ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.integral_norm_eq_lintegral_nnnorm
theorem integral_norm_eq_lintegral_nnnorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
    (hf : AEStronglyMeasurable f μ) : ∫ x, ‖f x‖ ∂μ = ENNReal.toReal (∫⁻ x, ‖f x‖₊ ∂μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {P : Type u_7} [NormedAddCommGroup P] {f : α → P} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: ∫ (x : α), ‖f x‖ ∂μ = (∫⁻ (x : α), ↑‖f x‖₊ ∂μ).toReal
-- MeasureTheory.ofReal_integral_norm_eq_lintegral_nnnorm
theorem ofReal_integral_norm_eq_lintegral_nnnorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
    (hf : Integrable f μ) : ENNReal.ofReal (∫ x, ‖f x‖ ∂μ) = ∫⁻ x, ‖f x‖₊ ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {P : Type u_7} [NormedAddCommGroup P] {f : α → P} (hf : MeasureTheory.Integrable f μ)
Goal: ENNReal.ofReal (∫ (x : α), ‖f x‖ ∂μ) = ∫⁻ (x : α), ↑‖f x‖₊ ∂μ
-- MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part
theorem integral_eq_integral_pos_part_sub_integral_neg_part {f : α → ℝ} (hf : Integrable f μ) :
    ∫ a, f a ∂μ = ∫ a, (Real.toNNReal (f a) : ℝ) ∂μ - ∫ a, (Real.toNNReal (-f a) : ℝ) ∂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), ↑(f a).toNNReal ∂μ - ∫ (a : α), ↑(-f a).toNNReal ∂μ
-- MeasureTheory.integral_nonneg_of_ae
theorem integral_nonneg_of_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f)
Goal: 0 ≤ ∫ (a : α), f a ∂μ
-- MeasureTheory.lintegral_coe_eq_integral
theorem lintegral_coe_eq_integral (f : α → ℝ≥0) (hfi : Integrable (fun x => (f x : ℝ)) μ) :
    ∫⁻ a, f a ∂μ = ENNReal.ofReal (∫ a, f a ∂μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0) (hfi : MeasureTheory.Integrable (fun x => ↑(f x)) μ)
Goal: ∫⁻ (a : α), ↑(f a) ∂μ = ENNReal.ofReal (∫ (a : α), ↑(f a) ∂μ)
-- MeasureTheory.ofReal_integral_eq_lintegral_ofReal
theorem ofReal_integral_eq_lintegral_ofReal {f : α → ℝ} (hfi : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
    ENNReal.ofReal (∫ x, f x ∂μ) = ∫⁻ x, ENNReal.ofReal (f x) ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hfi : MeasureTheory.Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f)
Goal: ENNReal.ofReal (∫ (x : α), f x ∂μ) = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ
-- MeasureTheory.integral_toReal
theorem integral_toReal {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hf : ∀ᵐ x ∂μ, f x < ∞) :
    ∫ a, (f a).toReal ∂μ = (∫⁻ a, f a ∂μ).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hf : ∀ᵐ (x : α) ∂μ, f x < ⊤)
Goal: ∫ (a : α), (f a).toReal ∂μ = (∫⁻ (a : α), f a ∂μ).toReal
-- MeasureTheory.lintegral_coe_le_coe_iff_integral_le
theorem lintegral_coe_le_coe_iff_integral_le {f : α → ℝ≥0} (hfi : Integrable (fun x => (f x : ℝ)) μ)
    {b : ℝ≥0} : ∫⁻ a, f a ∂μ ≤ b ↔ ∫ a, (f a : ℝ) ∂μ ≤ b 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0} (hfi : MeasureTheory.Integrable (fun x => ↑(f x)) μ) {b : ℝ≥0}
Goal: ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b ↔ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b
-- MeasureTheory.integral_coe_le_of_lintegral_coe_le
theorem integral_coe_le_of_lintegral_coe_le {f : α → ℝ≥0} {b : ℝ≥0} (h : ∫⁻ a, f a ∂μ ≤ b) :
    ∫ a, (f a : ℝ) ∂μ ≤ b 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0} {b : ℝ≥0} (h : ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b)
Goal: ∫ (a : α), ↑(f a) ∂μ ≤ ↑b
-- MeasureTheory.integral_nonneg
