ∫ a, f a * r ∂μ = (∫ a, f a ∂μ) * r 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {L : Type u_6} [RCLike L] (r : L) (f : α → L)
Goal: ∫ (a : α), f a * r ∂μ = (∫ (a : α), f a ∂μ) * r
-- MeasureTheory.integral_div
theorem integral_div {L : Type*} [RCLike L] (r : L) (f : α → L) :
    ∫ a, f a / r ∂μ = (∫ a, f a ∂μ) / r 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {L : Type u_6} [RCLike L] (r : L) (f : α → L)
Goal: ∫ (a : α), f a / r ∂μ = (∫ (a : α), f a ∂μ) / r
-- MeasureTheory.integral_congr_ae
theorem integral_congr_ae {f g : α → G} (h : f =ᵐ[μ] g) : ∫ a, f a ∂μ = ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → G} (h : f =ᵐ[μ] g)
Goal: ∫ (a : α), f a ∂μ = ∫ (a : α), g a ∂μ
-- MeasureTheory.L1.integral_of_fun_eq_integral
@[simp, nolint simpNF]
theorem L1.integral_of_fun_eq_integral {f : α → G} (hf : Integrable f μ) :
    ∫ a, (hf.toL1 f) a ∂μ = ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → G} (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (a : α), ↑↑(MeasureTheory.Integrable.toL1 f hf) a ∂μ = ∫ (a : α), f a ∂μ
-- MeasureTheory.continuous_integral
@[continuity]
theorem continuous_integral : Continuous fun f : α →₁[μ] G => ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: Continuous fun f => ∫ (a : α), ↑↑f a ∂μ
-- MeasureTheory.norm_integral_le_lintegral_norm
theorem norm_integral_le_lintegral_norm (f : α → G) :
    ‖∫ a, f a ∂μ‖ ≤ ENNReal.toReal (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → G)
Goal: ‖∫ (a : α), f a ∂μ‖ ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal
-- MeasureTheory.ennnorm_integral_le_lintegral_ennnorm
theorem ennnorm_integral_le_lintegral_ennnorm (f : α → G) :
    (‖∫ a, f a ∂μ‖₊ : ℝ≥0∞) ≤ ∫⁻ a, ‖f a‖₊ ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → G)
Goal: ↑‖∫ (a : α), f a ∂μ‖₊ ≤ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ
-- MeasureTheory.integral_eq_zero_of_ae
theorem integral_eq_zero_of_ae {f : α → G} (hf : f =ᵐ[μ] 0) : ∫ a, f a ∂μ = 0 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → G} (hf : f =ᵐ[μ] 0)
Goal: ∫ (a : α), f a ∂μ = 0
-- MeasureTheory.HasFiniteIntegral.tendsto_setIntegral_nhds_zero
theorem HasFiniteIntegral.tendsto_setIntegral_nhds_zero {ι} {f : α → G}
    (hf : HasFiniteIntegral f μ) {l : Filter ι} {s : ι → Set α} (hs : Tendsto (μ ∘ s) l (𝓝 0)) :
    Tendsto (fun i => ∫ x in s i, f x ∂μ) l (𝓝 0) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_6} {f : α → G} (hf : MeasureTheory.HasFiniteIntegral f μ) {l : Filter ι} {s : ι → Set α} (hs : Filter.Tendsto (⇑μ ∘ s) l (𝓝 0))
Goal: Filter.Tendsto (fun i => ∫ (x : α) in s i, f x ∂μ) l (𝓝 0)
-- MeasureTheory.Integrable.tendsto_setIntegral_nhds_zero
theorem Integrable.tendsto_setIntegral_nhds_zero {ι} {f : α → G} (hf : Integrable f μ)
    {l : Filter ι} {s : ι → Set α} (hs : Tendsto (μ ∘ s) l (𝓝 0)) :
    Tendsto (fun i => ∫ x in s i, f x ∂μ) l (𝓝 0) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_6} {f : α → G} (hf : MeasureTheory.Integrable f μ) {l : Filter ι} {s : ι → Set α} (hs : Filter.Tendsto (⇑μ ∘ s) l (𝓝 0))
