ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsUnifLocDoublingMeasure.exists_measure_closedBall_le_mul''
class IsUnifLocDoublingMeasure {α : Type*} [MetricSpace α] [MeasurableSpace α]
  (μ : Measure α) : Prop where
  exists_measure_closedBall_le_mul'' :
    ∃ C : ℝ≥0, ∀ᶠ ε in 𝓝[>] 0, ∀ x, μ (closedBall x (2 * ε)) ≤ C * μ (closedBall x ε)
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [self : IsUnifLocDoublingMeasure μ]
Goal: ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (Metric.closedBall x (2 * ε)) ≤ ↑C * μ (Metric.closedBall x ε)
-- IsUnifLocDoublingMeasure.exists_measure_closedBall_le_mul
theorem exists_measure_closedBall_le_mul :
    ∃ C : ℝ≥0, ∀ᶠ ε in 𝓝[>] 0, ∀ x, μ (closedBall x (2 * ε)) ≤ C * μ (closedBall x ε) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ]
Goal: ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (Metric.closedBall x (2 * ε)) ≤ ↑C * μ (Metric.closedBall x ε)
-- IsUnifLocDoublingMeasure.exists_measure_closedBall_le_mul'
theorem exists_measure_closedBall_le_mul' :
    ∀ᶠ ε in 𝓝[>] 0, ∀ x, μ (closedBall x (2 * ε)) ≤ doublingConstant μ * μ (closedBall x ε) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ]
Goal: ∀ᶠ (ε : ℝ) in 𝓝[>] 0,
  ∀ (x : α),
    μ (Metric.closedBall x (2 * ε)) ≤ ↑(IsUnifLocDoublingMeasure.doublingConstant μ) * μ (Metric.closedBall x ε)
-- IsUnifLocDoublingMeasure.exists_eventually_forall_measure_closedBall_le_mul
theorem exists_eventually_forall_measure_closedBall_le_mul (K : ℝ) :
    ∃ C : ℝ≥0, ∀ᶠ ε in 𝓝[>] 0, ∀ x, ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ C * μ (closedBall x ε) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] (K : ℝ)
Goal: ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (Metric.closedBall x (t * ε)) ≤ ↑C * μ (Metric.closedBall x ε)
-- IsUnifLocDoublingMeasure.one_le_scalingConstantOf
@[simp]
theorem one_le_scalingConstantOf (K : ℝ) : 1 ≤ scalingConstantOf μ K 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] (K : ℝ)
Goal: 1 ≤ IsUnifLocDoublingMeasure.scalingConstantOf μ K
-- IsUnifLocDoublingMeasure.eventually_measure_mul_le_scalingConstantOf_mul
theorem eventually_measure_mul_le_scalingConstantOf_mul (K : ℝ) :
    ∃ R : ℝ,
      0 < R ∧
        ∀ x t r, t ∈ Ioc 0 K → r ≤ R →
          μ (closedBall x (t * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] (K : ℝ)
Goal: ∃ R,
  0 < R ∧
    ∀ (x : α) (t r : ℝ),
      t ∈ Set.Ioc 0 K →
        r ≤ R →
          μ (Metric.closedBall x (t * r)) ≤
            ↑(IsUnifLocDoublingMeasure.scalingConstantOf μ K) * μ (Metric.closedBall x r)
-- IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul
theorem eventually_measure_le_scaling_constant_mul (K : ℝ) :
    ∀ᶠ r in 𝓝[>] 0, ∀ x, μ (closedBall x (K * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] (K : ℝ)
Goal: ∀ᶠ (r : ℝ) in 𝓝[>] 0,
  ∀ (x : α),
    μ (Metric.closedBall x (K * r)) ≤ ↑(IsUnifLocDoublingMeasure.scalingConstantOf μ K) * μ (Metric.closedBall x r)
-- IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'
theorem eventually_measure_le_scaling_constant_mul' (K : ℝ) (hK : 0 < K) :
    ∀ᶠ r in 𝓝[>] 0, ∀ x,
      μ (closedBall x r) ≤ scalingConstantOf μ K⁻¹ * μ (closedBall x (K * r)) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] (K : ℝ) (hK : 0 < K)
Goal: ∀ᶠ (r : ℝ) in 𝓝[>] 0,
  ∀ (x : α),
    μ (Metric.closedBall x r) ≤ ↑(IsUnifLocDoublingMeasure.scalingConstantOf μ K⁻¹) * μ (Metric.closedBall x (K * r))
-- IsUnifLocDoublingMeasure.scalingScaleOf_pos
theorem scalingScaleOf_pos (K : ℝ) : 0 < scalingScaleOf μ K 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] (K : ℝ)
Goal: 0 < IsUnifLocDoublingMeasure.scalingScaleOf μ K
-- IsUnifLocDoublingMeasure.measure_mul_le_scalingConstantOf_mul
