ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.exp_llr_of_ac
lemma exp_llr_of_ac (μ ν : Measure α) [SigmaFinite μ] [Measure.HaveLebesgueDecomposition μ ν]
    (hμν : μ ≪ ν) :
    (fun x ↦ exp (llr μ ν x)) =ᵐ[μ] fun x ↦ (μ.rnDeriv ν x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [μ.HaveLebesgueDecomposition ν] (hμν : μ.AbsolutelyContinuous ν)
Goal: (fun x => rexp (MeasureTheory.llr μ ν x)) =ᵐ[μ] fun x => (μ.rnDeriv ν x).toReal
-- MeasureTheory.exp_llr_of_ac'
lemma exp_llr_of_ac' (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] (hμν : ν ≪ μ) :
    (fun x ↦ exp (llr μ ν x)) =ᵐ[ν] fun x ↦ (μ.rnDeriv ν x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : ν.AbsolutelyContinuous μ)
Goal: (fun x => rexp (MeasureTheory.llr μ ν x)) =ᵐ[ν] fun x => (μ.rnDeriv ν x).toReal
-- MeasureTheory.neg_llr
lemma neg_llr [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    - llr μ ν =ᵐ[μ] llr ν μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν)
Goal: -MeasureTheory.llr μ ν =ᵐ[μ] MeasureTheory.llr ν μ
-- MeasureTheory.exp_neg_llr
lemma exp_neg_llr [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    (fun x ↦ exp (- llr μ ν x)) =ᵐ[μ] fun x ↦ (ν.rnDeriv μ x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν)
Goal: (fun x => rexp (-MeasureTheory.llr μ ν x)) =ᵐ[μ] fun x => (ν.rnDeriv μ x).toReal
-- MeasureTheory.exp_neg_llr'
lemma exp_neg_llr' [SigmaFinite μ] [SigmaFinite ν] (hμν : ν ≪ μ) :
    (fun x ↦ exp (- llr μ ν x)) =ᵐ[ν] fun x ↦ (ν.rnDeriv μ x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : ν.AbsolutelyContinuous μ)
Goal: (fun x => rexp (-MeasureTheory.llr μ ν x)) =ᵐ[ν] fun x => (ν.rnDeriv μ x).toReal
-- MeasureTheory.measurable_llr
@[measurability]
lemma measurable_llr (μ ν : Measure α) : Measurable (llr μ ν) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: Measurable (MeasureTheory.llr μ ν)
-- MeasureTheory.stronglyMeasurable_llr
@[measurability]
lemma stronglyMeasurable_llr (μ ν : Measure α) : StronglyMeasurable (llr μ ν) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: MeasureTheory.StronglyMeasurable (MeasureTheory.llr μ ν)
-- MeasureTheory.llr_smul_left
lemma llr_smul_left [IsFiniteMeasure μ] [Measure.HaveLebesgueDecomposition μ ν]
    (hμν : μ ≪ ν) (c : ℝ≥0∞) (hc : c ≠ 0) (hc_ne_top : c ≠ ∞) :
    llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log c.toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [μ.HaveLebesgueDecomposition ν] (hμν : μ.AbsolutelyContinuous ν) (c : ℝ≥0∞) (hc : c ≠ 0) (hc_ne_top : c ≠ ⊤)
Goal: MeasureTheory.llr (c • μ) ν =ᵐ[μ] fun x => MeasureTheory.llr μ ν x + Real.log c.toReal
-- MeasureTheory.llr_smul_right
lemma llr_smul_right [IsFiniteMeasure μ] [Measure.HaveLebesgueDecomposition μ ν]
    (hμν : μ ≪ ν) (c : ℝ≥0∞) (hc : c ≠ 0) (hc_ne_top : c ≠ ∞) :
    llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log c.toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [μ.HaveLebesgueDecomposition ν] (hμν : μ.AbsolutelyContinuous ν) (c : ℝ≥0∞) (hc : c ≠ 0) (hc_ne_top : c ≠ ⊤)
Goal: MeasureTheory.llr μ (c • ν) =ᵐ[μ] fun x => MeasureTheory.llr μ ν x - Real.log c.toReal
-- MeasureTheory.llr_tilted_left
lemma llr_tilted_left [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν)
    (hf : Integrable (fun x ↦ exp (f x)) μ) (hfν : AEMeasurable f ν) :
    (llr (μ.tilted f) ν) =ᵐ[μ] fun x ↦ f x - log (∫ z, exp (f z) ∂μ) + llr μ ν x 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν) (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ) (hfν : AEMeasurable f ν)
Goal: MeasureTheory.llr (μ.tilted f) ν =ᵐ[μ] fun x => f x - Real.log (∫ (z : α), rexp (f z) ∂μ) + MeasureTheory.llr μ ν x
-- MeasureTheory.integrable_llr_tilted_left
lemma integrable_llr_tilted_left [IsFiniteMeasure μ] [SigmaFinite ν]
    (hμν : μ ≪ ν) (hf : Integrable f μ) (h_int : Integrable (llr μ ν) μ)
    (hfμ : Integrable (fun x ↦ exp (f x)) μ) (hfν : AEMeasurable f ν) :
    Integrable (llr (μ.tilted f) ν) μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν) (hf : MeasureTheory.Integrable f μ) (h_int : MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ) (hfμ : MeasureTheory.Integrable (fun x => rexp (f x)) μ) (hfν : AEMeasurable f ν)
Goal: MeasureTheory.Integrable (MeasureTheory.llr (μ.tilted f) ν) μ
-- MeasureTheory.integral_llr_tilted_left
lemma integral_llr_tilted_left [IsProbabilityMeasure μ] [SigmaFinite ν]
    (hμν : μ ≪ ν) (hf : Integrable f μ) (h_int : Integrable (llr μ ν) μ)
    (hfμ : Integrable (fun x ↦ exp (f x)) μ) (hfν : AEMeasurable f ν) :
    ∫ x, llr (μ.tilted f) ν x ∂μ = ∫ x, llr μ ν x ∂μ + ∫ x, f x ∂μ - log (∫ x, exp (f x) ∂μ) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν) (hf : MeasureTheory.Integrable f μ) (h_int : MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ) (hfμ : MeasureTheory.Integrable (fun x => rexp (f x)) μ) (hfν : AEMeasurable f ν)
Goal: ∫ (x : α), MeasureTheory.llr (μ.tilted f) ν x ∂μ =
  ∫ (x : α), MeasureTheory.llr μ ν x ∂μ + ∫ (x : α), f x ∂μ - Real.log (∫ (x : α), rexp (f x) ∂μ)
-- MeasureTheory.llr_tilted_right
lemma llr_tilted_right [SigmaFinite μ] [SigmaFinite ν]
    (hμν : μ ≪ ν) (hf : Integrable (fun x ↦ exp (f x)) ν) :
    (llr μ (ν.tilted f)) =ᵐ[μ] fun x ↦ - f x + log (∫ z, exp (f z) ∂ν) + llr μ ν x 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν) (hf : MeasureTheory.Integrable (fun x => rexp (f x)) ν)
Goal: MeasureTheory.llr μ (ν.tilted f) =ᵐ[μ] fun x => -f x + Real.log (∫ (z : α), rexp (f z) ∂ν) + MeasureTheory.llr μ ν x
-- MeasureTheory.integrable_llr_tilted_right
lemma integrable_llr_tilted_right [IsFiniteMeasure μ] [SigmaFinite ν]
    (hμν : μ ≪ ν) (hfμ : Integrable f μ)
    (h_int : Integrable (llr μ ν) μ) (hfν : Integrable (fun x ↦ exp (f x)) ν) :
    Integrable (llr μ (ν.tilted f)) μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν) (hfμ : MeasureTheory.Integrable f μ) (h_int : MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ) (hfν : MeasureTheory.Integrable (fun x => rexp (f x)) ν)
Goal: MeasureTheory.Integrable (MeasureTheory.llr μ (ν.tilted f)) μ
-- MeasureTheory.integral_llr_tilted_right
lemma integral_llr_tilted_right [IsProbabilityMeasure μ] [SigmaFinite ν]
    (hμν : μ ≪ ν) (hfμ : Integrable f μ) (hfν : Integrable (fun x ↦ exp (f x)) ν)
    (h_int : Integrable (llr μ ν) μ) :
    ∫ x, llr μ (ν.tilted f) x ∂μ = ∫ x, llr μ ν x ∂μ - ∫ x, f x ∂μ + log (∫ x, exp (f x) ∂ν) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.SigmaFinite ν] (hμν : μ.AbsolutelyContinuous ν) (hfμ : MeasureTheory.Integrable f μ) (hfν : MeasureTheory.Integrable (fun x => rexp (f x)) ν) (h_int : MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ)
Goal: ∫ (x : α), MeasureTheory.llr μ (ν.tilted f) x ∂μ =
  ∫ (x : α), MeasureTheory.llr μ ν x ∂μ - ∫ (x : α), f x ∂μ + Real.log (∫ (x : α), rexp (f x) ∂ν)
-- MeasureTheory.ae_eq_zero_of_forall_inner
theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
    [SecondCountableTopology E] {f : α → E} (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) :
    f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [SecondCountableTopology E] {f : α → E} (hf : ∀ (c : E), (fun x => ⟪c, f x⟫_𝕜) =ᵐ[μ] 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable
theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    {t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : α → E}
    (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) (h't : ∀ᵐ x ∂μ, f x ∈ t) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} (𝕜 : Type u_3) {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : α → E} (hf : ∀ (c : NormedSpace.Dual 𝕜 E), (fun x => c (f x)) =ᵐ[μ] 0) (h't : ∀ᵐ (x : α) ∂μ, f x ∈ t)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.ae_eq_zero_of_forall_dual
theorem ae_eq_zero_of_forall_dual [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    [SecondCountableTopology E] {f : α → E} (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) :
    f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} (𝕜 : Type u_3) {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [SecondCountableTopology E] {f : α → E} (hf : ∀ (c : NormedSpace.Dual 𝕜 E), (fun x => c (f x)) =ᵐ[μ] 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.ae_const_le_iff_forall_lt_measure_zero
theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [TopologicalSpace β]
    [OrderTopology β] [FirstCountableTopology β] (f : α → β) (c : β) :
    (∀ᵐ x ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3} [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [FirstCountableTopology β] (f : α → β) (c : β)
Goal: (∀ᵐ (x : α) ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0
-- MeasureTheory.ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀
theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀ [SigmaFinite μ]
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ) :
    f ≤ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ)
Goal: f ≤ᵐ[μ] g
-- MeasureTheory.ae_le_of_forall_setLIntegral_le_of_sigmaFinite
theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
    (hf : Measurable f)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in s, g x ∂μ) : f ≤ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : Measurable f) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ)
Goal: f ≤ᵐ[μ] g
-- MeasureTheory.ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite₀
theorem ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite₀ [SigmaFinite μ]
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite
theorem ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
    (hf : Measurable f) (hg : Measurable g)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg
theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ)
Goal: 0 ≤ᵐ[μ] f
-- MeasureTheory.ae_le_of_forall_setIntegral_le
theorem ae_le_of_forall_setIntegral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
    (hf_le : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) ≤ ∫ x in s, g x ∂μ) :
    f ≤ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hf_le : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ ≤ ∫ (x : α) in s, g x ∂μ)
Goal: f ≤ᵐ[μ] g
-- MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter
theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter {f : α → ℝ} {t : Set α}
    (hf : IntegrableOn f t μ)
    (hf_zero : ∀ s, MeasurableSet s → μ (s ∩ t) < ∞ → 0 ≤ ∫ x in s ∩ t, f x ∂μ) :
    0 ≤ᵐ[μ.restrict t] f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} {t : Set α} (hf : MeasureTheory.IntegrableOn f t μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ)
Goal: 0 ≤ᵐ[μ.restrict t] f
-- MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite
theorem ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : α → ℝ} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ)
Goal: 0 ≤ᵐ[μ] f
-- MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg
theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg {f : α → ℝ}
    (hf : AEFinStronglyMeasurable f μ)
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ)
Goal: 0 ≤ᵐ[μ] f
-- MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg
theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg {f : α → ℝ}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) {t : Set α}
    (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : 0 ≤ᵐ[μ.restrict t] f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ⊤)
Goal: 0 ≤ᵐ[μ.restrict t] f
-- MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real
theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real {f : α → ℝ}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
    (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ⊤)
Goal: f =ᵐ[μ.restrict t] 0
-- MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero
theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
    (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ⊤)
Goal: f =ᵐ[μ.restrict t] 0
-- MeasureTheory.ae_eq_restrict_of_forall_setIntegral_eq
theorem ae_eq_restrict_of_forall_setIntegral_eq {f g : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : α → E} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn g s μ) (hfg_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ⊤)
Goal: f =ᵐ[μ.restrict t] g
-- MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite
theorem ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [MeasureTheory.SigmaFinite μ] {f : α → E} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite
theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [MeasureTheory.SigmaFinite μ] {f g : α → E} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn g s μ) (hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero
theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0) (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq
theorem AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq {f g : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    (hf : AEFinStronglyMeasurable f μ) (hg : AEFinStronglyMeasurable g μ) : f =ᵐ[μ] g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : α → E} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn g s μ) (hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ) (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hg : MeasureTheory.AEFinStronglyMeasurable g μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.Lp.ae_eq_zero_of_forall_setIntegral_eq_zero
theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {p : ℝ≥0∞} (f : ↥(MeasureTheory.Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑f) s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0)
Goal: ↑↑f =ᵐ[μ] 0
-- MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq
theorem Lp.ae_eq_of_forall_setIntegral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {p : ℝ≥0∞} (f g : ↥(MeasureTheory.Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑f) s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑g) s μ) (hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = ∫ (x : α) in s, ↑↑g x ∂μ)
Goal: ↑↑f =ᵐ[μ] ↑↑g