ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.hasSum_lintegral_measure
theorem hasSum_lintegral_measure {ι} {_ : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
    HasSum (fun i => ∫⁻ a, f a ∂μ i) (∫⁻ a, f a ∂Measure.sum μ) 
Arguments: {α : Type u_1} {ι : Type u_5}
Goal: ∀ {x : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → MeasureTheory.Measure α),
  HasSum (fun i => ∫⁻ (a : α), f a ∂μ i) (∫⁻ (a : α), f a ∂MeasureTheory.Measure.sum μ)
-- MeasureTheory.lintegral_add_measure
@[simp]
theorem lintegral_add_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ ν : Measure α) :
    ∫⁻ a, f a ∂(μ + ν) = ∫⁻ a, f a ∂μ + ∫⁻ a, f a ∂ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ ν : MeasureTheory.Measure α)
Goal: ∫⁻ (a : α), f a ∂(μ + ν) = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), f a ∂ν
-- MeasureTheory.lintegral_finset_sum_measure
@[simp]
theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞)
    (μ : ι → Measure α) : ∫⁻ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫⁻ a, f a ∂μ i 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞) (μ : ι → MeasureTheory.Measure α)
Goal: ∫⁻ (a : α), f a ∂∑ i ∈ s, μ i = ∑ i ∈ s, ∫⁻ (a : α), f a ∂μ i
-- MeasureTheory.lintegral_zero_measure
@[simp]
theorem lintegral_zero_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂(0 : Measure α) = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a ∂0 = 0
-- MeasureTheory.lintegral_of_isEmpty
@[simp]
theorem lintegral_of_isEmpty {α} [MeasurableSpace α] [IsEmpty α] (μ : Measure α) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = 0 
Arguments: {α : Type u_5} [MeasurableSpace α] [IsEmpty α] (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = 0
-- MeasureTheory.setLIntegral_empty
theorem setLIntegral_empty (f : α → ℝ≥0∞) : ∫⁻ x in ∅, f x ∂μ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α) in ∅, f x ∂μ = 0
-- MeasureTheory.setLIntegral_univ
theorem setLIntegral_univ (f : α → ℝ≥0∞) : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α) in Set.univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.setLIntegral_measure_zero
theorem setLIntegral_measure_zero (s : Set α) (f : α → ℝ≥0∞) (hs' : μ s = 0) :
    ∫⁻ x in s, f x ∂μ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) (f : α → ℝ≥0∞) (hs' : μ s = 0)
Goal: ∫⁻ (x : α) in s, f x ∂μ = 0
-- MeasureTheory.lintegral_finset_sum'
theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞}
    (hf : ∀ b ∈ s, AEMeasurable (f b) μ) :
    ∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, AEMeasurable (f b) μ)
Goal: ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
-- MeasureTheory.lintegral_finset_sum
theorem lintegral_finset_sum (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, Measurable (f b)) :
    ∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, Measurable (f b))
Goal: ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
-- MeasureTheory.lintegral_const_mul
@[simp]
theorem lintegral_const_mul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.lintegral_const_mul''
theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.lintegral_const_mul_le
theorem lintegral_const_mul_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
    r * ∫⁻ a, f a ∂μ ≤ ∫⁻ a, r * f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) (f : α → ℝ≥0∞)
Goal: r * ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), r * f a ∂μ
-- MeasureTheory.lintegral_const_mul'
theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
    ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ⊤)
Goal: ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.lintegral_mul_const
theorem lintegral_mul_const (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r
-- MeasureTheory.lintegral_mul_const''
theorem lintegral_mul_const'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r
-- MeasureTheory.lintegral_mul_const_le
theorem lintegral_mul_const_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
    (∫⁻ a, f a ∂μ) * r ≤ ∫⁻ a, f a * r ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) (f : α → ℝ≥0∞)
Goal: (∫⁻ (a : α), f a ∂μ) * r ≤ ∫⁻ (a : α), f a * r ∂μ
-- MeasureTheory.lintegral_mul_const'
theorem lintegral_mul_const' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
    ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ⊤)
Goal: ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r
-- MeasureTheory.lintegral_lintegral_mul
theorem lintegral_lintegral_mul {β} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ≥0∞}
    {g : β → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g ν) :
    ∫⁻ x, ∫⁻ y, f x * g y ∂ν ∂μ = (∫⁻ x, f x ∂μ) * ∫⁻ y, g y ∂ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g ν)
Goal: ∫⁻ (x : α), ∫⁻ (y : β), f x * g y ∂ν ∂μ = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν
-- MeasureTheory.lintegral_rw₁
theorem lintegral_rw₁ {f f' : α → β} (h : f =ᵐ[μ] f') (g : β → ℝ≥0∞) :
    ∫⁻ a, g (f a) ∂μ = ∫⁻ a, g (f' a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f f' : α → β} (h : f =ᵐ[μ] f') (g : β → ℝ≥0∞)
Goal: ∫⁻ (a : α), g (f a) ∂μ = ∫⁻ (a : α), g (f' a) ∂μ
-- MeasureTheory.lintegral_rw₂
theorem lintegral_rw₂ {f₁ f₁' : α → β} {f₂ f₂' : α → γ} (h₁ : f₁ =ᵐ[μ] f₁') (h₂ : f₂ =ᵐ[μ] f₂')
    (g : β → γ → ℝ≥0∞) : ∫⁻ a, g (f₁ a) (f₂ a) ∂μ = ∫⁻ a, g (f₁' a) (f₂' a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f₁ f₁' : α → β} {f₂ f₂' : α → γ} (h₁ : f₁ =ᵐ[μ] f₁') (h₂ : f₂ =ᵐ[μ] f₂') (g : β → γ → ℝ≥0∞)
Goal: ∫⁻ (a : α), g (f₁ a) (f₂ a) ∂μ = ∫⁻ (a : α), g (f₁' a) (f₂' a) ∂μ
-- MeasureTheory.lintegral_indicator_le
theorem lintegral_indicator_le (f : α → ℝ≥0∞) (s : Set α) :
    ∫⁻ a, s.indicator f a ∂μ ≤ ∫⁻ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) (s : Set α)
Goal: ∫⁻ (a : α), s.indicator f a ∂μ ≤ ∫⁻ (a : α) in s, f a ∂μ
-- MeasureTheory.lintegral_indicator
@[simp]
theorem lintegral_indicator (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (a : α), s.indicator f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ
-- MeasureTheory.lintegral_indicator₀
theorem lintegral_indicator₀ (f : α → ℝ≥0∞) {s : Set α} (hs : NullMeasurableSet s μ) :
    ∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: ∫⁻ (a : α), s.indicator f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ
-- MeasureTheory.lintegral_indicator_const_le
theorem lintegral_indicator_const_le (s : Set α) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) a ∂μ ≤ c * μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) (c : ℝ≥0∞)
Goal: ∫⁻ (a : α), s.indicator (fun x => c) a ∂μ ≤ c * μ s
-- MeasureTheory.lintegral_indicator_const₀
theorem lintegral_indicator_const₀ {s : Set α} (hs : NullMeasurableSet s μ) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (c : ℝ≥0∞)
Goal: ∫⁻ (a : α), s.indicator (fun x => c) a ∂μ = c * μ s
-- MeasureTheory.lintegral_indicator_const
theorem lintegral_indicator_const {s : Set α} (hs : MeasurableSet s) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (c : ℝ≥0∞)
Goal: ∫⁻ (a : α), s.indicator (fun x => c) a ∂μ = c * μ s
-- MeasureTheory.setLIntegral_eq_of_support_subset
lemma setLIntegral_eq_of_support_subset {s : Set α} {f : α → ℝ≥0∞} (hsf : f.support ⊆ s) :
    ∫⁻ 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∞} (hsf : Function.support f ⊆ s)
Goal: ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.setLIntegral_eq_const
theorem setLIntegral_eq_const {f : α → ℝ≥0∞} (hf : Measurable f) (r : ℝ≥0∞) :
    ∫⁻ x in { x | f x = r }, f x ∂μ = r * μ { x | f x = r } 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (r : ℝ≥0∞)
Goal: ∫⁻ (x : α) in {x | f x = r}, f x ∂μ = r * μ {x | f x = r}
-- MeasureTheory.lintegral_indicator_one_le
theorem lintegral_indicator_one_le (s : Set α) : ∫⁻ a, s.indicator 1 a ∂μ ≤ μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α)
Goal: ∫⁻ (a : α), s.indicator 1 a ∂μ ≤ μ s
-- MeasureTheory.lintegral_indicator_one₀
@[simp]
theorem lintegral_indicator_one₀ {s : Set α} (hs : NullMeasurableSet s μ) :
    ∫⁻ a, s.indicator 1 a ∂μ = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: ∫⁻ (a : α), s.indicator 1 a ∂μ = μ s
-- MeasureTheory.lintegral_indicator_one
@[simp]
theorem lintegral_indicator_one {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ a, s.indicator 1 a ∂μ = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (a : α), s.indicator 1 a ∂μ = μ s
-- MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral
theorem lintegral_add_mul_meas_add_le_le_lintegral {f g : α → ℝ≥0∞} (hle : f ≤ᵐ[μ] g)
    (hg : AEMeasurable g μ) (ε : ℝ≥0∞) :
    ∫⁻ a, f a ∂μ + ε * μ { x | f x + ε ≤ g x } ≤ ∫⁻ a, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hle : f ≤ᵐ[μ] g) (hg : AEMeasurable g μ) (ε : ℝ≥0∞)
Goal: ∫⁻ (a : α), f a ∂μ + ε * μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.mul_meas_ge_le_lintegral₀
theorem mul_meas_ge_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (ε : ℝ≥0∞) :
    ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (ε : ℝ≥0∞)
Goal: ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.mul_meas_ge_le_lintegral
theorem mul_meas_ge_le_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) (ε : ℝ≥0∞) :
    ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (ε : ℝ≥0∞)
Goal: ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.meas_le_lintegral₀
lemma meas_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    {s : Set α} (hs : ∀ x ∈ s, 1 ≤ f x) : μ s ≤ ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {s : Set α} (hs : ∀ x ∈ s, 1 ≤ f x)
Goal: μ s ≤ ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.lintegral_le_meas
