ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.ae_le_set_inter
theorem ae_le_set_inter {s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t') :
    (s ∩ s' : Set α) ≤ᵐ[μ] (t ∩ t' : Set α) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t')
Goal: s ∩ s' ≤ᵐ[μ] t ∩ t'
-- MeasureTheory.ae_le_set_union
theorem ae_le_set_union {s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t') :
    (s ∪ s' : Set α) ≤ᵐ[μ] (t ∪ t' : Set α) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t')
Goal: s ∪ s' ≤ᵐ[μ] t ∪ t'
-- MeasureTheory.union_ae_eq_right
theorem union_ae_eq_right : (s ∪ t : Set α) =ᵐ[μ] t ↔ μ (s \ t) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: s ∪ t =ᵐ[μ] t ↔ μ (s \ t) = 0
-- MeasureTheory.diff_ae_eq_self
theorem diff_ae_eq_self : (s \ t : Set α) =ᵐ[μ] s ↔ μ (s ∩ t) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: s \ t =ᵐ[μ] s ↔ μ (s ∩ t) = 0
-- MeasureTheory.diff_null_ae_eq_self
theorem diff_null_ae_eq_self (ht : μ t = 0) : (s \ t : Set α) =ᵐ[μ] s 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (ht : μ t = 0)
Goal: s \ t =ᵐ[μ] s
-- MeasureTheory.ae_eq_set
theorem ae_eq_set {s t : Set α} : s =ᵐ[μ] t ↔ μ (s \ t) = 0 ∧ μ (t \ s) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: s =ᵐ[μ] t ↔ μ (s \ t) = 0 ∧ μ (t \ s) = 0
-- MeasureTheory.measure_symmDiff_eq_zero_iff
@[simp]
theorem measure_symmDiff_eq_zero_iff {s t : Set α} : μ (s ∆ t) = 0 ↔ s =ᵐ[μ] t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: μ (symmDiff s t) = 0 ↔ s =ᵐ[μ] t
-- MeasureTheory.ae_eq_set_compl_compl
@[simp]
theorem ae_eq_set_compl_compl {s t : Set α} : sᶜ =ᵐ[μ] tᶜ ↔ s =ᵐ[μ] t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: sᶜ =ᵐ[μ] tᶜ ↔ s =ᵐ[μ] t
-- MeasureTheory.ae_eq_set_compl
theorem ae_eq_set_compl {s t : Set α} : sᶜ =ᵐ[μ] t ↔ s =ᵐ[μ] tᶜ 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: sᶜ =ᵐ[μ] t ↔ s =ᵐ[μ] tᶜ
-- MeasureTheory.ae_eq_set_inter
theorem ae_eq_set_inter {s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t') :
    (s ∩ s' : Set α) =ᵐ[μ] (t ∩ t' : Set α) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t')
Goal: s ∩ s' =ᵐ[μ] t ∩ t'
-- MeasureTheory.ae_eq_set_union
theorem ae_eq_set_union {s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t') :
    (s ∪ s' : Set α) =ᵐ[μ] (t ∪ t' : Set α) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t')
Goal: s ∪ s' =ᵐ[μ] t ∪ t'
-- MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left
theorem union_ae_eq_univ_of_ae_eq_univ_left (h : s =ᵐ[μ] univ) : (s ∪ t : Set α) =ᵐ[μ] univ 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : s =ᵐ[μ] Set.univ)
Goal: s ∪ t =ᵐ[μ] Set.univ
-- MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_right
theorem union_ae_eq_univ_of_ae_eq_univ_right (h : t =ᵐ[μ] univ) : (s ∪ t : Set α) =ᵐ[μ] univ 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : t =ᵐ[μ] Set.univ)
Goal: s ∪ t =ᵐ[μ] Set.univ
-- MeasureTheory.union_ae_eq_right_of_ae_eq_empty
theorem union_ae_eq_right_of_ae_eq_empty (h : s =ᵐ[μ] (∅ : Set α)) : (s ∪ t : Set α) =ᵐ[μ] t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : s =ᵐ[μ] ∅)
Goal: s ∪ t =ᵐ[μ] t
-- MeasureTheory.union_ae_eq_left_of_ae_eq_empty
theorem union_ae_eq_left_of_ae_eq_empty (h : t =ᵐ[μ] (∅ : Set α)) : (s ∪ t : Set α) =ᵐ[μ] s 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : t =ᵐ[μ] ∅)
Goal: s ∪ t =ᵐ[μ] s
-- MeasureTheory.inter_ae_eq_right_of_ae_eq_univ
theorem inter_ae_eq_right_of_ae_eq_univ (h : s =ᵐ[μ] univ) : (s ∩ t : Set α) =ᵐ[μ] t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : s =ᵐ[μ] Set.univ)
Goal: s ∩ t =ᵐ[μ] t
-- MeasureTheory.inter_ae_eq_left_of_ae_eq_univ
theorem inter_ae_eq_left_of_ae_eq_univ (h : t =ᵐ[μ] univ) : (s ∩ t : Set α) =ᵐ[μ] s 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : t =ᵐ[μ] Set.univ)