theorem measure_mul_le_scalingConstantOf_mul {K : ℝ} {x : α} {t r : ℝ} (ht : t ∈ Ioc 0 K)
    (hr : r ≤ scalingScaleOf μ K) :
    μ (closedBall x (t * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] {K : ℝ} {x : α} {t r : ℝ} (ht : t ∈ Set.Ioc 0 K) (hr : r ≤ IsUnifLocDoublingMeasure.scalingScaleOf μ K)
Goal: μ (Metric.closedBall x (t * r)) ≤ ↑(IsUnifLocDoublingMeasure.scalingConstantOf μ K) * μ (Metric.closedBall x r)
-- MeasureTheory.lintegral_def
irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ 
Arguments: {α : Type u_5}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞),
  MeasureTheory.lintegral μ f = ⨆ g, ⨆ (_ : ↑g ≤ f), g.lintegral μ
-- MeasureTheory.SimpleFunc.lintegral_eq_lintegral
theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) :
    ∫⁻ a, f a ∂μ = f.lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) (μ : MeasureTheory.Measure α)
Goal: ∫⁻ (a : α), ↑f a ∂μ = f.lintegral μ
-- MeasureTheory.lintegral_mono'
@[mono]
theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄
    (hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} ⦃μ ν : MeasureTheory.Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g)
Goal: ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂ν
-- MeasureTheory.lintegral_mono_fn'
@[gcongr] theorem lintegral_mono_fn' ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) (h2 : μ ≤ ν) :
    lintegral μ f ≤ lintegral ν g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ (x : α), f x ≤ g x) (h2 : μ ≤ ν)
Goal: MeasureTheory.lintegral μ f ≤ MeasureTheory.lintegral ν g
-- MeasureTheory.lintegral_mono
theorem lintegral_mono ⦃f g : α → ℝ≥0∞⦄ (hfg : 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≥0∞⦄ (hfg : f ≤ g)
Goal: ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_mono_fn
@[gcongr] theorem lintegral_mono_fn ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) :
    lintegral μ f ≤ lintegral μ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ (x : α), f x ≤ g x)
Goal: MeasureTheory.lintegral μ f ≤ MeasureTheory.lintegral μ g
-- MeasureTheory.lintegral_mono_nnreal
theorem lintegral_mono_nnreal {f g : α → ℝ≥0} (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≥0} (h : f ≤ g)
Goal: ∫⁻ (a : α), ↑(f a) ∂μ ≤ ∫⁻ (a : α), ↑(g a) ∂μ
-- MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral
theorem iSup_lintegral_measurable_le_eq_lintegral (f : α → ℝ≥0∞) :
    ⨆ (g : α → ℝ≥0∞) (_ : Measurable g) (_ : g ≤ f), ∫⁻ a, g a ∂μ = ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞)
Goal: ⨆ g, ⨆ (_ : Measurable g), ⨆ (_ : g ≤ f), ∫⁻ (a : α), g a ∂μ = ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.lintegral_mono_set
theorem lintegral_mono_set {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
    (hst : s ⊆ t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} ⦃μ : MeasureTheory.Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞},
  s ⊆ t → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in t, f x ∂μ
-- MeasureTheory.lintegral_mono_set'
theorem lintegral_mono_set' {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
    (hst : s ≤ᵐ[μ] t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} ⦃μ : MeasureTheory.Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞},
  s ≤ᵐ[μ] t → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in t, f x ∂μ
-- MeasureTheory.monotone_lintegral
theorem monotone_lintegral {_ : MeasurableSpace α} (μ : Measure α) : Monotone (lintegral μ) 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α), Monotone (MeasureTheory.lintegral μ)
-- MeasureTheory.lintegral_const
@[simp]
theorem lintegral_const (c : ℝ≥0∞) : ∫⁻ _, c ∂μ = c * μ univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ℝ≥0∞)
Goal: ∫⁻ (x : α), c ∂μ = c * μ Set.univ
-- MeasureTheory.lintegral_zero
theorem lintegral_zero : ∫⁻ _ : α, 0 ∂μ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: ∫⁻ (x : α), 0 ∂μ = 0
-- MeasureTheory.lintegral_zero_fun
theorem lintegral_zero_fun : lintegral μ (0 : α → ℝ≥0∞) = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.lintegral μ 0 = 0
-- MeasureTheory.lintegral_one
theorem lintegral_one : ∫⁻ _, (1 : ℝ≥0∞) ∂μ = μ univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: ∫⁻ (x : α), 1 ∂μ = μ Set.univ
-- MeasureTheory.setLIntegral_const
theorem setLIntegral_const (s : Set α) (c : ℝ≥0∞) : ∫⁻ _ in s, c ∂μ = c * μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) (c : ℝ≥0∞)
Goal: ∫⁻ (x : α) in s, c ∂μ = c * μ s
-- MeasureTheory.setLIntegral_one
theorem setLIntegral_one (s) : ∫⁻ _ in s, 1 ∂μ = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α)
Goal: ∫⁻ (x : α) in s, 1 ∂μ = μ s
-- MeasureTheory.setLIntegral_const_lt_top
theorem setLIntegral_const_lt_top [IsFiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ∞) :
    ∫⁻ _ in s, c ∂μ < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: ∫⁻ (x : α) in s, c ∂μ < ⊤
-- MeasureTheory.lintegral_const_lt_top
theorem lintegral_const_lt_top [IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ∞) : ∫⁻ _, c ∂μ < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: ∫⁻ (x : α), c ∂μ < ⊤
-- MeasureTheory.exists_measurable_le_lintegral_eq
theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) :
    ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_eq_nnreal
theorem lintegral_eq_nnreal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) :
    ∫⁻ a, f a ∂μ =
      ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ x, ↑(φ x) ≤ f x), (φ.map ((↑) : ℝ≥0 → ℝ≥0∞)).lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : MeasureTheory.Measure α)
Goal: ∫⁻ (a : α), f a ∂μ =
  ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), (MeasureTheory.SimpleFunc.map ENNReal.ofNNReal φ).lintegral μ
-- MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos
theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ φ : α →ₛ ℝ≥0,
      (∀ x, ↑(φ x) ≤ f x) ∧
        ∀ ψ : α →ₛ ℝ≥0, (∀ x, ↑(ψ x) ≤ f x) → (map (↑) (ψ - φ)).lintegral μ < ε 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (h : ∫⁻ (x : α), f x ∂μ ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ φ,
  (∀ (x : α), ↑(↑φ x) ≤ f x) ∧
    ∀ (ψ : MeasureTheory.SimpleFunc α ℝ≥0),
      (∀ (x : α), ↑(↑ψ x) ≤ f x) → (MeasureTheory.SimpleFunc.map ENNReal.ofNNReal (ψ - φ)).lintegral μ < ε
-- MeasureTheory.iSup_lintegral_le
theorem iSup_lintegral_le {ι : Sort*} (f : ι → α → ℝ≥0∞) :
    ⨆ i, ∫⁻ a, f i a ∂μ ≤ ∫⁻ a, ⨆ i, f i a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} (f : ι → α → ℝ≥0∞)
Goal: ⨆ i, ∫⁻ (a : α), f i a ∂μ ≤ ∫⁻ (a : α), ⨆ i, f i a ∂μ
-- MeasureTheory.iSup₂_lintegral_le
theorem iSup₂_lintegral_le {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
    ⨆ (i) (j), ∫⁻ a, f i j a ∂μ ≤ ∫⁻ a, ⨆ (i) (j), f i j a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} {ι' : ι → Sort u_6} (f : (i : ι) → ι' i → α → ℝ≥0∞)
Goal: ⨆ i, ⨆ j, ∫⁻ (a : α), f i j a ∂μ ≤ ∫⁻ (a : α), ⨆ i, ⨆ j, f i j a ∂μ
-- MeasureTheory.le_iInf_lintegral
theorem le_iInf_lintegral {ι : Sort*} (f : ι → α → ℝ≥0∞) :
    ∫⁻ a, ⨅ i, f i a ∂μ ≤ ⨅ i, ∫⁻ a, f i a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} (f : ι → α → ℝ≥0∞)
Goal: ∫⁻ (a : α), ⨅ i, f i a ∂μ ≤ ⨅ i, ∫⁻ (a : α), f i a ∂μ
-- MeasureTheory.le_iInf₂_lintegral
theorem le_iInf₂_lintegral {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
    ∫⁻ a, ⨅ (i) (h : ι' i), f i h a ∂μ ≤ ⨅ (i) (h : ι' i), ∫⁻ a, f i h a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} {ι' : ι → Sort u_6} (f : (i : ι) → ι' i → α → ℝ≥0∞)
