ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.average_zero_measure
@[simp]
theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : α → E)
Goal: ⨍ (x : α), f x ∂0 = 0
-- MeasureTheory.average_neg
@[simp]
theorem average_neg (f : α → E) : ⨍ x, -f x ∂μ = -⨍ x, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E)
Goal: ⨍ (x : α), -f x ∂μ = -⨍ (x : α), f x ∂μ
-- MeasureTheory.average_eq'
theorem average_eq' (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂(μ univ)⁻¹ • μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E)
Goal: ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂(μ Set.univ)⁻¹ • μ
-- MeasureTheory.average_eq
theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E)
Goal: ⨍ (x : α), f x ∂μ = (μ Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂μ
-- MeasureTheory.average_eq_integral
theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (f : α → E)
Goal: ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ
-- MeasureTheory.measure_smul_average
@[simp]
theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
    (μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E)
Goal: (μ Set.univ).toReal • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ
-- MeasureTheory.setAverage_eq
theorem setAverage_eq (f : α → E) (s : Set α) :
    ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) (s : Set α)
Goal: ⨍ (x : α) in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ (x : α) in s, f x ∂μ
-- MeasureTheory.setAverage_eq'
theorem setAverage_eq' (f : α → E) (s : Set α) :
    ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) (s : Set α)
Goal: ⨍ (x : α) in s, f x ∂μ = ∫ (x : α), f x ∂(μ s)⁻¹ • μ.restrict s
-- MeasureTheory.average_congr
theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f g : α → E} (h : f =ᵐ[μ] g)
Goal: ⨍ (x : α), f x ∂μ = ⨍ (x : α), g x ∂μ
-- MeasureTheory.setAverage_congr
theorem setAverage_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {s t : Set α} {f : α → E} (h : s =ᵐ[μ] t)
Goal: ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in t, f x ∂μ
-- MeasureTheory.setAverage_congr_fun
theorem setAverage_congr_fun (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: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → E} (hs : MeasurableSet s) (h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x)
Goal: ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in s, g x ∂μ
-- MeasureTheory.average_add_measure
theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
    (hμ : Integrable f μ) (hν : Integrable f ν) :
    ⨍ x, f x ∂(μ + ν) =
      ((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ +
        ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure ν] {f : α → E} (hμ : MeasureTheory.Integrable f μ) (hν : MeasureTheory.Integrable f ν)
Goal: ⨍ (x : α), f x ∂(μ + ν) =
  ((μ Set.univ).toReal / ((μ Set.univ).toReal + (ν Set.univ).toReal)) • ⨍ (x : α), f x ∂μ +
    ((ν Set.univ).toReal / ((μ Set.univ).toReal + (ν Set.univ).toReal)) • ⨍ (x : α), f x ∂ν
-- MeasureTheory.average_pair
theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi : Integrable g μ) :
    ⨍ x, (f x, g x) ∂μ = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : MeasureTheory.Measure α} {f : α → E} {g : α → F} (hfi : MeasureTheory.Integrable f μ) (hgi : MeasureTheory.Integrable g μ)
Goal: ⨍ (x : α), (f x, g x) ∂μ = (⨍ (x : α), f x ∂μ, ⨍ (x : α), g x ∂μ)
-- MeasureTheory.measure_smul_setAverage
theorem measure_smul_setAverage (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
    (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} (f : α → E) {s : Set α} (h : μ s ≠ ⊤)
Goal: (μ s).toReal • ⨍ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ
-- MeasureTheory.average_union
theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
    (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
    ⨍ x in s ∪ t, f x ∂μ =
      ((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ +
        ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in t, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f : α → E} {s t : Set α} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ) (hsμ : μ s ≠ ⊤) (htμ : μ t ≠ ⊤) (hfs : MeasureTheory.IntegrableOn f s μ) (hft : MeasureTheory.IntegrableOn f t μ)
Goal: ⨍ (x : α) in s ∪ t, f x ∂μ =
  ((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ (x : α) in s, f x ∂μ +
    ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ (x : α) in t, f x ∂μ
-- MeasureTheory.average_union_mem_openSegment
theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
    (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞)
    (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
    ⨍ x in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f : α → E} {s t : Set α} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ⊤) (htμ : μ t ≠ ⊤) (hfs : MeasureTheory.IntegrableOn f s μ) (hft : MeasureTheory.IntegrableOn f t μ)
Goal: ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ)
-- MeasureTheory.average_union_mem_segment
theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
    (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ)
    (hft : IntegrableOn f t μ) :
    ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f : α → E} {s t : Set α} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ) (hsμ : μ s ≠ ⊤) (htμ : μ t ≠ ⊤) (hfs : MeasureTheory.IntegrableOn f s μ) (hft : MeasureTheory.IntegrableOn f t μ)
Goal: ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ-[ℝ]⨍ (x : α) in t, f x ∂μ]
-- MeasureTheory.average_mem_openSegment_compl_self
theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}
    (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) (hfi : Integrable f μ) :
    ⨍ x, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {f : α → E} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) (hfi : MeasureTheory.Integrable f μ)
Goal: ⨍ (x : α), f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in sᶜ, f x ∂μ)
-- MeasureTheory.average_const
@[simp]
theorem average_const (μ : Measure α) [IsFiniteMeasure μ] [h : NeZero μ] (c : E) :
    ⨍ _x, c ∂μ = c 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [h : NeZero μ] (c : E)
Goal: ⨍ (_x : α), c ∂μ = c
-- MeasureTheory.setAverage_const
theorem setAverage_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) :
    ⨍ _ in s, c ∂μ = c 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ⊤) (c : E)
Goal: ⨍ (x : α) in s, c ∂μ = c
-- MeasureTheory.integral_average
theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
    ∫ _, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E)
Goal: ∫ (x : α), ⨍ (a : α), f a ∂μ ∂μ = ∫ (x : α), f x ∂μ
-- MeasureTheory.setIntegral_setAverage
theorem setIntegral_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
    ∫ _ in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E) (s : Set α)
Goal: ∫ (x : α) in s, ⨍ (a : α) in s, f a ∂μ ∂μ = ∫ (x : α) in s, f x ∂μ
-- MeasureTheory.integral_sub_average
theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
    ∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E)
Goal: ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0
-- MeasureTheory.setAverage_sub_setAverage
theorem setAverage_sub_setAverage (hs : μ s ≠ ∞) (f : α → E) :
    ∫ x in s, f x - ⨍ a in s, f a ∂μ ∂μ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set α} (hs : μ s ≠ ⊤) (f : α → E)
Goal: ∫ (x : α) in s, f x - ⨍ (a : α) in s, f a ∂μ ∂μ = 0
-- MeasureTheory.integral_average_sub
theorem integral_average_sub [IsFiniteMeasure μ] (hf : Integrable f μ) :
    ∫ x, ⨍ a, f a ∂μ - f x ∂μ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), ⨍ (a : α), f a ∂μ - f x ∂μ = 0
-- MeasureTheory.setIntegral_setAverage_sub
theorem setIntegral_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
    ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → E} (hs : μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s μ)
Goal: ∫ (x : α) in s, ⨍ (a : α) in s, f a ∂μ - f x ∂μ = 0
-- MeasureTheory.ofReal_average
theorem ofReal_average {f : α → ℝ} (hf : Integrable f μ) (hf₀ : 0 ≤ᵐ[μ] f) :
    ENNReal.ofReal (⨍ x, f x ∂μ) = (∫⁻ x, ENNReal.ofReal (f x) ∂μ) / μ univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) (hf₀ : 0 ≤ᵐ[μ] f)