Goal: s ∩ t =ᵐ[μ] s
-- MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_left
theorem inter_ae_eq_empty_of_ae_eq_empty_left (h : s =ᵐ[μ] (∅ : Set α)) :
    (s ∩ t : Set α) =ᵐ[μ] (∅ : Set α) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : s =ᵐ[μ] ∅)
Goal: s ∩ t =ᵐ[μ] ∅
-- MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_right
theorem inter_ae_eq_empty_of_ae_eq_empty_right (h : t =ᵐ[μ] (∅ : Set α)) :
    (s ∩ t : Set α) =ᵐ[μ] (∅ : Set α) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : t =ᵐ[μ] ∅)
Goal: s ∩ t =ᵐ[μ] ∅
-- Set.mulIndicator_ae_eq_one
@[to_additive]
theorem _root_.Set.mulIndicator_ae_eq_one {M : Type*} [One M] {f : α → M} {s : Set α} :
    s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ f.mulSupport) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {M : Type u_4} [One M] {f : α → M} {s : Set α}
Goal: s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ Function.mulSupport f) = 0
-- Set.indicator_ae_eq_zero
@[to_additive]
theorem _root_.Set.mulIndicator_ae_eq_one {M : Type*} [One M] {f : α → M} {s : Set α} :
    s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ f.mulSupport) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {M : Type u_4} [Zero M] {f : α → M} {s : Set α}
Goal: s.indicator f =ᵐ[μ] 0 ↔ μ (s ∩ Function.support f) = 0
-- MeasureTheory.measure_mono_ae
@[mono]
theorem measure_mono_ae (H : s ≤ᵐ[μ] t) : μ s ≤ μ t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (H : s ≤ᵐ[μ] t)
Goal: μ s ≤ μ t
-- MeasureTheory.measure_congr
theorem measure_congr (H : s =ᵐ[μ] t) : μ s = μ t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (H : s =ᵐ[μ] t)
Goal: μ s = μ t
-- MeasureTheory.measure_mono_null_ae
theorem measure_mono_null_ae (H : s ≤ᵐ[μ] t) (ht : μ t = 0) : μ s = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (H : s ≤ᵐ[μ] t) (ht : μ t = 0)
Goal: μ s = 0
-- MeasureTheory.condexp_ae_eq_restrict_zero
theorem condexp_ae_eq_restrict_zero (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict s] 0) :
    μ[f|m] =ᵐ[μ.restrict s] 0 
Arguments: {α : Type u_1} {E : Type u_3} {m m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α} (hs : MeasurableSet s) (hf : f =ᵐ[μ.restrict s] 0)
Goal: μ[f|m] =ᵐ[μ.restrict s] 0
-- MeasureTheory.condexp_indicator_aux
theorem condexp_indicator_aux (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict sᶜ] 0) :
    μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m]) 
Arguments: {α : Type u_1} {E : Type u_3} {m m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α} (hs : MeasurableSet s) (hf : f =ᵐ[μ.restrict sᶜ] 0)
Goal: μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m])
-- MeasureTheory.condexp_indicator
theorem condexp_indicator (hf_int : Integrable f μ) (hs : MeasurableSet[m] s) :
    μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m]) 
Arguments: {α : Type u_1} {E : Type u_3} {m m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α} (hf_int : MeasureTheory.Integrable f μ) (hs : MeasurableSet s)
Goal: μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m])
-- MeasureTheory.condexp_restrict_ae_eq_restrict
theorem condexp_restrict_ae_eq_restrict (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (hs_m : MeasurableSet[m] s) (hf_int : Integrable f μ) :
    (μ.restrict s)[f|m] =ᵐ[μ.restrict s] μ[f|m] 
Arguments: {α : Type u_1} {E : Type u_3} {m m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hs_m : MeasurableSet s) (hf_int : MeasureTheory.Integrable f μ)
Goal: μ.restrict s[f|m] =ᵐ[μ.restrict s] μ[f|m]
-- MeasureTheory.condexp_ae_eq_restrict_of_measurableSpace_eq_on
theorem condexp_ae_eq_restrict_of_measurableSpace_eq_on {m m₂ m0 : MeasurableSpace α}
    {μ : Measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm)]
    [SigmaFinite (μ.trim hm₂)] (hs_m : MeasurableSet[m] s)
    (hs : ∀ t, MeasurableSet[m] (s ∩ t) ↔ MeasurableSet[m₂] (s ∩ t)) :
    μ[f|m] =ᵐ[μ.restrict s] μ[f|m₂] 
Arguments: {α : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} {s : Set α} {m m₂ m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] [MeasureTheory.SigmaFinite (μ.trim hm₂)] (hs_m : MeasurableSet s) (hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t))
Goal: μ[f|m] =ᵐ[μ.restrict s] μ[f|m₂]
-- VitaliFamily.ae_eventually_measure_pos
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α]
Goal: ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
-- VitaliFamily.eventually_measure_lt_top
theorem eventually_measure_lt_top [IsLocallyFiniteMeasure μ] (x : α) :
    ∀ᶠ a in v.filterAt x, μ a < ∞ 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [MeasureTheory.IsLocallyFiniteMeasure μ] (x : α)
Goal: ∀ᶠ (a : Set α) in v.filterAt x, μ a < ⊤
-- VitaliFamily.measure_le_of_frequently_le
theorem measure_le_of_frequently_le [SecondCountableTopology α] [BorelSpace α] {ρ : Measure α}
    (ν : Measure α) [IsLocallyFiniteMeasure ν] (hρ : ρ ≪ μ) (s : Set α)
    (hs : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, ρ a ≤ ν a) : ρ s ≤ ν s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] {ρ : MeasureTheory.Measure α} (ν : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure ν] (hρ : ρ ≪ μ) (s : Set α) (hs : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ν a)
Goal: ρ s ≤ ν s
-- VitaliFamily.ae_eventually_measure_zero_of_singular
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ⟂ₘ μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0)
-- VitaliFamily.null_of_frequently_le_of_frequently_ge
theorem null_of_frequently_le_of_frequently_ge {c d : ℝ≥0} (hcd : c < d) (s : Set α)
    (hc : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, ρ a ≤ c * μ a)
    (hd : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, (d : ℝ≥0∞) * μ a ≤ ρ a) : μ s = 0 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ) {c d : ℝ≥0} (hcd : c < d) (s : Set α) (hc : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ a) (hd : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a)
Goal: μ s = 0
-- VitaliFamily.ae_tendsto_div
theorem ae_tendsto_div : ∀ᵐ x ∂μ, ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: ∀ᵐ (x : α) ∂μ, ∃ c, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)
-- VitaliFamily.ae_tendsto_limRatio
theorem ae_tendsto_limRatio :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x))
-- VitaliFamily.exists_measurable_supersets_limRatio
theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
    ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
      {x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ) {p q : ℝ≥0} (hpq : p < q)
Goal: ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x | v.limRatio ρ x < ↑p} ⊆ a ∧ {x | ↑q < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0
-- VitaliFamily.aemeasurable_limRatio
theorem aemeasurable_limRatio : AEMeasurable (v.limRatio ρ) μ 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: AEMeasurable (v.limRatio ρ) μ
-- VitaliFamily.limRatioMeas_measurable
theorem limRatioMeas_measurable : Measurable (v.limRatioMeas hρ) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: Measurable (v.limRatioMeas hρ)
-- VitaliFamily.ae_tendsto_limRatioMeas
theorem ae_tendsto_limRatioMeas :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))
-- VitaliFamily.measure_le_mul_of_subset_limRatioMeas_lt
theorem measure_le_mul_of_subset_limRatioMeas_lt {p : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | v.limRatioMeas hρ x < p}) : ρ s ≤ p * μ s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ) {p : ℝ≥0} {s : Set α} (h : s ⊆ {x | v.limRatioMeas hρ x < ↑p})