theorem integral_nonneg {f : α → ℝ} (hf : 0 ≤ f) : 0 ≤ ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : 0 ≤ f)
Goal: 0 ≤ ∫ (a : α), f a ∂μ
-- MeasureTheory.integral_nonpos_of_ae
theorem integral_nonpos_of_ae {f : α → ℝ} (hf : f ≤ᵐ[μ] 0) : ∫ a, f a ∂μ ≤ 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : f ≤ᵐ[μ] 0)
Goal: ∫ (a : α), f a ∂μ ≤ 0
-- MeasureTheory.integral_nonpos
theorem integral_nonpos {f : α → ℝ} (hf : f ≤ 0) : ∫ a, f a ∂μ ≤ 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : f ≤ 0)
Goal: ∫ (a : α), f a ∂μ ≤ 0
-- MeasureTheory.integral_eq_zero_iff_of_nonneg_ae
theorem integral_eq_zero_iff_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : Integrable f μ) :
    ∫ x, f x ∂μ = 0 ↔ f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.integral_eq_zero_iff_of_nonneg
theorem integral_eq_zero_iff_of_nonneg {f : α → ℝ} (hf : 0 ≤ f) (hfi : Integrable f μ) :
    ∫ x, f x ∂μ = 0 ↔ f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : 0 ≤ f) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.integral_eq_iff_of_ae_le
lemma integral_eq_iff_of_ae_le {f g : α → ℝ}
    (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    ∫ a, f a ∂μ = ∫ a, g a ∂μ ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hfg : f ≤ᵐ[μ] g)
Goal: ∫ (a : α), f a ∂μ = ∫ (a : α), g a ∂μ ↔ f =ᵐ[μ] g
-- MeasureTheory.integral_pos_iff_support_of_nonneg_ae
theorem integral_pos_iff_support_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : Integrable f μ) :
    (0 < ∫ x, f x ∂μ) ↔ 0 < μ (Function.support f) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : MeasureTheory.Integrable f μ)
Goal: 0 < ∫ (x : α), f x ∂μ ↔ 0 < μ (Function.support f)
-- MeasureTheory.integral_pos_iff_support_of_nonneg
theorem integral_pos_iff_support_of_nonneg {f : α → ℝ} (hf : 0 ≤ f) (hfi : Integrable f μ) :
    (0 < ∫ x, f x ∂μ) ↔ 0 < μ (Function.support f) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : 0 ≤ f) (hfi : MeasureTheory.Integrable f μ)
Goal: 0 < ∫ (x : α), f x ∂μ ↔ 0 < μ (Function.support f)
-- MeasureTheory.integral_exp_pos
lemma integral_exp_pos {μ : Measure α} {f : α → ℝ} [hμ : NeZero μ]
    (hf : Integrable (fun x ↦ Real.exp (f x)) μ) :
    0 < ∫ x, Real.exp (f x) ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [hμ : NeZero μ] (hf : MeasureTheory.Integrable (fun x => Real.exp (f x)) μ)
Goal: 0 < ∫ (x : α), Real.exp (f x) ∂μ
-- MeasureTheory.integral_tendsto_of_tendsto_of_monotone
lemma integral_tendsto_of_tendsto_of_monotone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
    (hf : ∀ n, Integrable (f n) μ) (hF : Integrable F μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n ↦ f n x)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))) :
    Tendsto (fun n ↦ ∫ x, f n x ∂μ) atTop (𝓝 (∫ x, F x ∂μ)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ} {F : α → ℝ} (hf : ∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) (hF : MeasureTheory.Integrable F μ) (h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x) (h_tendsto : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (F x)))
Goal: Filter.Tendsto (fun n => ∫ (x : α), f n x ∂μ) Filter.atTop (𝓝 (∫ (x : α), F x ∂μ))
-- MeasureTheory.integral_tendsto_of_tendsto_of_antitone
lemma integral_tendsto_of_tendsto_of_antitone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
    (hf : ∀ n, Integrable (f n) μ) (hF : Integrable F μ) (h_mono : ∀ᵐ x ∂μ, Antitone fun n ↦ f n x)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))) :
    Tendsto (fun n ↦ ∫ x, f n x ∂μ) atTop (𝓝 (∫ x, F x ∂μ)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ} {F : α → ℝ} (hf : ∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) (hF : MeasureTheory.Integrable F μ) (h_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n => f n x) (h_tendsto : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (F x)))
Goal: Filter.Tendsto (fun n => ∫ (x : α), f n x ∂μ) Filter.atTop (𝓝 (∫ (x : α), F x ∂μ))
-- MeasureTheory.tendsto_of_integral_tendsto_of_monotone
lemma tendsto_of_integral_tendsto_of_monotone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
    (hf_int : ∀ n, Integrable (f n) μ) (hF_int : Integrable F μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫ a, f i a ∂μ) atTop (𝓝 (∫ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
    (hf_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ} {F : α → ℝ} (hf_int : ∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) (hF_int : MeasureTheory.Integrable F μ) (hf_tendsto : Filter.Tendsto (fun i => ∫ (a : α), f i a ∂μ) Filter.atTop (𝓝 (∫ (a : α), F a ∂μ))) (hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a) (hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a)
Goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))
-- MeasureTheory.tendsto_of_integral_tendsto_of_antitone
lemma tendsto_of_integral_tendsto_of_antitone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
    (hf_int : ∀ n, Integrable (f n) μ) (hF_int : Integrable F μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫ a, f i a ∂μ) atTop (𝓝 (∫ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Antitone (fun i ↦ f i a))
    (hf_bound : ∀ᵐ a ∂μ, ∀ i, F a ≤ f i a) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ} {F : α → ℝ} (hf_int : ∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) (hF_int : MeasureTheory.Integrable F μ) (hf_tendsto : Filter.Tendsto (fun i => ∫ (a : α), f i a ∂μ) Filter.atTop (𝓝 (∫ (a : α), F a ∂μ))) (hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a) (hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a)
Goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))
-- MeasureTheory.L1.norm_eq_integral_norm
theorem L1.norm_eq_integral_norm (f : α →₁[μ] H) : ‖f‖ = ∫ a, ‖f a‖ ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {H : Type u_7} [NormedAddCommGroup H] (f : ↥(MeasureTheory.Lp H 1 μ))
Goal: ‖f‖ = ∫ (a : α), ‖↑↑f a‖ ∂μ
-- MeasureTheory.L1.dist_eq_integral_dist
theorem L1.dist_eq_integral_dist (f g : α →₁[μ] H) : dist f g = ∫ a, dist (f a) (g a) ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {H : Type u_7} [NormedAddCommGroup H] (f g : ↥(MeasureTheory.Lp H 1 μ))
Goal: dist f g = ∫ (a : α), dist (↑↑f a) (↑↑g a) ∂μ
-- MeasureTheory.L1.norm_of_fun_eq_integral_norm
theorem L1.norm_of_fun_eq_integral_norm {f : α → H} (hf : Integrable f μ) :
    ‖hf.toL1 f‖ = ∫ a, ‖f a‖ ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {H : Type u_7} [NormedAddCommGroup H] {f : α → H} (hf : MeasureTheory.Integrable f μ)
Goal: ‖MeasureTheory.Integrable.toL1 f hf‖ = ∫ (a : α), ‖f a‖ ∂μ
-- MeasureTheory.Memℒp.eLpNorm_eq_integral_rpow_norm
theorem Memℒp.eLpNorm_eq_integral_rpow_norm {f : α → H} {p : ℝ≥0∞} (hp1 : p ≠ 0) (hp2 : p ≠ ∞)
    (hf : Memℒp f p μ) :
    eLpNorm f p μ = ENNReal.ofReal ((∫ a, ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {H : Type u_7} [NormedAddCommGroup H] {f : α → H} {p : ℝ≥0∞} (hp1 : p ≠ 0) (hp2 : p ≠ ⊤) (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.eLpNorm f p μ = ENNReal.ofReal ((∫ (a : α), ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹)
-- MeasureTheory.integral_mono_ae
theorem integral_mono_ae {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) (h : f ≤ᵐ[μ] g) :
    ∫ a, f a ∂μ ≤ ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (h : f ≤ᵐ[μ] g)
Goal: ∫ (a : α), f a ∂μ ≤ ∫ (a : α), g a ∂μ
-- MeasureTheory.integral_mono
@[mono]
theorem integral_mono {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) (h : f ≤ g) :
    ∫ a, f a ∂μ ≤ ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (h : f ≤ g)
Goal: ∫ (a : α), f a ∂μ ≤ ∫ (a : α), g a ∂μ
-- MeasureTheory.integral_mono_of_nonneg
theorem integral_mono_of_nonneg {f g : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hgi : Integrable g μ)
    (h : f ≤ᵐ[μ] g) : ∫ a, f a ∂μ ≤ ∫ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hgi : MeasureTheory.Integrable g μ) (h : f ≤ᵐ[μ] g)
Goal: ∫ (a : α), f a ∂μ ≤ ∫ (a : α), g a ∂μ
-- MeasureTheory.integral_mono_measure
theorem integral_mono_measure {f : α → ℝ} {ν} (hle : μ ≤ ν) (hf : 0 ≤ᵐ[ν] f)
    (hfi : Integrable f ν) : ∫ a, f a ∂μ ≤ ∫ a, f a ∂ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} {ν : MeasureTheory.Measure α} (hle : μ ≤ ν) (hf : 0 ≤ᵐ[ν] f) (hfi : MeasureTheory.Integrable f ν)
Goal: ∫ (a : α), f a ∂μ ≤ ∫ (a : α), f a ∂ν
-- MeasureTheory.norm_integral_le_integral_norm
theorem norm_integral_le_integral_norm (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.norm_integral_le_of_norm_le
theorem norm_integral_le_of_norm_le {f : α → G} {g : α → ℝ} (hg : Integrable g μ)
    (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ g x) : ‖∫ x, f x ∂μ‖ ≤ ∫ x, g x ∂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} {g : α → ℝ} (hg : MeasureTheory.Integrable g μ) (h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ g x)
Goal: ‖∫ (x : α), f x ∂μ‖ ≤ ∫ (x : α), g x ∂μ
-- MeasureTheory.SimpleFunc.integral_eq_integral
theorem SimpleFunc.integral_eq_integral (f : α →ₛ E) (hfi : Integrable f μ) :
    f.integral μ = ∫ x, f x ∂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.SimpleFunc α E) (hfi : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.integral μ f = ∫ (x : α), ↑f x ∂μ
-- MeasureTheory.SimpleFunc.integral_eq_sum
theorem SimpleFunc.integral_eq_sum (f : α →ₛ E) (hfi : Integrable f μ) :
    ∫ x, f x ∂μ = ∑ x ∈ f.range, ENNReal.toReal (μ (f ⁻¹' {x})) • x 
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.SimpleFunc α E) (hfi : MeasureTheory.Integrable (↑f) μ)
Goal: ∫ (x : α), ↑f x ∂μ = ∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal • x
-- MeasureTheory.integral_const
@[simp]
theorem integral_const (c : E) : ∫ _ : α, c ∂μ = (μ univ).toReal • c 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : E)
Goal: ∫ (x : α), c ∂μ = (μ Set.univ).toReal • c
-- MeasureTheory.norm_integral_le_of_norm_le_const
theorem norm_integral_le_of_norm_le_const [IsFiniteMeasure μ] {f : α → G} {C : ℝ}
    (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : ‖∫ x, f x ∂μ‖ ≤ C * (μ univ).toReal 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {f : α → G} {C : ℝ} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: ‖∫ (x : α), f x ∂μ‖ ≤ C * (μ Set.univ).toReal
-- MeasureTheory.tendsto_integral_approxOn_of_measurable
theorem tendsto_integral_approxOn_of_measurable [MeasurableSpace E] [BorelSpace E] {f : α → E}
    {s : Set E} [SeparableSpace s] (hfi : Integrable f μ) (hfm : Measurable f)
    (hs : ∀ᵐ x ∂μ, f x ∈ closure s) {y₀ : E} (h₀ : y₀ ∈ s) (h₀i : Integrable (fun _ => y₀) μ) :
    Tendsto (fun n => (SimpleFunc.approxOn f hfm s y₀ h₀ n).integral μ)
      atTop (𝓝 <| ∫ x, f x ∂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} [MeasurableSpace E] [BorelSpace E] {f : α → E} {s : Set E} [TopologicalSpace.SeparableSpace ↑s] (hfi : MeasureTheory.Integrable f μ) (hfm : Measurable f) (hs : ∀ᵐ (x : α) ∂μ, f x ∈ closure s) {y₀ : E} (h₀ : y₀ ∈ s) (h₀i : MeasureTheory.Integrable (fun x => y₀) μ)
Goal: Filter.Tendsto (fun n => MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.approxOn f hfm s y₀ h₀ n))
  Filter.atTop (𝓝 (∫ (x : α), f x ∂μ))
-- MeasureTheory.tendsto_integral_approxOn_of_measurable_of_range_subset
theorem tendsto_integral_approxOn_of_measurable_of_range_subset [MeasurableSpace E] [BorelSpace E]
    {f : α → E} (fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [SeparableSpace s]
    (hs : range f ∪ {0} ⊆ s) :
    Tendsto (fun n => (SimpleFunc.approxOn f fmeas s 0 (hs <| by simp) n).integral μ) atTop
      (𝓝 <| ∫ x, f x ∂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} [MeasurableSpace E] [BorelSpace E] {f : α → E} (fmeas : Measurable f) (hf : MeasureTheory.Integrable f μ) (s : Set E) [TopologicalSpace.SeparableSpace ↑s] (hs : Set.range f ∪ {0} ⊆ s)
Goal: Filter.Tendsto (fun n => MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.approxOn f fmeas s 0 ⋯ n))
  Filter.atTop (𝓝 (∫ (x : α), f x ∂μ))
-- MeasureTheory.tendsto_integral_norm_approxOn_sub
theorem tendsto_integral_norm_approxOn_sub [MeasurableSpace E] [BorelSpace E] {f : α → E}
    (fmeas : Measurable f) (hf : Integrable f μ) [SeparableSpace (range f ∪ {0} : Set E)] :
    Tendsto (fun n ↦ ∫ x, ‖SimpleFunc.approxOn f fmeas (range f ∪ {0}) 0 (by simp) n x - f x‖ ∂μ)
      atTop (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace E] [BorelSpace E] {f : α → E} (fmeas : Measurable f) (hf : MeasureTheory.Integrable f μ) [TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})]
Goal: Filter.Tendsto (fun n => ∫ (x : α), ‖↑(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) x - f x‖ ∂μ)
  Filter.atTop (𝓝 0)
-- MeasureTheory.integral_add_measure
theorem integral_add_measure {f : α → G} (hμ : Integrable f μ) (hν : Integrable f ν) :
    ∫ x, f x ∂(μ + ν) = ∫ x, f x ∂μ + ∫ x, f x ∂ν 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → G} (hμ : MeasureTheory.Integrable f μ) (hν : MeasureTheory.Integrable f ν)
Goal: ∫ (x : α), f x ∂(μ + ν) = ∫ (x : α), f x ∂μ + ∫ (x : α), f x ∂ν
-- MeasureTheory.integral_zero_measure
@[simp]
theorem integral_zero_measure {m : MeasurableSpace α} (f : α → G) :
    (∫ x, f x ∂(0 : Measure α)) = 0 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} (f : α → G)
Goal: ∫ (x : α), f x ∂0 = 0
-- MeasureTheory.integral_finset_sum_measure
theorem integral_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
    {s : Finset ι} (hf : ∀ i ∈ s, Integrable f (μ i)) :
    ∫ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫ a, f a ∂μ i 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {ι : Type u_7} {m : MeasurableSpace α} {f : α → G} {μ : ι → MeasureTheory.Measure α} {s : Finset ι} (hf : ∀ i ∈ s, MeasureTheory.Integrable f (μ i))
Goal: ∫ (a : α), f a ∂∑ i ∈ s, μ i = ∑ i ∈ s, ∫ (a : α), f a ∂μ i
-- MeasureTheory.nndist_integral_add_measure_le_lintegral
theorem nndist_integral_add_measure_le_lintegral
    {f : α → G} (h₁ : Integrable f μ) (h₂ : Integrable f ν) :
    (nndist (∫ x, f x ∂μ) (∫ x, f x ∂(μ + ν)) : ℝ≥0∞) ≤ ∫⁻ x, ‖f x‖₊ ∂ν 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → G} (h₁ : MeasureTheory.Integrable f μ) (h₂ : MeasureTheory.Integrable f ν)
Goal: ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂(μ + ν))) ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂ν
-- MeasureTheory.hasSum_integral_measure
theorem hasSum_integral_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
    (hf : Integrable f (Measure.sum μ)) :
    HasSum (fun i => ∫ a, f a ∂μ i) (∫ a, f a ∂Measure.sum μ) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {ι : Type u_7} {m : MeasurableSpace α} {f : α → G} {μ : ι → MeasureTheory.Measure α} (hf : MeasureTheory.Integrable f (MeasureTheory.Measure.sum μ))
Goal: HasSum (fun i => ∫ (a : α), f a ∂μ i) (∫ (a : α), f a ∂MeasureTheory.Measure.sum μ)
-- MeasureTheory.integral_sum_measure
theorem integral_sum_measure {ι} {_ : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
    (hf : Integrable f (Measure.sum μ)) : ∫ a, f a ∂Measure.sum μ = ∑' i, ∫ a, f a ∂μ i 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {ι : Type u_7}
Goal: ∀ {x : MeasurableSpace α} {f : α → G} {μ : ι → MeasureTheory.Measure α},
  MeasureTheory.Integrable f (MeasureTheory.Measure.sum μ) →
    ∫ (a : α), f a ∂MeasureTheory.Measure.sum μ = ∑' (i : ι), ∫ (a : α), f a ∂μ i
-- MeasureTheory.integral_smul_measure
@[simp]
theorem integral_smul_measure (f : α → G) (c : ℝ≥0∞) :
    ∫ x, f x ∂c • μ = c.toReal • ∫ 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} (f : α → G) (c : ℝ≥0∞)
Goal: ∫ (x : α), f x ∂c • μ = c.toReal • ∫ (x : α), f x ∂μ
-- MeasureTheory.integral_smul_nnreal_measure
@[simp]
theorem integral_smul_nnreal_measure (f : α → G) (c : ℝ≥0) :
    ∫ x, f x ∂(c • μ) = c • ∫ 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} (f : α → G) (c : ℝ≥0)
Goal: ∫ (x : α), f x ∂c • μ = c • ∫ (x : α), f x ∂μ
-- MeasureTheory.integral_map_of_stronglyMeasurable
theorem integral_map_of_stronglyMeasurable {β} [MeasurableSpace β] {φ : α → β} (hφ : Measurable φ)
    {f : β → G} (hfm : StronglyMeasurable f) : ∫ y, f y ∂Measure.map φ μ = ∫ x, f (φ x) ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [MeasurableSpace β] {φ : α → β} (hφ : Measurable φ) {f : β → G} (hfm : MeasureTheory.StronglyMeasurable f)
Goal: ∫ (y : β), f y ∂MeasureTheory.Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ
-- MeasureTheory.integral_map
theorem integral_map {β} [MeasurableSpace β] {φ : α → β} (hφ : AEMeasurable φ μ) {f : β → G}
    (hfm : AEStronglyMeasurable f (Measure.map φ μ)) :
    ∫ y, f y ∂Measure.map φ μ = ∫ x, f (φ x) ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [MeasurableSpace β] {φ : α → β} (hφ : AEMeasurable φ μ) {f : β → G} (hfm : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map φ μ))
Goal: ∫ (y : β), f y ∂MeasureTheory.Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ
-- MeasurableEmbedding.integral_map
theorem _root_.MeasurableEmbedding.integral_map {β} {_ : MeasurableSpace β} {f : α → β}
    (hf : MeasurableEmbedding f) (g : β → G) : ∫ y, g y ∂Measure.map f μ = ∫ x, g (f x) ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7}
Goal: ∀ {x : MeasurableSpace β} {f : α → β},
  MeasurableEmbedding f → ∀ (g : β → G), ∫ (y : β), g y ∂MeasureTheory.Measure.map f μ = ∫ (x : α), g (f x) ∂μ