Goal: ENNReal.ofReal (⨍ (x : α), f x ∂μ) = (∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ) / μ Set.univ
-- MeasureTheory.ofReal_setAverage
theorem ofReal_setAverage {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f) :
    ENNReal.ofReal (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) / μ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hf : MeasureTheory.IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f)
Goal: ENNReal.ofReal (⨍ (x : α) in s, f x ∂μ) = (∫⁻ (x : α) in s, ENNReal.ofReal (f x) ∂μ) / μ s
-- MeasureTheory.toReal_laverage
theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
    (⨍⁻ x, f x ∂μ).toReal = ⨍ x, (f x).toReal ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤)
Goal: (⨍⁻ (x : α), f x ∂μ).toReal = ⨍ (x : α), (f x).toReal ∂μ
-- MeasureTheory.toReal_setLaverage
theorem toReal_setLaverage {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
    (hf' : ∀ᵐ x ∂μ.restrict s, f x ≠ ∞) :
    (⨍⁻ x in s, f x ∂μ).toReal = ⨍ x in s, (f x).toReal ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s)) (hf' : ∀ᵐ (x : α) ∂μ.restrict s, f x ≠ ⊤)
Goal: (⨍⁻ (x : α) in s, f x ∂μ).toReal = ⨍ (x : α) in s, (f x).toReal ∂μ
-- MeasureTheory.measure_le_setAverage_pos
theorem measure_le_setAverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
    0 < μ ({x ∈ s | f x ≤ ⨍ a in s, f a ∂μ}) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s μ)
Goal: 0 < μ {x | x ∈ s ∧ f x ≤ ⨍ (a : α) in s, f a ∂μ}
-- MeasureTheory.measure_setAverage_le_pos
theorem measure_setAverage_le_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
    0 < μ ({x ∈ s | ⨍ a in s, f a ∂μ ≤ f x}) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s μ)
Goal: 0 < μ {x | x ∈ s ∧ ⨍ (a : α) in s, f a ∂μ ≤ f x}
-- MeasureTheory.exists_le_setAverage
theorem exists_le_setAverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
    ∃ x ∈ s, f x ≤ ⨍ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s μ)
Goal: ∃ x ∈ s, f x ≤ ⨍ (a : α) in s, f a ∂μ
-- MeasureTheory.exists_setAverage_le
theorem exists_setAverage_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
    ∃ x ∈ s, ⨍ a in s, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s μ)
Goal: ∃ x ∈ s, ⨍ (a : α) in s, f a ∂μ ≤ f x
-- MeasureTheory.measure_le_average_pos
theorem measure_le_average_pos (hμ : μ ≠ 0) (hf : Integrable f μ) :
    0 < μ {x | f x ≤ ⨍ a, f a ∂μ} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f μ)
Goal: 0 < μ {x | f x ≤ ⨍ (a : α), f a ∂μ}
-- MeasureTheory.measure_average_le_pos
theorem measure_average_le_pos (hμ : μ ≠ 0) (hf : Integrable f μ) :
    0 < μ {x | ⨍ a, f a ∂μ ≤ f x} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f μ)
Goal: 0 < μ {x | ⨍ (a : α), f a ∂μ ≤ f x}
-- MeasureTheory.exists_le_average
theorem exists_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, f x ≤ ⨍ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f μ)
Goal: ∃ x, f x ≤ ⨍ (a : α), f a ∂μ
-- MeasureTheory.exists_average_le
theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f μ)
Goal: ∃ x, ⨍ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.exists_not_mem_null_le_average
theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ f x ≤ ⨍ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f μ) (hN : μ N = 0)
Goal: ∃ x ∉ N, f x ≤ ⨍ (a : α), f a ∂μ
-- MeasureTheory.exists_not_mem_null_average_le
theorem exists_not_mem_null_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ ⨍ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f μ) (hN : μ N = 0)
Goal: ∃ x ∉ N, ⨍ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.measure_le_integral_pos
theorem measure_le_integral_pos (hf : Integrable f μ) : 0 < μ {x | f x ≤ ∫ a, f a ∂μ} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f μ)
Goal: 0 < μ {x | f x ≤ ∫ (a : α), f a ∂μ}
-- MeasureTheory.measure_integral_le_pos
theorem measure_integral_le_pos (hf : Integrable f μ) : 0 < μ {x | ∫ a, f a ∂μ ≤ f x} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f μ)
Goal: 0 < μ {x | ∫ (a : α), f a ∂μ ≤ f x}
-- MeasureTheory.exists_le_integral
theorem exists_le_integral (hf : Integrable f μ) : ∃ x, f x ≤ ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f μ)
Goal: ∃ x, f x ≤ ∫ (a : α), f a ∂μ
-- MeasureTheory.exists_integral_le
theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f μ)
Goal: ∃ x, ∫ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.exists_not_mem_null_le_integral
theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ f x ≤ ∫ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f μ) (hN : μ N = 0)
Goal: ∃ x ∉ N, f x ≤ ∫ (a : α), f a ∂μ
-- MeasureTheory.exists_not_mem_null_integral_le
theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ ∫ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f μ) (hN : μ N = 0)
Goal: ∃ x ∉ N, ∫ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.measure_le_setLaverage_pos
theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
    (hf : AEMeasurable f (μ.restrict s)) : 0 < μ {x ∈ s | f x ≤ ⨍⁻ a in s, f a ∂μ} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ⊤) (hf : AEMeasurable f (μ.restrict s))
Goal: 0 < μ {x | x ∈ s ∧ f x ≤ ⨍⁻ (a : α) in s, f a ∂μ}
-- MeasureTheory.measure_setLaverage_le_pos
theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
    (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : 0 < μ {x ∈ s | ⨍⁻ a in s, f a ∂μ ≤ f x} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hμ : μ s ≠ 0) (hs : MeasureTheory.NullMeasurableSet s μ) (hint : ∫⁻ (a : α) in s, f a ∂μ ≠ ⊤)
Goal: 0 < μ {x | x ∈ s ∧ ⨍⁻ (a : α) in s, f a ∂μ ≤ f x}
-- MeasureTheory.exists_le_setLaverage
theorem exists_le_setLaverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : AEMeasurable f (μ.restrict s)) :
    ∃ x ∈ s, f x ≤ ⨍⁻ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ⊤) (hf : AEMeasurable f (μ.restrict s))
Goal: ∃ x ∈ s, f x ≤ ⨍⁻ (a : α) in s, f a ∂μ
-- MeasureTheory.exists_setLaverage_le
theorem exists_setLaverage_le (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
    (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : ∃ x ∈ s, ⨍⁻ a in s, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hμ : μ s ≠ 0) (hs : MeasureTheory.NullMeasurableSet s μ) (hint : ∫⁻ (a : α) in s, f a ∂μ ≠ ⊤)
Goal: ∃ x ∈ s, ⨍⁻ (a : α) in s, f a ∂μ ≤ f x
-- MeasureTheory.measure_laverage_le_pos
theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) :
    0 < μ {x | ⨍⁻ a, f a ∂μ ≤ f x} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hμ : μ ≠ 0) (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤)
Goal: 0 < μ {x | ⨍⁻ (a : α), f a ∂μ ≤ f x}
-- MeasureTheory.exists_laverage_le
theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ⨍⁻ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hμ : μ ≠ 0) (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤)
Goal: ∃ x, ⨍⁻ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.exists_not_mem_null_laverage_le
theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α, f a ∂μ ≠ ∞) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ ⨍⁻ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ≥0∞} (hμ : μ ≠ 0) (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) (hN : μ N = 0)
Goal: ∃ x ∉ N, ⨍⁻ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.measure_le_laverage_pos
theorem measure_le_laverage_pos (hμ : μ ≠ 0) (hf : AEMeasurable f μ) :
    0 < μ {x | f x ≤ ⨍⁻ a, f a ∂μ} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : AEMeasurable f μ)
Goal: 0 < μ {x | f x ≤ ⨍⁻ (a : α), f a ∂μ}
-- MeasureTheory.exists_le_laverage
theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f x ≤ ⨍⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : AEMeasurable f μ)
Goal: ∃ x, f x ≤ ⨍⁻ (a : α), f a ∂μ
-- MeasureTheory.exists_not_mem_null_le_laverage
theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ f x ≤ ⨍⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ≥0∞} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0)
Goal: ∃ x ∉ N, f x ≤ ⨍⁻ (a : α), f a ∂μ
-- MeasureTheory.measure_le_lintegral_pos
theorem measure_le_lintegral_pos (hf : AEMeasurable f μ) : 0 < μ {x | f x ≤ ∫⁻ a, f a ∂μ} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} [MeasureTheory.IsProbabilityMeasure μ] (hf : AEMeasurable f μ)
Goal: 0 < μ {x | f x ≤ ∫⁻ (a : α), f a ∂μ}