Goal: ρ s ≤ ↑p * μ s
-- VitaliFamily.mul_measure_le_of_subset_lt_limRatioMeas
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ) {q : ℝ≥0} {s : Set α} (h : s ⊆ {x | ↑q < v.limRatioMeas hρ x})
Goal: ↑q * μ s ≤ ρ s
-- VitaliFamily.measure_limRatioMeas_top
theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: μ {x | v.limRatioMeas hρ x = ⊤} = 0
-- VitaliFamily.measure_limRatioMeas_zero
theorem measure_limRatioMeas_zero : ρ {x | v.limRatioMeas hρ x = 0} = 0 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: ρ {x | v.limRatioMeas hρ x = 0} = 0
-- VitaliFamily.withDensity_le_mul
theorem withDensity_le_mul {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    μ.withDensity (v.limRatioMeas hρ) s ≤ (t : ℝ≥0∞) ^ 2 * ρ s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ) {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t)
Goal: (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑t ^ 2 * ρ s
-- VitaliFamily.le_mul_withDensity
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ) {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t)
Goal: ρ s ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s
-- VitaliFamily.withDensity_limRatioMeas_eq
theorem withDensity_limRatioMeas_eq : μ.withDensity (v.limRatioMeas hρ) = ρ 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: μ.withDensity (v.limRatioMeas hρ) = ρ
-- VitaliFamily.ae_tendsto_rnDeriv_of_absolutelyContinuous
theorem ae_tendsto_rnDeriv_of_absolutelyContinuous :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ ≪ μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))
-- VitaliFamily.ae_tendsto_rnDeriv
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] (ρ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure ρ]
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))
-- VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet
theorem ae_tendsto_measure_inter_div_of_measurableSet {s : Set α} (hs : MeasurableSet s) :
    ∀ᵐ x ∂μ, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {s : Set α} (hs : MeasurableSet s)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x))
-- VitaliFamily.ae_tendsto_measure_inter_div
theorem ae_tendsto_measure_inter_div (s : Set α) :
    ∀ᵐ x ∂μ.restrict s, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 1) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] (s : Set α)
Goal: ∀ᵐ (x : α) ∂μ.restrict s, Filter.Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 1)
-- VitaliFamily.ae_tendsto_lintegral_div'
theorem ae_tendsto_lintegral_div' {f : α → ℝ≥0∞} (hf : Measurable f) (h'f : (∫⁻ y, f y ∂μ) ≠ ∞) :
    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → ℝ≥0∞} (hf : Measurable f) (h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x))
-- VitaliFamily.ae_tendsto_lintegral_div
theorem ae_tendsto_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h'f : (∫⁻ y, f y ∂μ) ≠ ∞) :
    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x))
-- VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable
theorem ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable {f : α → E} (hf : Integrable f μ)
    (h'f : StronglyMeasurable f) :
    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → E} (hf : MeasureTheory.Integrable f μ) (h'f : MeasureTheory.StronglyMeasurable f)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
-- VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div_of_integrable
theorem ae_tendsto_lintegral_nnnorm_sub_div_of_integrable {f : α → E} (hf : Integrable f μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → E} (hf : MeasureTheory.Integrable f μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)