Goal: ∫⁻ (a : α), ⨅ i, ⨅ h, f i h a ∂μ ≤ ⨅ i, ⨅ h, ∫⁻ (a : α), f i h a ∂μ
-- MeasureTheory.lintegral_mono_ae
theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ g a) :
    ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (h : ∀ᵐ (a : α) ∂μ, f a ≤ g a)
Goal: ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.setLIntegral_mono_ae
theorem setLIntegral_mono_ae {s : Set α} {f g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.restrict s))
    (hfg : ∀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} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.restrict s)) (hfg : ∀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.setLIntegral_mono
theorem setLIntegral_mono {s : Set α} {f g : α → ℝ≥0∞} (hg : Measurable g)
    (hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → ℝ≥0∞} (hg : Measurable g) (hfg : ∀ x ∈ s, f x ≤ g x)
Goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
-- MeasureTheory.setLIntegral_mono_ae'
theorem setLIntegral_mono_ae' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
    (hfg : ∀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} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s) (hfg : ∀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.setLIntegral_mono'
theorem setLIntegral_mono' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
    (hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s) (hfg : ∀ x ∈ s, f x ≤ g x)
Goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
-- MeasureTheory.setLIntegral_le_lintegral
theorem setLIntegral_le_lintegral (s : Set α) (f : α → ℝ≥0∞) :
    ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.lintegral_congr_ae
theorem lintegral_congr_ae {f g : α → ℝ≥0∞} (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≥0∞} (h : f =ᵐ[μ] g)
Goal: ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_congr
theorem lintegral_congr {f g : α → ℝ≥0∞} (h : ∀ a, f a = g a) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (h : ∀ (a : α), f a = g a)
Goal: ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.setLIntegral_congr
theorem setLIntegral_congr {f : α → ℝ≥0∞} {s t : Set α} (h : s =ᵐ[μ] t) :
    ∫⁻ x in s, f x ∂μ = ∫⁻ x in t, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {s t : Set α} (h : s =ᵐ[μ] t)
Goal: ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in t, f x ∂μ
-- MeasureTheory.setLIntegral_congr_fun
theorem setLIntegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s)
    (hfg : ∀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} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s) (hfg : ∀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.lintegral_ofReal_le_lintegral_nnnorm
theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) :
    ∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ x, ‖f x‖₊ ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ)
Goal: ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂μ
-- MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg
theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) :
    ∫⁻ 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} (h_nonneg : 0 ≤ᵐ[μ] f)
Goal: ∫⁻ (x : α), ↑‖f x‖₊ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ
-- MeasureTheory.lintegral_nnnorm_eq_of_nonneg
theorem lintegral_nnnorm_eq_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) :
    ∫⁻ 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} (h_nonneg : 0 ≤ f)
Goal: ∫⁻ (x : α), ↑‖f x‖₊ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ
-- MeasureTheory.lintegral_iSup
theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
    ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ≥0∞} (hf : ∀ (n : ℕ), Measurable (f n)) (h_mono : Monotone f)
Goal: ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ
-- MeasureTheory.lintegral_iSup'
theorem lintegral_iSup' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ)
    (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ≥0∞} (hf : ∀ (n : ℕ), AEMeasurable (f n) μ) (h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x)
Goal: ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ
-- MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone
theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
    (hf : ∀ n, AEMeasurable (f n) 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≥0∞} {F : α → ℝ≥0∞} (hf : ∀ (n : ℕ), AEMeasurable (f n) 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.lintegral_eq_iSup_eapprox_lintegral
theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, f a ∂μ = ⨆ n, (eapprox f n).lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α), f a ∂μ = ⨆ n, (MeasureTheory.SimpleFunc.eapprox f n).lintegral μ
-- MeasureTheory.exists_pos_setLIntegral_lt_of_measure_lt
theorem exists_pos_setLIntegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞}
    (hε : ε ≠ 0) : ∃ δ > 0, ∀ s, μ s < δ → ∫⁻ x in s, f x ∂μ < ε 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (h : ∫⁻ (x : α), f x ∂μ ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ δ > 0, ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε
-- MeasureTheory.tendsto_setLIntegral_zero
theorem tendsto_setLIntegral_zero {ι} {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {l : Filter ι}
    {s : ι → Set α} (hl : Tendsto (μ ∘ s) l (𝓝 0)) :
    Tendsto (fun i => ∫⁻ x in s i, f x ∂μ) l (𝓝 0) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_5} {f : α → ℝ≥0∞} (h : ∫⁻ (x : α), f x ∂μ ≠ ⊤) {l : Filter ι} {s : ι → Set α} (hl : Filter.Tendsto (⇑μ ∘ s) l (𝓝 0))
Goal: Filter.Tendsto (fun i => ∫⁻ (x : α) in s i, f x ∂μ) l (𝓝 0)
-- MeasureTheory.le_lintegral_add
theorem le_lintegral_add (f g : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ ≤ ∫⁻ a, f a + g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ ≤ ∫⁻ (a : α), f a + g a ∂μ
-- MeasureTheory.lintegral_add_aux
theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g)
Goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_add_left
@[simp]
theorem lintegral_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_add_left'
theorem lintegral_add_left' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (g : α → ℝ≥0∞) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (g : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_add_right'
theorem lintegral_add_right' (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
Goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_add_right
@[simp]
theorem lintegral_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g)
Goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_smul_measure
@[simp]
theorem lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) : ∫⁻ a, f a ∂c • μ = c * ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ℝ≥0∞) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a ∂c • μ = c * ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.setLIntegral_smul_measure
lemma setLIntegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) (s : Set α) :
    ∫⁻ a in s, f a ∂(c • μ) = c * ∫⁻ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ℝ≥0∞) (f : α → ℝ≥0∞) (s : Set α)
Goal: ∫⁻ (a : α) in s, f a ∂c • μ = c * ∫⁻ (a : α) in s, f a ∂μ
-- MeasureTheory.lintegral_sum_measure
@[simp]
theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
    ∫⁻ a, f a ∂Measure.sum μ = ∑' i, ∫⁻ a, f a ∂μ i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {ι : Type u_5} (f : α → ℝ≥0∞) (μ : ι → MeasureTheory.Measure α)
Goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.sum μ = ∑' (i : ι), ∫⁻ (a : α), f a ∂μ i