Goal: Filter.Tendsto (fun i => ∫ (x : α) in s i, f x ∂μ) l (𝓝 0)
-- MeasureTheory.tendsto_integral_of_L1
theorem tendsto_integral_of_L1 {ι} (f : α → G) (hfi : Integrable f μ) {F : ι → α → G} {l : Filter ι}
    (hFi : ∀ᶠ i in l, Integrable (F i) μ)
    (hF : Tendsto (fun i => ∫⁻ x, ‖F i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => ∫ x, F i x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_6} (f : α → G) (hfi : MeasureTheory.Integrable f μ) {F : ι → α → G} {l : Filter ι} (hFi : ∀ᶠ (i : ι) in l, MeasureTheory.Integrable (F i) μ) (hF : Filter.Tendsto (fun i => ∫⁻ (x : α), ↑‖F i x - f x‖₊ ∂μ) l (𝓝 0))
Goal: Filter.Tendsto (fun i => ∫ (x : α), F i x ∂μ) l (𝓝 (∫ (x : α), f x ∂μ))
-- MeasureTheory.tendsto_integral_of_L1'
lemma tendsto_integral_of_L1' {ι} (f : α → G) (hfi : Integrable f μ) {F : ι → α → G} {l : Filter ι}
    (hFi : ∀ᶠ i in l, Integrable (F i) μ) (hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ) l (𝓝 0)) :
    Tendsto (fun i ↦ ∫ x, F i x ∂μ) l (𝓝 (∫ x, f x ∂μ)) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_6} (f : α → G) (hfi : MeasureTheory.Integrable f μ) {F : ι → α → G} {l : Filter ι} (hFi : ∀ᶠ (i : ι) in l, MeasureTheory.Integrable (F i) μ) (hF : Filter.Tendsto (fun i => MeasureTheory.eLpNorm (F i - f) 1 μ) l (𝓝 0))
Goal: Filter.Tendsto (fun i => ∫ (x : α), F i x ∂μ) l (𝓝 (∫ (x : α), f x ∂μ))
-- MeasureTheory.tendsto_setIntegral_of_L1
lemma tendsto_setIntegral_of_L1 {ι} (f : α → G) (hfi : Integrable f μ) {F : ι → α → G}
    {l : Filter ι}
    (hFi : ∀ᶠ i in l, Integrable (F i) μ) (hF : Tendsto (fun i ↦ ∫⁻ x, ‖F i x - f x‖₊ ∂μ) l (𝓝 0))
    (s : Set α) :
    Tendsto (fun i ↦ ∫ x in s, F i x ∂μ) l (𝓝 (∫ x in s, f x ∂μ)) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_6} (f : α → G) (hfi : MeasureTheory.Integrable f μ) {F : ι → α → G} {l : Filter ι} (hFi : ∀ᶠ (i : ι) in l, MeasureTheory.Integrable (F i) μ) (hF : Filter.Tendsto (fun i => ∫⁻ (x : α), ↑‖F i x - f x‖₊ ∂μ) l (𝓝 0)) (s : Set α)
Goal: Filter.Tendsto (fun i => ∫ (x : α) in s, F i x ∂μ) l (𝓝 (∫ (x : α) in s, f x ∂μ))
-- MeasureTheory.tendsto_setIntegral_of_L1'
lemma tendsto_setIntegral_of_L1' {ι} (f : α → G) (hfi : Integrable f μ) {F : ι → α → G}
    {l : Filter ι}
    (hFi : ∀ᶠ i in l, Integrable (F i) μ) (hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ) l (𝓝 0))
    (s : Set α) :
    Tendsto (fun i ↦ ∫ x in s, F i x ∂μ) l (𝓝 (∫ x in s, f x ∂μ)) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_6} (f : α → G) (hfi : MeasureTheory.Integrable f μ) {F : ι → α → G} {l : Filter ι} (hFi : ∀ᶠ (i : ι) in l, MeasureTheory.Integrable (F i) μ) (hF : Filter.Tendsto (fun i => MeasureTheory.eLpNorm (F i - f) 1 μ) l (𝓝 0)) (s : Set α)
Goal: Filter.Tendsto (fun i => ∫ (x : α) in s, F i x ∂μ) l (𝓝 (∫ (x : α) in s, f x ∂μ))
-- MeasureTheory.continuousWithinAt_of_dominated
theorem continuousWithinAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousWithinAt (fun x => F x a) s x₀) :