lemma lintegral_le_meas {s : Set α} {f : α → ℝ≥0∞} (hf : ∀ a, f a ≤ 1) (h'f : ∀ a ∈ sᶜ, f a = 0) :
    ∫⁻ a, f a ∂μ ≤ μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hf : ∀ (a : α), f a ≤ 1) (h'f : ∀ a ∈ sᶜ, f a = 0)
Goal: ∫⁻ (a : α), f a ∂μ ≤ μ s
-- MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero
theorem lintegral_eq_top_of_measure_eq_top_ne_zero {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hμf : μ {x | f x = ∞} ≠ 0) : ∫⁻ x, f x ∂μ = ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hμf : μ {x | f x = ⊤} ≠ 0)
Goal: ∫⁻ (x : α), f x ∂μ = ⊤
-- MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero
theorem setLintegral_eq_top_of_measure_eq_top_ne_zero {f : α → ℝ≥0∞} {s : Set α}
    (hf : AEMeasurable f (μ.restrict s)) (hμf : μ ({x ∈ s | f x = ∞}) ≠ 0) :
    ∫⁻ x in s, f x ∂μ = ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f (μ.restrict s)) (hμf : μ {x | x ∈ s ∧ f x = ⊤} ≠ 0)
Goal: ∫⁻ (x : α) in s, f x ∂μ = ⊤
-- MeasureTheory.measure_eq_top_of_lintegral_ne_top
theorem measure_eq_top_of_lintegral_ne_top {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hμf : ∫⁻ x, f x ∂μ ≠ ∞) : μ {x | f x = ∞} = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hμf : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: μ {x | f x = ⊤} = 0
-- MeasureTheory.measure_eq_top_of_setLintegral_ne_top
theorem measure_eq_top_of_setLintegral_ne_top {f : α → ℝ≥0∞} {s : Set α}
    (hf : AEMeasurable f (μ.restrict s)) (hμf : ∫⁻ x in s, f x ∂μ ≠ ∞) :
    μ ({x ∈ s | f x = ∞}) = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f (μ.restrict s)) (hμf : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤)
Goal: μ {x | x ∈ s ∧ f x = ⊤} = 0
-- MeasureTheory.meas_ge_le_lintegral_div
theorem meas_ge_le_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
    (hε' : ε ≠ ∞) : μ { x | ε ≤ f x } ≤ (∫⁻ a, f a ∂μ) / ε 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) (hε' : ε ≠ ⊤)
Goal: μ {x | ε ≤ f x} ≤ (∫⁻ (a : α), f a ∂μ) / ε
-- MeasureTheory.ae_eq_of_ae_le_of_lintegral_le
theorem ae_eq_of_ae_le_of_lintegral_le {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) (hf : ∫⁻ x, f x ∂μ ≠ ∞)
    (hg : AEMeasurable g μ) (hgf : ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ) : f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) (hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (hg : AEMeasurable g μ) (hgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.lintegral_eq_zero_iff'
@[simp]
theorem lintegral_eq_zero_iff' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: ∫⁻ (a : α), f a ∂μ = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.lintegral_eq_zero_iff
@[simp]
theorem lintegral_eq_zero_iff {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 
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 = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.lintegral_pos_iff_support
theorem lintegral_pos_iff_support {f : α → ℝ≥0∞} (hf : Measurable f) :
    (0 < ∫⁻ a, f a ∂μ) ↔ 0 < μ (Function.support f) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: 0 < ∫⁻ (a : α), f a ∂μ ↔ 0 < μ (Function.support f)
-- MeasureTheory.setLintegral_pos_iff
theorem setLintegral_pos_iff {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} :
    0 < ∫⁻ a in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α}
Goal: 0 < ∫⁻ (a : α) in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s)
-- MeasureTheory.lintegral_iSup_ae
theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n))
    (h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f n.succ a) : ∫⁻ 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 : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a)
Goal: ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ
-- MeasureTheory.lintegral_sub'
theorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)
    (h_le : g ≤ᵐ[μ] f) : ∫⁻ 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∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤) (h_le : g ≤ᵐ[μ] f)
Goal: ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_sub
theorem lintegral_sub {f g : α → ℝ≥0∞} (hg : Measurable g) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)
    (h_le : g ≤ᵐ[μ] f) : ∫⁻ 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∞} (hg : Measurable g) (hg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤) (h_le : g ≤ᵐ[μ] f)
Goal: ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ
-- MeasureTheory.lintegral_sub_le'
theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) :
    ∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ)
Goal: ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ
-- MeasureTheory.lintegral_sub_le
theorem lintegral_sub_le (f g : α → ℝ≥0∞) (hf : Measurable f) :
    ∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : α → ℝ≥0∞) (hf : Measurable f)
Goal: ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ
-- MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt
theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
    (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) (h : ∃ᵐ x ∂μ, f x ≠ g x) :
    ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (h_le : f ≤ᵐ[μ] g) (h : ∃ᵐ (x : α) ∂μ, f x ≠ g x)
Goal: ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ
-- MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on
theorem lintegral_strict_mono_of_ae_le_of_ae_lt_on {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
    (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) {s : Set α} (hμs : μ s ≠ 0)
    (h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (h_le : f ≤ᵐ[μ] g) {s : Set α} (hμs : μ s ≠ 0) (h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x < g x)
Goal: ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ
-- MeasureTheory.lintegral_strict_mono
theorem lintegral_strict_mono {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : AEMeasurable g μ)
    (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h : ∀ᵐ x ∂μ, f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : AEMeasurable g μ) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (h : ∀ᵐ (x : α) ∂μ, f x < g x)
Goal: ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ
-- MeasureTheory.setLIntegral_strict_mono
theorem setLIntegral_strict_mono {f g : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s)
    (hs : μ s ≠ 0) (hg : Measurable g) (hfi : ∫⁻ x in s, f x ∂μ ≠ ∞)
    (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} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s) (hs : μ s ≠ 0) (hg : Measurable g) (hfi : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤) (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.lintegral_iInf_ae
theorem lintegral_iInf_ae {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n))
    (h_mono : ∀ n : ℕ, f n.succ ≤ᵐ[μ] f n) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
    ∫⁻ 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∞} (h_meas : ∀ (n : ℕ), Measurable (f n)) (h_mono : ∀ (n : ℕ), f n.succ ≤ᵐ[μ] f n) (h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤)
Goal: ∫⁻ (a : α), ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
-- MeasureTheory.lintegral_iInf
theorem lintegral_iInf {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) (h_anti : Antitone f)
    (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) : ∫⁻ 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∞} (h_meas : ∀ (n : ℕ), Measurable (f n)) (h_anti : Antitone f) (h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤)
Goal: ∫⁻ (a : α), ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
-- MeasureTheory.lintegral_iInf'
theorem lintegral_iInf' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ)
    (h_anti : ∀ᵐ a ∂μ, Antitone (fun i ↦ f i a)) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
    ∫⁻ 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∞} (h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ) (h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a) (h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤)
Goal: ∫⁻ (a : α), ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
-- MeasureTheory.lintegral_iInf_directed_of_measurable
theorem lintegral_iInf_directed_of_measurable {mα : MeasurableSpace α} [Countable β]
    {f : β → α → ℝ≥0∞} {μ : Measure α} (hμ : μ ≠ 0) (hf : ∀ b, Measurable (f b))
    (hf_int : ∀ b, ∫⁻ a, f b a ∂μ ≠ ∞) (h_directed : Directed (· ≥ ·) f) :
    ∫⁻ a, ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ a, f b a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [Countable β] {f : β → α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hμ : μ ≠ 0) (hf : ∀ (b : β), Measurable (f b)) (hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤) (h_directed : Directed (fun x x_1 => x ≥ x_1) f)
Goal: ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
-- MeasureTheory.lintegral_liminf_le'
theorem lintegral_liminf_le' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ) :
    ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ)
Goal: ∫⁻ (a : α), Filter.liminf (fun n => f n a) Filter.atTop ∂μ ≤ Filter.liminf (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop
-- MeasureTheory.lintegral_liminf_le
theorem lintegral_liminf_le {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) :
    ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ (n : ℕ), Measurable (f n))
Goal: ∫⁻ (a : α), Filter.liminf (fun n => f n a) Filter.atTop ∂μ ≤ Filter.liminf (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop