ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.haveLebesgueDecomposition_spec
theorem haveLebesgueDecomposition_spec (μ ν : Measure α) [h : HaveLebesgueDecomposition μ ν] :
    Measurable (μ.rnDeriv ν) ∧
      μ.singularPart ν ⟂ₘ ν ∧ μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [h : μ.HaveLebesgueDecomposition ν]
Goal: Measurable (μ.rnDeriv ν) ∧ μ.singularPart ν ⟂ₘ ν ∧ μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
-- MeasureTheory.Measure.rnDeriv_of_not_haveLebesgueDecomposition
lemma rnDeriv_of_not_haveLebesgueDecomposition (h : ¬ HaveLebesgueDecomposition μ ν) :
    μ.rnDeriv ν = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : ¬μ.HaveLebesgueDecomposition ν)
Goal: μ.rnDeriv ν = 0
-- MeasureTheory.Measure.singularPart_of_not_haveLebesgueDecomposition
lemma singularPart_of_not_haveLebesgueDecomposition (h : ¬ HaveLebesgueDecomposition μ ν) :
    μ.singularPart ν = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : ¬μ.HaveLebesgueDecomposition ν)
Goal: μ.singularPart ν = 0
-- MeasureTheory.Measure.measurable_rnDeriv
@[measurability, fun_prop]
theorem measurable_rnDeriv (μ ν : Measure α) : Measurable <| μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: Measurable (μ.rnDeriv ν)
-- MeasureTheory.Measure.mutuallySingular_singularPart
theorem mutuallySingular_singularPart (μ ν : Measure α) : μ.singularPart ν ⟂ₘ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: μ.singularPart ν ⟂ₘ ν
-- MeasureTheory.Measure.haveLebesgueDecomposition_add
theorem haveLebesgueDecomposition_add (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] :
    μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν]
Goal: μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
-- MeasureTheory.Measure.singularPart_add_rnDeriv
lemma singularPart_add_rnDeriv (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] :
    μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν]
Goal: μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = μ
-- MeasureTheory.Measure.rnDeriv_add_singularPart
lemma rnDeriv_add_singularPart (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] :
    ν.withDensity (μ.rnDeriv ν) + μ.singularPart ν = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν]
Goal: ν.withDensity (μ.rnDeriv ν) + μ.singularPart ν = μ
-- MeasureTheory.Measure.haveLebesgueDecomposition_withDensity
theorem haveLebesgueDecomposition_withDensity (μ : Measure α) {f : α → ℝ≥0∞} (hf : Measurable f) :
    (μ.withDensity f).HaveLebesgueDecomposition μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: (μ.withDensity f).HaveLebesgueDecomposition μ
-- MeasureTheory.Measure.singularPart_le
theorem singularPart_le (μ ν : Measure α) : μ.singularPart ν ≤ μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: μ.singularPart ν ≤ μ
-- MeasureTheory.Measure.withDensity_rnDeriv_le
theorem withDensity_rnDeriv_le (μ ν : Measure α) : ν.withDensity (μ.rnDeriv ν) ≤ μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: ν.withDensity (μ.rnDeriv ν) ≤ μ
-- AEMeasurable.singularPart
lemma _root_.AEMeasurable.singularPart {β : Type*} {_ : MeasurableSpace β} {f : α → β}
    (hf : AEMeasurable f μ) (ν : Measure α) :
    AEMeasurable f (μ.singularPart ν) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3}
Goal: ∀ {x : MeasurableSpace β} {f : α → β},
  AEMeasurable f μ → ∀ (ν : MeasureTheory.Measure α), AEMeasurable f (μ.singularPart ν)
-- AEMeasurable.withDensity_rnDeriv
lemma _root_.AEMeasurable.withDensity_rnDeriv {β : Type*} {_ : MeasurableSpace β} {f : α → β}
    (hf : AEMeasurable f μ) (ν : Measure α) :
    AEMeasurable f (ν.withDensity (μ.rnDeriv ν)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3}
Goal: ∀ {x : MeasurableSpace β} {f : α → β},
  AEMeasurable f μ → ∀ (ν : MeasureTheory.Measure α), AEMeasurable f (ν.withDensity (μ.rnDeriv ν))
-- MeasureTheory.Measure.MutuallySingular.singularPart
lemma MutuallySingular.singularPart (h : μ ⟂ₘ ν) (ν' : Measure α) :
    μ.singularPart ν' ⟂ₘ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : μ ⟂ₘ ν) (ν' : MeasureTheory.Measure α)
Goal: μ.singularPart ν' ⟂ₘ ν
-- MeasureTheory.Measure.absolutelyContinuous_withDensity_rnDeriv
lemma absolutelyContinuous_withDensity_rnDeriv [HaveLebesgueDecomposition ν μ] (hμν : μ ≪ ν) :
    μ ≪ μ.withDensity (ν.rnDeriv μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [ν.HaveLebesgueDecomposition μ] (hμν : μ ≪ ν)
Goal: μ ≪ μ.withDensity (ν.rnDeriv μ)
-- MeasureTheory.Measure.singularPart_eq_zero_of_ac
lemma singularPart_eq_zero_of_ac (h : μ ≪ ν) : μ.singularPart ν = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : μ ≪ ν)
Goal: μ.singularPart ν = 0
-- MeasureTheory.Measure.singularPart_zero
@[simp]
theorem singularPart_zero (ν : Measure α) : (0 : Measure α).singularPart ν = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.singularPart 0 ν = 0
-- MeasureTheory.Measure.singularPart_zero_right
@[simp]
lemma singularPart_zero_right (μ : Measure α) : μ.singularPart 0 = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: μ.singularPart 0 = μ
-- MeasureTheory.Measure.singularPart_eq_zero
lemma singularPart_eq_zero (μ ν : Measure α) [μ.HaveLebesgueDecomposition ν] :
    μ.singularPart ν = 0 ↔ μ ≪ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν]
Goal: μ.singularPart ν = 0 ↔ μ ≪ ν
-- MeasureTheory.Measure.withDensity_rnDeriv_eq_zero
@[simp]
lemma withDensity_rnDeriv_eq_zero (μ ν : Measure α) [μ.HaveLebesgueDecomposition ν] :
    ν.withDensity (μ.rnDeriv ν) = 0 ↔ μ ⟂ₘ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν]
Goal: ν.withDensity (μ.rnDeriv ν) = 0 ↔ μ ⟂ₘ ν
-- MeasureTheory.Measure.rnDeriv_eq_zero
@[simp]
lemma rnDeriv_eq_zero (μ ν : Measure α) [μ.HaveLebesgueDecomposition ν] :
    μ.rnDeriv ν =ᵐ[ν] 0 ↔ μ ⟂ₘ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν]
Goal: μ.rnDeriv ν =ᵐ[ν] 0 ↔ μ ⟂ₘ ν
-- MeasureTheory.Measure.rnDeriv_zero
lemma rnDeriv_zero (ν : Measure α) : (0 : Measure α).rnDeriv ν =ᵐ[ν] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.rnDeriv 0 ν =ᵐ[ν] 0
-- MeasureTheory.Measure.MutuallySingular.rnDeriv_ae_eq_zero
lemma MutuallySingular.rnDeriv_ae_eq_zero (hμν : μ ⟂ₘ ν) :
    μ.rnDeriv ν =ᵐ[ν] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (hμν : μ ⟂ₘ ν)
Goal: μ.rnDeriv ν =ᵐ[ν] 0
-- MeasureTheory.Measure.singularPart_withDensity
@[simp]
theorem singularPart_withDensity (ν : Measure α) (f : α → ℝ≥0∞) :
    (ν.withDensity f).singularPart ν = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: (ν.withDensity f).singularPart ν = 0
-- MeasureTheory.Measure.rnDeriv_singularPart
lemma rnDeriv_singularPart (μ ν : Measure α) :
    (μ.singularPart ν).rnDeriv ν =ᵐ[ν] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: (μ.singularPart ν).rnDeriv ν =ᵐ[ν] 0
-- MeasureTheory.Measure.singularPart_self
@[simp]
lemma singularPart_self (μ : Measure α) : μ.singularPart μ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: μ.singularPart μ = 0
-- MeasureTheory.Measure.rnDeriv_self
lemma rnDeriv_self (μ : Measure α) [SigmaFinite μ] : μ.rnDeriv μ =ᵐ[μ] fun _ ↦ 1 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: μ.rnDeriv μ =ᵐ[μ] fun x => 1
-- MeasureTheory.Measure.singularPart_eq_self
lemma singularPart_eq_self [μ.HaveLebesgueDecomposition ν] : μ.singularPart ν = μ ↔ μ ⟂ₘ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν]
Goal: μ.singularPart ν = μ ↔ μ ⟂ₘ ν
-- MeasureTheory.Measure.singularPart_singularPart
@[simp]
lemma singularPart_singularPart (μ ν : Measure α) :
    (μ.singularPart ν).singularPart ν = μ.singularPart ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: (μ.singularPart ν).singularPart ν = μ.singularPart ν
-- MeasureTheory.Measure.lintegral_rnDeriv_lt_top_of_measure_ne_top
theorem lintegral_rnDeriv_lt_top_of_measure_ne_top (ν : Measure α) {s : Set α} (hs : μ s ≠ ∞) :
    ∫⁻ x in s, μ.rnDeriv ν x ∂ν < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (ν : MeasureTheory.Measure α) {s : Set α} (hs : μ s ≠ ⊤)
Goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν < ⊤
-- MeasureTheory.Measure.lintegral_rnDeriv_lt_top
theorem lintegral_rnDeriv_lt_top (μ ν : Measure α) [IsFiniteMeasure μ] :
    ∫⁻ x, μ.rnDeriv ν x ∂ν < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ]
Goal: ∫⁻ (x : α), μ.rnDeriv ν x ∂ν < ⊤
-- MeasureTheory.Measure.integrable_toReal_rnDeriv
lemma integrable_toReal_rnDeriv [IsFiniteMeasure μ] :
    Integrable (fun x ↦ (μ.rnDeriv ν x).toReal) ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ]
Goal: MeasureTheory.Integrable (fun x => (μ.rnDeriv ν x).toReal) ν
-- MeasureTheory.Measure.rnDeriv_lt_top
theorem rnDeriv_lt_top (μ ν : Measure α) [SigmaFinite μ] : ∀ᵐ x ∂ν, μ.rnDeriv ν x < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: ∀ᵐ (x : α) ∂ν, μ.rnDeriv ν x < ⊤
-- MeasureTheory.Measure.rnDeriv_ne_top
lemma rnDeriv_ne_top (μ ν : Measure α) [SigmaFinite μ] : ∀ᵐ x ∂ν, μ.rnDeriv ν x ≠ ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: ∀ᵐ (x : α) ∂ν, μ.rnDeriv ν x ≠ ⊤
-- MeasureTheory.Measure.eq_singularPart
theorem eq_singularPart {s : Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν)
    (hadd : μ = s + ν.withDensity f) : s = μ.singularPart ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν s : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f)
Goal: s = μ.singularPart ν
-- MeasureTheory.Measure.singularPart_smul
theorem singularPart_smul (μ ν : Measure α) (r : ℝ≥0) :
    (r • μ).singularPart ν = r • μ.singularPart ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) (r : ℝ≥0)
Goal: (r • μ).singularPart ν = r • μ.singularPart ν
-- MeasureTheory.Measure.singularPart_smul_right
theorem singularPart_smul_right (μ ν : Measure α) (r : ℝ≥0) (hr : r ≠ 0) :
    μ.singularPart (r • ν) = μ.singularPart ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) (r : ℝ≥0) (hr : r ≠ 0)
Goal: μ.singularPart (r • ν) = μ.singularPart ν
-- MeasureTheory.Measure.singularPart_add
theorem singularPart_add (μ₁ μ₂ ν : Measure α) [HaveLebesgueDecomposition μ₁ ν]
    [HaveLebesgueDecomposition μ₂ ν] :
    (μ₁ + μ₂).singularPart ν = μ₁.singularPart ν + μ₂.singularPart ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ₁ μ₂ ν : MeasureTheory.Measure α) [μ₁.HaveLebesgueDecomposition ν] [μ₂.HaveLebesgueDecomposition ν]
Goal: (μ₁ + μ₂).singularPart ν = μ₁.singularPart ν + μ₂.singularPart ν
-- MeasureTheory.Measure.singularPart_restrict
lemma singularPart_restrict (μ ν : Measure α) [HaveLebesgueDecomposition μ ν]
    {s : Set α} (hs : MeasurableSet s) :
    (μ.restrict s).singularPart ν = (μ.singularPart ν).restrict s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] {s : Set α} (hs : MeasurableSet s)
Goal: (μ.restrict s).singularPart ν = (μ.singularPart ν).restrict s
-- MeasureTheory.Measure.eq_withDensity_rnDeriv
theorem eq_withDensity_rnDeriv {s : Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν)
    (hadd : μ = s + ν.withDensity f) : ν.withDensity f = ν.withDensity (μ.rnDeriv ν) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν s : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f)
Goal: ν.withDensity f = ν.withDensity (μ.rnDeriv ν)
-- MeasureTheory.Measure.eq_withDensity_rnDeriv₀
theorem eq_withDensity_rnDeriv₀ {s : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f ν) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f) :
    ν.withDensity f = ν.withDensity (μ.rnDeriv ν) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν s : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f ν) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f)
Goal: ν.withDensity f = ν.withDensity (μ.rnDeriv ν)
-- MeasureTheory.Measure.eq_rnDeriv₀
theorem eq_rnDeriv₀ [SigmaFinite ν] {s : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f ν) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f) :
    f =ᵐ[ν] μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite ν] {s : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f ν) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f)
Goal: f =ᵐ[ν] μ.rnDeriv ν
-- MeasureTheory.Measure.eq_rnDeriv
theorem eq_rnDeriv [SigmaFinite ν] {s : Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν)
    (hadd : μ = s + ν.withDensity f) : f =ᵐ[ν] μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite ν] {s : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f)
Goal: f =ᵐ[ν] μ.rnDeriv ν
-- MeasureTheory.Measure.rnDeriv_withDensity₀
theorem rnDeriv_withDensity₀ (ν : Measure α) [SigmaFinite ν] {f : α → ℝ≥0∞}
    (hf : AEMeasurable f ν) :
    (ν.withDensity f).rnDeriv ν =ᵐ[ν] f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] {f : α → ℝ≥0∞} (hf : AEMeasurable f ν)
Goal: (ν.withDensity f).rnDeriv ν =ᵐ[ν] f
-- MeasureTheory.Measure.rnDeriv_withDensity
theorem rnDeriv_withDensity (ν : Measure α) [SigmaFinite ν] {f : α → ℝ≥0∞} (hf : Measurable f) :
    (ν.withDensity f).rnDeriv ν =ᵐ[ν] f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: (ν.withDensity f).rnDeriv ν =ᵐ[ν] f
-- MeasureTheory.Measure.rnDeriv_restrict
lemma rnDeriv_restrict (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] [SigmaFinite ν]
    {s : Set α} (hs : MeasurableSet s) :
    (μ.restrict s).rnDeriv ν =ᵐ[ν] s.indicator (μ.rnDeriv ν) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] [MeasureTheory.SigmaFinite ν] {s : Set α} (hs : MeasurableSet s)
Goal: (μ.restrict s).rnDeriv ν =ᵐ[ν] s.indicator (μ.rnDeriv ν)
-- MeasureTheory.Measure.rnDeriv_restrict_self
theorem rnDeriv_restrict_self (ν : Measure α) [SigmaFinite ν] {s : Set α} (hs : MeasurableSet s) :
    (ν.restrict s).rnDeriv ν =ᵐ[ν] s.indicator 1 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] {s : Set α} (hs : MeasurableSet s)
Goal: (ν.restrict s).rnDeriv ν =ᵐ[ν] s.indicator 1
-- MeasureTheory.Measure.rnDeriv_smul_left
theorem rnDeriv_smul_left (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] (r : ℝ≥0) :
    (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [ν.HaveLebesgueDecomposition μ] (r : ℝ≥0)
Goal: (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_smul_left_of_ne_top
theorem rnDeriv_smul_left_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ ∞) :
    (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ ⊤)
Goal: (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_smul_right
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0)
Goal: ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_smul_right_of_ne_top
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ⊤)
Goal: ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_add
lemma rnDeriv_add (ν₁ ν₂ μ : Measure α) [IsFiniteMeasure ν₁] [IsFiniteMeasure ν₂]
    [ν₁.HaveLebesgueDecomposition μ] [ν₂.HaveLebesgueDecomposition μ]
    [(ν₁ + ν₂).HaveLebesgueDecomposition μ] :
    (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ + ν₂.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν₁ ν₂ μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν₁] [MeasureTheory.IsFiniteMeasure ν₂] [ν₁.HaveLebesgueDecomposition μ] [ν₂.HaveLebesgueDecomposition μ] [(ν₁ + ν₂).HaveLebesgueDecomposition μ]
Goal: (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ + ν₂.rnDeriv μ
-- MeasureTheory.Measure.exists_positive_of_not_mutuallySingular
theorem exists_positive_of_not_mutuallySingular (μ ν : Measure α) [IsFiniteMeasure μ]
    [IsFiniteMeasure ν] (h : ¬μ ⟂ₘ ν) :
    ∃ ε : ℝ≥0, 0 < ε ∧
      ∃ E : Set α,
        MeasurableSet E ∧ 0 < ν E ∧ 0 ≤[E] μ.toSignedMeasure - (ε • ν).toSignedMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (h : ¬μ ⟂ₘ ν)
Goal: ∃ ε,
  0 < ε ∧
    ∃ E,
      MeasurableSet E ∧
        0 < ν E ∧
          MeasureTheory.VectorMeasure.restrict 0 E ≤
            MeasureTheory.VectorMeasure.restrict (μ.toSignedMeasure - (ε • ν).toSignedMeasure) E
-- MeasureTheory.Measure.LebesgueDecomposition.zero_mem_measurableLE
theorem zero_mem_measurableLE : (0 : α → ℝ≥0∞) ∈ measurableLE μ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α}
Goal: 0 ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν
-- MeasureTheory.Measure.LebesgueDecomposition.sup_mem_measurableLE
theorem sup_mem_measurableLE {f g : α → ℝ≥0∞} (hf : f ∈ measurableLE μ ν)
    (hg : g ∈ measurableLE μ ν) : (fun a ↦ f a ⊔ g a) ∈ measurableLE μ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hf : f ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν) (hg : g ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν)
Goal: (fun a => f a ⊔ g a) ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν
-- MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup
theorem iSup_succ_eq_sup {α} (f : ℕ → α → ℝ≥0∞) (m : ℕ) (a : α) :
    ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a = f m.succ a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a 
Arguments: {α : Sort u_3} (f : ℕ → α → ℝ≥0∞) (m : ℕ) (a : α)
Goal: ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = f m.succ a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a
-- MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE
theorem iSup_mem_measurableLE (f : ℕ → α → ℝ≥0∞) (hf : ∀ n, f n ∈ measurableLE μ ν) (n : ℕ) :
    (fun x ↦ ⨆ (k) (_ : k ≤ n), f k x) ∈ measurableLE μ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (f : ℕ → α → ℝ≥0∞) (hf : ∀ (n : ℕ), f n ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν) (n : ℕ)
Goal: (fun x => ⨆ k, ⨆ (_ : k ≤ n), f k x) ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν
-- MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE'
theorem iSup_mem_measurableLE' (f : ℕ → α → ℝ≥0∞) (hf : ∀ n, f n ∈ measurableLE μ ν) (n : ℕ) :
    (⨆ (k) (_ : k ≤ n), f k) ∈ measurableLE μ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (f : ℕ → α → ℝ≥0∞) (hf : ∀ (n : ℕ), f n ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν) (n : ℕ)
Goal: ⨆ k, ⨆ (_ : k ≤ n), f k ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν
-- MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone
theorem iSup_monotone {α : Type*} (f : ℕ → α → ℝ≥0∞) :
    Monotone fun n x ↦ ⨆ (k) (_ : k ≤ n), f k x 
Arguments: {α : Type u_3} (f : ℕ → α → ℝ≥0∞)
Goal: Monotone fun n x => ⨆ k, ⨆ (_ : k ≤ n), f k x
-- MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone'
theorem iSup_monotone' {α : Type*} (f : ℕ → α → ℝ≥0∞) (x : α) :
    Monotone fun n ↦ ⨆ (k) (_ : k ≤ n), f k x 
Arguments: {α : Type u_3} (f : ℕ → α → ℝ≥0∞) (x : α)
Goal: Monotone fun n => ⨆ k, ⨆ (_ : k ≤ n), f k x
-- MeasureTheory.Measure.LebesgueDecomposition.iSup_le_le
theorem iSup_le_le {α : Type*} (f : ℕ → α → ℝ≥0∞) (n k : ℕ) (hk : k ≤ n) :
    f k ≤ fun x ↦ ⨆ (k) (_ : k ≤ n), f k x 
Arguments: {α : Type u_3} (f : ℕ → α → ℝ≥0∞) (n k : ℕ) (hk : k ≤ n)
Goal: f k ≤ fun x => ⨆ k, ⨆ (_ : k ≤ n), f k x
-- MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure
theorem haveLebesgueDecomposition_of_finiteMeasure [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    HaveLebesgueDecomposition μ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
Goal: μ.HaveLebesgueDecomposition ν
-- MeasureTheory.Measure.HaveLebesgueDecomposition.sfinite_of_isFiniteMeasure
theorem HaveLebesgueDecomposition.sfinite_of_isFiniteMeasure [SFinite μ]
    (_h : ∀ (μ : Measure α) [IsFiniteMeasure μ], HaveLebesgueDecomposition μ ν) :
    HaveLebesgueDecomposition μ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (_h : ∀ (μ : MeasureTheory.Measure α) [inst : MeasureTheory.IsFiniteMeasure μ], μ.HaveLebesgueDecomposition ν)
Goal: μ.HaveLebesgueDecomposition ν
-- MeasureTheory.Measure.rnDeriv_smul_left'
theorem rnDeriv_smul_left' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ] (r : ℝ≥0) :
    (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] (r : ℝ≥0)
Goal: (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_smul_left_of_ne_top'
theorem rnDeriv_smul_left_of_ne_top' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ]
    {r : ℝ≥0∞} (hr : r ≠ ∞) :
    (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] {r : ℝ≥0∞} (hr : r ≠ ⊤)
Goal: (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_smul_right'
theorem rnDeriv_smul_right' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ]
    {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] {r : ℝ≥0} (hr : r ≠ 0)
Goal: ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