ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.setIntegral_trim
theorem setIntegral_trim {X} {m m0 : MeasurableSpace X} {μ : Measure X} (hm : m ≤ m0) {f : X → E}
    (hf_meas : StronglyMeasurable[m] f) {s : Set X} (hs : MeasurableSet[m] s) :
    ∫ x in s, f x ∂μ = ∫ x in s, f x ∂μ.trim hm 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {X : Type u_5} {m m0 : MeasurableSpace X} {μ : MeasureTheory.Measure X} (hm : m ≤ m0) {f : X → E} (hf_meas : MeasureTheory.StronglyMeasurable f) {s : Set X} (hs : MeasurableSet s)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, f x ∂μ.trim hm
-- MeasureTheory.integral_Icc_eq_integral_Ioc'
theorem integral_Icc_eq_integral_Ioc' (hx : μ {x} = 0) :
    ∫ t in Icc x y, f t ∂μ = ∫ t in Ioc x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} (hx : μ {x} = 0)
Goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioc x y, f t ∂μ
-- MeasureTheory.integral_Icc_eq_integral_Ico'
theorem integral_Icc_eq_integral_Ico' (hy : μ {y} = 0) :
    ∫ t in Icc x y, f t ∂μ = ∫ t in Ico x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} (hy : μ {y} = 0)
Goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ico x y, f t ∂μ
-- MeasureTheory.integral_Ioc_eq_integral_Ioo'
theorem integral_Ioc_eq_integral_Ioo' (hy : μ {y} = 0) :
    ∫ t in Ioc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} (hy : μ {y} = 0)
Goal: ∫ (t : X) in Set.Ioc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ
-- MeasureTheory.integral_Ico_eq_integral_Ioo'
theorem integral_Ico_eq_integral_Ioo' (hx : μ {x} = 0) :
    ∫ t in Ico x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} (hx : μ {x} = 0)
Goal: ∫ (t : X) in Set.Ico x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ
-- MeasureTheory.integral_Icc_eq_integral_Ioo'
theorem integral_Icc_eq_integral_Ioo' (hx : μ {x} = 0) (hy : μ {y} = 0) :
    ∫ t in Icc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} (hx : μ {x} = 0) (hy : μ {y} = 0)
Goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ
-- MeasureTheory.integral_Iic_eq_integral_Iio'
theorem integral_Iic_eq_integral_Iio' (hx : μ {x} = 0) :
    ∫ t in Iic x, f t ∂μ = ∫ t in Iio x, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x : X} (hx : μ {x} = 0)
Goal: ∫ (t : X) in Set.Iic x, f t ∂μ = ∫ (t : X) in Set.Iio x, f t ∂μ
-- MeasureTheory.integral_Ici_eq_integral_Ioi'
theorem integral_Ici_eq_integral_Ioi' (hx : μ {x} = 0) :
    ∫ t in Ici x, f t ∂μ = ∫ t in Ioi x, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x : X} (hx : μ {x} = 0)
Goal: ∫ (t : X) in Set.Ici x, f t ∂μ = ∫ (t : X) in Set.Ioi x, f t ∂μ
-- MeasureTheory.integral_Icc_eq_integral_Ioc
theorem integral_Icc_eq_integral_Ioc : ∫ t in Icc x y, f t ∂μ = ∫ t in Ioc x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} [MeasureTheory.NoAtoms μ]
Goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioc x y, f t ∂μ
-- MeasureTheory.integral_Icc_eq_integral_Ico
theorem integral_Icc_eq_integral_Ico : ∫ t in Icc x y, f t ∂μ = ∫ t in Ico x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} [MeasureTheory.NoAtoms μ]
Goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ico x y, f t ∂μ
-- MeasureTheory.integral_Ioc_eq_integral_Ioo
theorem integral_Ioc_eq_integral_Ioo : ∫ t in Ioc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} [MeasureTheory.NoAtoms μ]
Goal: ∫ (t : X) in Set.Ioc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ
-- MeasureTheory.integral_Ico_eq_integral_Ioo
theorem integral_Ico_eq_integral_Ioo : ∫ t in Ico x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} [MeasureTheory.NoAtoms μ]
Goal: ∫ (t : X) in Set.Ico x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ
-- MeasureTheory.integral_Icc_eq_integral_Ioo
theorem integral_Icc_eq_integral_Ioo : ∫ t in Icc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x y : X} [MeasureTheory.NoAtoms μ]
Goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ
-- MeasureTheory.integral_Iic_eq_integral_Iio
theorem integral_Iic_eq_integral_Iio : ∫ t in Iic x, f t ∂μ = ∫ t in Iio x, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x : X} [MeasureTheory.NoAtoms μ]
Goal: ∫ (t : X) in Set.Iic x, f t ∂μ = ∫ (t : X) in Set.Iio x, f t ∂μ
-- MeasureTheory.integral_Ici_eq_integral_Ioi
theorem integral_Ici_eq_integral_Ioi : ∫ t in Ici x, f t ∂μ = ∫ t in Ioi x, f t ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} [PartialOrder X] {x : X} [MeasureTheory.NoAtoms μ]
Goal: ∫ (t : X) in Set.Ici x, f t ∂μ = ∫ (t : X) in Set.Ioi x, f t ∂μ
-- MeasureTheory.setIntegral_mono_ae_restrict
theorem setIntegral_mono_ae_restrict (h : f ≤ᵐ[μ.restrict s] g) :
    ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f g : X → ℝ} {s : Set X} (hf : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.IntegrableOn g s μ) (h : f ≤ᵐ[μ.restrict s] g)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_mono_ae
theorem setIntegral_mono_ae (h : f ≤ᵐ[μ] g) : ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f g : X → ℝ} {s : Set X} (hf : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.IntegrableOn g s μ) (h : f ≤ᵐ[μ] g)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_mono_on
theorem setIntegral_mono_on (hs : MeasurableSet s) (h : ∀ x ∈ s, f x ≤ g x) :
    ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f g : X → ℝ} {s : Set X} (hf : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.IntegrableOn g s μ) (hs : MeasurableSet s) (h : ∀ x ∈ s, f x ≤ g x)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_mono_on_ae
theorem setIntegral_mono_on_ae (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) :
    ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f g : X → ℝ} {s : Set X} (hf : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.IntegrableOn g s μ) (hs : MeasurableSet s) (h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_mono
theorem setIntegral_mono (h : f ≤ g) : ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f g : X → ℝ} {s : Set X} (hf : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.IntegrableOn g s μ) (h : f ≤ g)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_mono_set
theorem setIntegral_mono_set (hfi : IntegrableOn f t μ) (hf : 0 ≤ᵐ[μ.restrict t] f)
    (hst : s ≤ᵐ[μ] t) : ∫ x in s, f x ∂μ ≤ ∫ x in t, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s t : Set X} (hfi : MeasureTheory.IntegrableOn f t μ) (hf : 0 ≤ᵐ[μ.restrict t] f) (hst : s ≤ᵐ[μ] t)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in t, f x ∂μ
-- MeasureTheory.setIntegral_le_integral
theorem setIntegral_le_integral (hfi : Integrable f μ) (hf : 0 ≤ᵐ[μ] f) :
    ∫ x in s, f x ∂μ ≤ ∫ x, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hfi : MeasureTheory.Integrable f μ) (hf : 0 ≤ᵐ[μ] f)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X), f x ∂μ
-- MeasureTheory.setIntegral_ge_of_const_le
theorem setIntegral_ge_of_const_le {c : ℝ} (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (hf : ∀ x ∈ s, c ≤ f x) (hfint : IntegrableOn (fun x : X => f x) s μ) :
    c * (μ s).toReal ≤ ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} {c : ℝ} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (hf : ∀ x ∈ s, c ≤ f x) (hfint : MeasureTheory.IntegrableOn (fun x => f x) s μ)
Goal: c * (μ s).toReal ≤ ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_nonneg_of_ae_restrict
theorem setIntegral_nonneg_of_ae_restrict (hf : 0 ≤ᵐ[μ.restrict s] f) : 0 ≤ ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hf : 0 ≤ᵐ[μ.restrict s] f)
Goal: 0 ≤ ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_nonneg_of_ae
theorem setIntegral_nonneg_of_ae (hf : 0 ≤ᵐ[μ] f) : 0 ≤ ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hf : 0 ≤ᵐ[μ] f)
Goal: 0 ≤ ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_nonneg
theorem setIntegral_nonneg (hs : MeasurableSet s) (hf : ∀ x, x ∈ s → 0 ≤ f x) :
    0 ≤ ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hs : MeasurableSet s) (hf : ∀ x ∈ s, 0 ≤ f x)
Goal: 0 ≤ ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_nonneg_ae
theorem setIntegral_nonneg_ae (hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ, x ∈ s → 0 ≤ f x) :
    0 ≤ ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hs : MeasurableSet s) (hf : ∀ᵐ (x : X) ∂μ, x ∈ s → 0 ≤ f x)
Goal: 0 ≤ ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_le_nonneg
theorem setIntegral_le_nonneg {s : Set X} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hfi : Integrable f μ) : ∫ x in s, f x ∂μ ≤ ∫ x in {y | 0 ≤ f y}, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hs : MeasurableSet s) (hf : MeasureTheory.StronglyMeasurable f) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in {y | 0 ≤ f y}, f x ∂μ
-- MeasureTheory.setIntegral_nonpos_of_ae_restrict
theorem setIntegral_nonpos_of_ae_restrict (hf : f ≤ᵐ[μ.restrict s] 0) : ∫ x in s, f x ∂μ ≤ 0 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hf : f ≤ᵐ[μ.restrict s] 0)
Goal: ∫ (x : X) in s, f x ∂μ ≤ 0
-- MeasureTheory.setIntegral_nonpos_of_ae
theorem setIntegral_nonpos_of_ae (hf : f ≤ᵐ[μ] 0) : ∫ x in s, f x ∂μ ≤ 0 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hf : f ≤ᵐ[μ] 0)
Goal: ∫ (x : X) in s, f x ∂μ ≤ 0
-- MeasureTheory.setIntegral_nonpos_ae
theorem setIntegral_nonpos_ae (hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ, x ∈ s → f x ≤ 0) :
    ∫ x in s, f x ∂μ ≤ 0 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hs : MeasurableSet s) (hf : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ 0)
Goal: ∫ (x : X) in s, f x ∂μ ≤ 0
-- MeasureTheory.setIntegral_nonpos
theorem setIntegral_nonpos (hs : MeasurableSet s) (hf : ∀ x, x ∈ s → f x ≤ 0) :
    ∫ x in s, f x ∂μ ≤ 0 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hs : MeasurableSet s) (hf : ∀ x ∈ s, f x ≤ 0)
Goal: ∫ (x : X) in s, f x ∂μ ≤ 0
-- MeasureTheory.setIntegral_nonpos_le
theorem setIntegral_nonpos_le {s : Set X} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hfi : Integrable f μ) : ∫ x in {y | f y ≤ 0}, f x ∂μ ≤ ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hs : MeasurableSet s) (hf : MeasureTheory.StronglyMeasurable f) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X) in {y | f y ≤ 0}, f x ∂μ ≤ ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.Integrable.measure_le_integral
lemma Integrable.measure_le_integral {f : X → ℝ} (f_int : Integrable f μ) (f_nonneg : 0 ≤ᵐ[μ] f)
    {s : Set X} (hs : ∀ x ∈ s, 1 ≤ f x) :
    μ s ≤ ENNReal.ofReal (∫ x, f x ∂μ) 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} (f_int : MeasureTheory.Integrable f μ) (f_nonneg : 0 ≤ᵐ[μ] f) {s : Set X} (hs : ∀ x ∈ s, 1 ≤ f x)
Goal: μ s ≤ ENNReal.ofReal (∫ (x : X), f x ∂μ)
-- MeasureTheory.integral_le_measure
lemma integral_le_measure {f : X → ℝ} {s : Set X}
    (hs : ∀ x ∈ s, f x ≤ 1) (h's : ∀ x ∈ sᶜ, f x ≤ 0) :
    ENNReal.ofReal (∫ x, f x ∂μ) ≤ μ s 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X} (hs : ∀ x ∈ s, f x ≤ 1) (h's : ∀ x ∈ sᶜ, f x ≤ 0)
Goal: ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s
-- MeasureTheory.integrableOn_iUnion_of_summable_integral_norm
theorem integrableOn_iUnion_of_summable_integral_norm {f : X → E} {s : ι → Set X}
    (hs : ∀ i : ι, MeasurableSet (s i)) (hi : ∀ i : ι, IntegrableOn f (s i) μ)
    (h : Summable fun i : ι => ∫ x : X in s i, ‖f x‖ ∂μ) : IntegrableOn f (iUnion s) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {ι : Type u_5} [Countable ι] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] {f : X → E} {s : ι → Set X} (hs : ∀ (i : ι), MeasurableSet (s i)) (hi : ∀ (i : ι), MeasureTheory.IntegrableOn f (s i) μ) (h : Summable fun i => ∫ (x : X) in s i, ‖f x‖ ∂μ)
Goal: MeasureTheory.IntegrableOn f (Set.iUnion s) μ
-- MeasureTheory.integrableOn_iUnion_of_summable_norm_restrict
theorem integrableOn_iUnion_of_summable_norm_restrict {f : C(X, E)} {s : ι → Compacts X}
    (hf : Summable fun i : ι => ‖f.restrict (s i)‖ * ENNReal.toReal (μ <| s i)) :
    IntegrableOn f (⋃ i : ι, s i) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {ι : Type u_5} [Countable ι] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [TopologicalSpace X] [BorelSpace X] [TopologicalSpace.MetrizableSpace X] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : C(X, E)} {s : ι → TopologicalSpace.Compacts X} (hf : Summable fun i => ‖ContinuousMap.restrict (↑(s i)) f‖ * (μ ↑(s i)).toReal)
Goal: MeasureTheory.IntegrableOn (⇑f) (⋃ i, ↑(s i)) μ
-- MeasureTheory.integrable_of_summable_norm_restrict
theorem integrable_of_summable_norm_restrict {f : C(X, E)} {s : ι → Compacts X}
    (hf : Summable fun i : ι => ‖f.restrict (s i)‖ * ENNReal.toReal (μ <| s i))
    (hs : ⋃ i : ι, ↑(s i) = (univ : Set X)) : Integrable f μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {ι : Type u_5} [Countable ι] {μ : MeasureTheory.Measure X} [NormedAddCommGroup E] [TopologicalSpace X] [BorelSpace X] [TopologicalSpace.MetrizableSpace X] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : C(X, E)} {s : ι → TopologicalSpace.Compacts X} (hf : Summable fun i => ‖ContinuousMap.restrict (↑(s i)) f‖ * (μ ↑(s i)).toReal) (hs : ⋃ i, ↑(s i) = Set.univ)
Goal: MeasureTheory.Integrable (⇑f) μ
-- MeasureTheory.Lp_toLp_restrict_add
theorem Lp_toLp_restrict_add (f g : Lp E p μ) (s : Set X) :
    ((Lp.memℒp (f + g)).restrict s).toLp (⇑(f + g)) =
      ((Lp.memℒp f).restrict s).toLp f + ((Lp.memℒp g).restrict s).toLp g 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure X} (f g : ↥(MeasureTheory.Lp E p μ)) (s : Set X)
Goal: MeasureTheory.Memℒp.toLp ↑↑(f + g) ⋯ = MeasureTheory.Memℒp.toLp ↑↑f ⋯ + MeasureTheory.Memℒp.toLp ↑↑g ⋯
-- MeasureTheory.Lp_toLp_restrict_smul
theorem Lp_toLp_restrict_smul (c : 𝕜) (f : Lp F p μ) (s : Set X) :
    ((Lp.memℒp (c • f)).restrict s).toLp (⇑(c • f)) = c • ((Lp.memℒp f).restrict s).toLp f 
Arguments: {X : Type u_1} {F : Type u_4} [MeasurableSpace X] {𝕜 : Type u_5} [NormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : ℝ≥0∞} {μ : MeasureTheory.Measure X} (c : 𝕜) (f : ↥(MeasureTheory.Lp F p μ)) (s : Set X)
Goal: MeasureTheory.Memℒp.toLp ↑↑(c • f) ⋯ = c • MeasureTheory.Memℒp.toLp ↑↑f ⋯
-- MeasureTheory.norm_Lp_toLp_restrict_le
theorem norm_Lp_toLp_restrict_le (s : Set X) (f : Lp E p μ) :
    ‖((Lp.memℒp f).restrict s).toLp f‖ ≤ ‖f‖ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure X} (s : Set X) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖MeasureTheory.Memℒp.toLp ↑↑f ⋯‖ ≤ ‖f‖
-- MeasureTheory.LpToLpRestrictCLM_coeFn
theorem LpToLpRestrictCLM_coeFn [Fact (1 ≤ p)] (s : Set X) (f : Lp F p μ) :
    LpToLpRestrictCLM X F 𝕜 μ p s f =ᵐ[μ.restrict s] f 
Arguments: {X : Type u_1} {F : Type u_4} [MeasurableSpace X] (𝕜 : Type u_5) [NormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : ℝ≥0∞} {μ : MeasureTheory.Measure X} [Fact (1 ≤ p)] (s : Set X) (f : ↥(MeasureTheory.Lp F p μ))
Goal: ↑↑((MeasureTheory.LpToLpRestrictCLM X F 𝕜 μ p s) f) =ᵐ[μ.restrict s] ↑↑f
-- MeasureTheory.continuous_setIntegral
@[continuity]
theorem continuous_setIntegral [NormedSpace ℝ E] (s : Set X) :
    Continuous fun f : X →₁[μ] E => ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [NormedSpace ℝ E] (s : Set X)
Goal: Continuous fun f => ∫ (x : X) in s, ↑↑f x ∂μ
-- Continuous.integral_pos_of_hasCompactSupport_nonneg_nonzero
theorem Continuous.integral_pos_of_hasCompactSupport_nonneg_nonzero [IsFiniteMeasureOnCompacts μ]
    {f : X → ℝ} {x : X} (f_cont : Continuous f) (f_comp : HasCompactSupport f) (f_nonneg : 0 ≤ f)
    (f_x : f x ≠ 0) : 0 < ∫ x, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ] {f : X → ℝ} {x : X} (f_cont : Continuous f) (f_comp : HasCompactSupport f) (f_nonneg : 0 ≤ f) (f_x : f x ≠ 0)
Goal: 0 < ∫ (x : X), f x ∂μ
-- Filter.Tendsto.integral_sub_linear_isLittleO_ae
theorem Filter.Tendsto.integral_sub_linear_isLittleO_ae
    {μ : Measure X} {l : Filter X} [l.IsMeasurablyGenerated] {f : X → E} {b : E}
    (h : Tendsto f (l ⊓ ae μ) (𝓝 b)) (hfm : StronglyMeasurableAtFilter f l μ)
    (hμ : μ.FiniteAtFilter l) {s : ι → Set X} {li : Filter ι} (hs : Tendsto s li l.smallSets)
    (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • b) =o[li] m 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {ι : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure X} {l : Filter X} [l.IsMeasurablyGenerated] {f : X → E} {b : E} (h : Filter.Tendsto f (l ⊓ MeasureTheory.ae μ) (𝓝 b)) (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {s : ι → Set X} {li : Filter ι} (hs : Filter.Tendsto s li l.smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal) (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl)
Goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • b) =o[li] m
-- ContinuousWithinAt.integral_sub_linear_isLittleO_ae
theorem ContinuousWithinAt.integral_sub_linear_isLittleO_ae [TopologicalSpace X]
    [OpensMeasurableSpace X] {μ : Measure X}
    [IsLocallyFiniteMeasure μ] {x : X} {t : Set X} {f : X → E} (hx : ContinuousWithinAt f t x)
    (ht : MeasurableSet t) (hfm : StronglyMeasurableAtFilter f (𝓝[t] x) μ) {s : ι → Set X}
    {li : Filter ι} (hs : Tendsto s li (𝓝[t] x).smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • f x) =o[li] m 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {ι : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [TopologicalSpace X] [OpensMeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsLocallyFiniteMeasure μ] {x : X} {t : Set X} {f : X → E} (hx : ContinuousWithinAt f t x) (ht : MeasurableSet t) (hfm : StronglyMeasurableAtFilter f (𝓝[t] x) μ) {s : ι → Set X} {li : Filter ι} (hs : Filter.Tendsto s li (𝓝[t] x).smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal) (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl)
Goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • f x) =o[li] m
-- ContinuousAt.integral_sub_linear_isLittleO_ae
theorem ContinuousAt.integral_sub_linear_isLittleO_ae [TopologicalSpace X] [OpensMeasurableSpace X]
    {μ : Measure X} [IsLocallyFiniteMeasure μ] {x : X}
    {f : X → E} (hx : ContinuousAt f x) (hfm : StronglyMeasurableAtFilter f (𝓝 x) μ) {s : ι → Set X}
    {li : Filter ι} (hs : Tendsto s li (𝓝 x).smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • f x) =o[li] m 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {ι : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [TopologicalSpace X] [OpensMeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsLocallyFiniteMeasure μ] {x : X} {f : X → E} (hx : ContinuousAt f x) (hfm : StronglyMeasurableAtFilter f (𝓝 x) μ) {s : ι → Set X} {li : Filter ι} (hs : Filter.Tendsto s li (𝓝 x).smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal) (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl)
Goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • f x) =o[li] m
-- ContinuousOn.integral_sub_linear_isLittleO_ae
theorem ContinuousOn.integral_sub_linear_isLittleO_ae [TopologicalSpace X] [OpensMeasurableSpace X]
    [SecondCountableTopologyEither X E] {μ : Measure X}
    [IsLocallyFiniteMeasure μ] {x : X} {t : Set X} {f : X → E} (hft : ContinuousOn f t) (hx : x ∈ t)
    (ht : MeasurableSet t) {s : ι → Set X} {li : Filter ι} (hs : Tendsto s li (𝓝[t] x).smallSets)
    (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • f x) =o[li] m 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] {ι : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [TopologicalSpace X] [OpensMeasurableSpace X] [SecondCountableTopologyEither X E] {μ : MeasureTheory.Measure X} [MeasureTheory.IsLocallyFiniteMeasure μ] {x : X} {t : Set X} {f : X → E} (hft : ContinuousOn f t) (hx : x ∈ t) (ht : MeasurableSet t) {s : ι → Set X} {li : Filter ι} (hs : Filter.Tendsto s li (𝓝[t] x).smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal) (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl)
Goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • f x) =o[li] m
-- ContinuousLinearMap.integral_compLp
theorem integral_compLp (L : E →L[𝕜] F) (φ : Lp E p μ) :
    ∫ x, (L.compLp φ) x ∂μ = ∫ 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] {p : ℝ≥0∞} [NormedSpace ℝ F] (L : E →L[𝕜] F) (φ : ↥(MeasureTheory.Lp E p μ))
Goal: ∫ (x : X), ↑↑(L.compLp φ) x ∂μ = ∫ (x : X), L (↑↑φ x) ∂μ