    ContinuousWithinAt (fun x => ∫ a, F x a ∂μ) s x₀ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {X : Type u_6} [TopologicalSpace X] [FirstCountableTopology X] {F : X → α → G} {x₀ : X} {bound : α → ℝ} {s : Set X} (hF_meas : ∀ᶠ (x : X) in 𝓝[s] x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (h_bound : ∀ᶠ (x : X) in 𝓝[s] x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => F x a) s x₀)
Goal: ContinuousWithinAt (fun x => ∫ (a : α), F x a ∂μ) s x₀
-- MeasureTheory.continuousAt_of_dominated
theorem continuousAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ}
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousAt (fun x => F x a) x₀) :
    ContinuousAt (fun x => ∫ a, F x a ∂μ) x₀ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {X : Type u_6} [TopologicalSpace X] [FirstCountableTopology X] {F : X → α → G} {x₀ : X} {bound : α → ℝ} (hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, ContinuousAt (fun x => F x a) x₀)
Goal: ContinuousAt (fun x => ∫ (a : α), F x a ∂μ) x₀
-- MeasureTheory.continuousOn_of_dominated
theorem continuousOn_of_dominated {F : X → α → G} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ x ∈ s, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousOn (fun x => F x a) s) :
    ContinuousOn (fun x => ∫ a, F x a ∂μ) s 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {X : Type u_6} [TopologicalSpace X] [FirstCountableTopology X] {F : X → α → G} {bound : α → ℝ} {s : Set X} (hF_meas : ∀ x ∈ s, MeasureTheory.AEStronglyMeasurable (F x) μ) (h_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => F x a) s)
Goal: ContinuousOn (fun x => ∫ (a : α), F x a ∂μ) s
-- MeasureTheory.continuous_of_dominated
theorem continuous_of_dominated {F : X → α → G} {bound : α → ℝ}
    (hF_meas : ∀ x, AEStronglyMeasurable (F x) μ) (h_bound : ∀ x, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a)
    (bound_integrable : Integrable bound μ) (h_cont : ∀ᵐ a ∂μ, Continuous fun x => F x a) :
    Continuous fun x => ∫ a, F x a ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {X : Type u_6} [TopologicalSpace X] [FirstCountableTopology X] {F : X → α → G} {bound : α → ℝ} (hF_meas : ∀ (x : X), MeasureTheory.AEStronglyMeasurable (F x) μ) (h_bound : ∀ (x : X), ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, Continuous fun x => F x a)
Goal: Continuous fun x => ∫ (a : α), F x a ∂μ
-- MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part
theorem integral_eq_lintegral_pos_part_sub_lintegral_neg_part {f : α → ℝ} (hf : Integrable f μ) :
    ∫ a, f a ∂μ =
      ENNReal.toReal (∫⁻ a, .ofReal (f a) ∂μ) - ENNReal.toReal (∫⁻ a, .ofReal (-f a) ∂μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal - (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal
-- MeasureTheory.integral_eq_lintegral_of_nonneg_ae
theorem integral_eq_lintegral_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f)
    (hfm : AEStronglyMeasurable f μ) :
    ∫ a, f a ∂μ = ENNReal.toReal (∫⁻ a, ENNReal.ofReal (f a) ∂μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfm : MeasureTheory.AEStronglyMeasurable f μ)
Goal: ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal