ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.measure_inter_lt_top_of_right_ne_top
theorem measure_inter_lt_top_of_right_ne_top (ht_finite : μ t ≠ ∞) : μ (s ∩ t) < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} (ht_finite : μ t ≠ ⊤)
Goal: μ (s ∩ t) < ⊤
-- MeasureTheory.measure_inter_null_of_null_right
theorem measure_inter_null_of_null_right (S : Set α) {T : Set α} (h : μ T = 0) : μ (S ∩ T) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (S : Set α) {T : Set α} (h : μ T = 0)
Goal: μ (S ∩ T) = 0
-- MeasureTheory.measure_inter_null_of_null_left
theorem measure_inter_null_of_null_left {S : Set α} (T : Set α) (h : μ S = 0) : μ (S ∩ T) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {S : Set α} (T : Set α) (h : μ S = 0)
Goal: μ (S ∩ T) = 0
-- MeasurableSpace.ae_induction_on_inter
theorem _root_.MeasurableSpace.ae_induction_on_inter {β} [MeasurableSpace β] {μ : Measure β}
    {C : β → Set α → Prop} {s : Set (Set α)} [m : MeasurableSpace α]
    (h_eq : m = MeasurableSpace.generateFrom s)
    (h_inter : IsPiSystem s) (h_empty : ∀ᵐ x ∂μ, C x ∅) (h_basic : ∀ᵐ x ∂μ, ∀ t ∈ s, C x t)
    (h_compl : ∀ᵐ x ∂μ, ∀ t, MeasurableSet t → C x t → C x tᶜ)
    (h_union : ∀ᵐ x ∂μ, ∀ f : ℕ → Set α,
        Pairwise (Disjoint on f) → (∀ i, MeasurableSet (f i)) → (∀ i, C x (f i)) → C x (⋃ i, f i)) :
    ∀ᵐ x ∂μ, ∀ ⦃t⦄, MeasurableSet t → C x t 
Arguments: {α : Type u_1} {β : Type u_6} [MeasurableSpace β] {μ : MeasureTheory.Measure β} {C : β → Set α → Prop} {s : Set (Set α)} [m : MeasurableSpace α] (h_eq : m = MeasurableSpace.generateFrom s) (h_inter : IsPiSystem s) (h_empty : ∀ᵐ (x : β) ∂μ, C x ∅) (h_basic : ∀ᵐ (x : β) ∂μ, ∀ t ∈ s, C x t) (h_compl : ∀ᵐ (x : β) ∂μ, ∀ (t : Set α), MeasurableSet t → C x t → C x tᶜ) (h_union :
  ∀ᵐ (x : β) ∂μ,
    ∀ (f : ℕ → Set α),
      Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C x (f i)) → C x (⋃ i, f i))
Goal: ∀ᵐ (x : β) ∂μ, ∀ ⦃t : Set α⦄, MeasurableSet t → C x t
-- MeasureTheory.toMeasurable_def
irreducible_def toMeasurable (μ : Measure α) (s : Set α) : Set α 
Arguments: {α : Type u_6} [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: MeasureTheory.toMeasurable μ s =
  if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose
  else
    if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose
    else ⋯.choose
-- MeasureTheory.subset_toMeasurable
theorem subset_toMeasurable (μ : Measure α) (s : Set α) : s ⊆ toMeasurable μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: s ⊆ MeasureTheory.toMeasurable μ s
-- MeasureTheory.ae_le_toMeasurable
theorem ae_le_toMeasurable : s ≤ᵐ[μ] toMeasurable μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α}
Goal: s ≤ᵐ[μ] MeasureTheory.toMeasurable μ s
-- MeasureTheory.measurableSet_toMeasurable
@[simp]
theorem measurableSet_toMeasurable (μ : Measure α) (s : Set α) :
    MeasurableSet (toMeasurable μ s) 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: MeasurableSet (MeasureTheory.toMeasurable μ s)
-- MeasureTheory.measure_toMeasurable
@[simp]
theorem measure_toMeasurable (s : Set α) : μ (toMeasurable μ s) = μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (s : Set α)
Goal: μ (MeasureTheory.toMeasurable μ s) = μ s
-- Measurable.aemeasurable
@[fun_prop, aesop unsafe 30% apply (rule_sets := [Measurable])]
theorem Measurable.aemeasurable (h : Measurable f) : AEMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : Measurable f)
Goal: AEMeasurable f μ
-- AEMeasurable.measurable_mk
@[measurability]
theorem measurable_mk (h : AEMeasurable f μ) : Measurable (h.mk f) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : AEMeasurable f μ)
Goal: Measurable (AEMeasurable.mk f h)
-- AEMeasurable.ae_eq_mk
theorem ae_eq_mk (h : AEMeasurable f μ) : f =ᵐ[μ] h.mk f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : AEMeasurable f μ)
Goal: f =ᵐ[μ] AEMeasurable.mk f h
-- AEMeasurable.congr
theorem congr (hf : AEMeasurable f μ) (h : f =ᵐ[μ] g) : AEMeasurable g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {f g : α → β} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ) (h : f =ᵐ[μ] g)
Goal: AEMeasurable g μ
-- aemeasurable_congr
theorem aemeasurable_congr (h : f =ᵐ[μ] g) : AEMeasurable f μ ↔ AEMeasurable g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {f g : α → β} {μ : MeasureTheory.Measure α} (h : f =ᵐ[μ] g)
Goal: AEMeasurable f μ ↔ AEMeasurable g μ
-- aemeasurable_const
@[simp, fun_prop, measurability]
theorem aemeasurable_const {b : β} : AEMeasurable (fun _a : α => b) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {b : β}
Goal: AEMeasurable (fun _a => b) μ
-- aemeasurable_id
@[measurability]
theorem aemeasurable_id : AEMeasurable id μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: AEMeasurable id μ
-- aemeasurable_id'
@[measurability]
theorem aemeasurable_id' : AEMeasurable (fun x => x) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: AEMeasurable (fun x => x) μ
-- Measurable.comp_aemeasurable
theorem Measurable.comp_aemeasurable [MeasurableSpace δ] {f : α → δ} {g : δ → β} (hg : Measurable g)
    (hf : AEMeasurable f μ) : AEMeasurable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasurableSpace δ] {f : α → δ} {g : δ → β} (hg : Measurable g) (hf : AEMeasurable f μ)
Goal: AEMeasurable (g ∘ f) μ
-- Measurable.comp_aemeasurable'
@[fun_prop, measurability]
theorem Measurable.comp_aemeasurable' [MeasurableSpace δ] {f : α → δ} {g : δ → β}
    (hg : Measurable g) (hf : AEMeasurable f μ) : AEMeasurable (fun x ↦ g (f x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasurableSpace δ] {f : α → δ} {g : δ → β} (hg : Measurable g) (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => g (f x)) μ
-- MeasureTheory.Measure.IsOpenPosMeasure.open_pos
class IsOpenPosMeasure : Prop where
  open_pos : ∀ U : Set X, IsOpen U → U.Nonempty → μ U ≠ 0
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [self : μ.IsOpenPosMeasure] (U : Set X)
Goal: IsOpen U → U.Nonempty → μ U ≠ 0
-- IsOpen.measure_ne_zero
theorem _root_.IsOpen.measure_ne_zero (hU : IsOpen U) (hne : U.Nonempty) : μ U ≠ 0 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {U : Set X} (hU : IsOpen U) (hne : U.Nonempty)
Goal: μ U ≠ 0
-- IsOpen.measure_pos
theorem _root_.IsOpen.measure_pos (hU : IsOpen U) (hne : U.Nonempty) : 0 < μ U 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {U : Set X} (hU : IsOpen U) (hne : U.Nonempty)
Goal: 0 < μ U
-- IsOpen.measure_pos_iff
theorem _root_.IsOpen.measure_pos_iff (hU : IsOpen U) : 0 < μ U ↔ U.Nonempty 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {U : Set X} (hU : IsOpen U)
Goal: 0 < μ U ↔ U.Nonempty
-- IsOpen.measure_eq_zero_iff
theorem _root_.IsOpen.measure_eq_zero_iff (hU : IsOpen U) : μ U = 0 ↔ U = ∅ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {U : Set X} (hU : IsOpen U)
Goal: μ U = 0 ↔ U = ∅
-- MeasureTheory.Measure.measure_pos_of_nonempty_interior
theorem measure_pos_of_nonempty_interior (h : (interior s).Nonempty) : 0 < μ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {s : Set X} (h : (interior s).Nonempty)
Goal: 0 < μ s
-- MeasureTheory.Measure.measure_pos_of_mem_nhds
theorem measure_pos_of_mem_nhds (h : s ∈ 𝓝 x) : 0 < μ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {s : Set X} {x : X} (h : s ∈ 𝓝 x)
Goal: 0 < μ s
-- MeasureTheory.Measure.isOpenPosMeasure_smul
theorem isOpenPosMeasure_smul {c : ℝ≥0∞} (h : c ≠ 0) : IsOpenPosMeasure (c • μ) 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {c : ℝ≥0∞} (h : c ≠ 0)
Goal: (c • μ).IsOpenPosMeasure
-- MeasureTheory.Measure.AbsolutelyContinuous.isOpenPosMeasure
protected theorem AbsolutelyContinuous.isOpenPosMeasure (h : μ ≪ ν) : IsOpenPosMeasure ν 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ ν : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] (h : μ ≪ ν)
Goal: ν.IsOpenPosMeasure
-- LE.le.isOpenPosMeasure
theorem _root_.LE.le.isOpenPosMeasure (h : μ ≤ ν) : IsOpenPosMeasure ν 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ ν : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] (h : μ ≤ ν)
Goal: ν.IsOpenPosMeasure
-- IsOpen.measure_zero_iff_eq_empty
theorem _root_.IsOpen.measure_zero_iff_eq_empty (hU : IsOpen U) :
    μ U = 0 ↔ U = ∅ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {U : Set X} (hU : IsOpen U)
Goal: μ U = 0 ↔ U = ∅
-- IsOpen.ae_eq_empty_iff_eq
theorem _root_.IsOpen.ae_eq_empty_iff_eq (hU : IsOpen U) :
    U =ᵐ[μ] (∅ : Set X) ↔ U = ∅ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {U : Set X} (hU : IsOpen U)
Goal: U =ᵐ[μ] ∅ ↔ U = ∅
-- IsOpen.eq_empty_of_measure_zero
theorem _root_.IsOpen.eq_empty_of_measure_zero (hU : IsOpen U) (h₀ : μ U = 0) : U = ∅ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {U : Set X} (hU : IsOpen U) (h₀ : μ U = 0)
Goal: U = ∅
-- IsClosed.ae_eq_univ_iff_eq
theorem _root_.IsClosed.ae_eq_univ_iff_eq (hF : IsClosed F) :
    F =ᵐ[μ] univ ↔ F = univ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {F : Set X} (hF : IsClosed F)
Goal: F =ᵐ[μ] Set.univ ↔ F = Set.univ
-- IsClosed.measure_eq_univ_iff_eq
theorem _root_.IsClosed.measure_eq_univ_iff_eq [OpensMeasurableSpace X] [IsFiniteMeasure μ]
    (hF : IsClosed F) :
    μ F = μ univ ↔ F = univ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {F : Set X} [OpensMeasurableSpace X] [MeasureTheory.IsFiniteMeasure μ] (hF : IsClosed F)
Goal: μ F = μ Set.univ ↔ F = Set.univ
-- IsClosed.measure_eq_one_iff_eq_univ
theorem _root_.IsClosed.measure_eq_one_iff_eq_univ [OpensMeasurableSpace X] [IsProbabilityMeasure μ]
    (hF : IsClosed F) :
    μ F = 1 ↔ F = univ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {F : Set X} [OpensMeasurableSpace X] [MeasureTheory.IsProbabilityMeasure μ] (hF : IsClosed F)
Goal: μ F = 1 ↔ F = Set.univ
-- MeasureTheory.Measure.interior_eq_empty_of_null
theorem interior_eq_empty_of_null (hs : μ s = 0) : interior s = ∅ 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {s : Set X} (hs : μ s = 0)
Goal: interior s = ∅
-- MeasureTheory.Measure.eqOn_open_of_ae_eq
theorem eqOn_open_of_ae_eq {f g : X → Y} (h : f =ᵐ[μ.restrict U] g) (hU : IsOpen U)
    (hf : ContinuousOn f U) (hg : ContinuousOn g U) : EqOn f g U 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {U : Set X} {f g : X → Y} (h : f =ᵐ[μ.restrict U] g) (hU : IsOpen U) (hf : ContinuousOn f U) (hg : ContinuousOn g U)
Goal: Set.EqOn f g U
-- MeasureTheory.Measure.eq_of_ae_eq
theorem eq_of_ae_eq {f g : X → Y} (h : f =ᵐ[μ] g) (hf : Continuous f) (hg : Continuous g) : f = g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {f g : X → Y} (h : f =ᵐ[μ] g) (hf : Continuous f) (hg : Continuous g)
Goal: f = g
-- MeasureTheory.Measure.eqOn_of_ae_eq
theorem eqOn_of_ae_eq {f g : X → Y} (h : f =ᵐ[μ.restrict s] g) (hf : ContinuousOn f s)
    (hg : ContinuousOn g s) (hU : s ⊆ closure (interior s)) : EqOn f g s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {s : Set X} {f g : X → Y} (h : f =ᵐ[μ.restrict s] g) (hf : ContinuousOn f s) (hg : ContinuousOn g s) (hU : s ⊆ closure (interior s))
Goal: Set.EqOn f g s
-- Continuous.ae_eq_iff_eq
theorem _root_.Continuous.ae_eq_iff_eq {f g : X → Y} (hf : Continuous f) (hg : Continuous g) :
    f =ᵐ[μ] g ↔ f = g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {f g : X → Y} (hf : Continuous f) (hg : Continuous g)
Goal: f =ᵐ[μ] g ↔ f = g
-- Continuous.isOpenPosMeasure_map
theorem _root_.Continuous.isOpenPosMeasure_map [OpensMeasurableSpace X]
    {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z]
    {f : X → Z} (hf : Continuous f) (hf_surj : Function.Surjective f) :
    (Measure.map f μ).IsOpenPosMeasure 
Arguments: {X : Type u_1} [TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] [OpensMeasurableSpace X] {Z : Type u_3} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] {f : X → Z} (hf : Continuous f) (hf_surj : Function.Surjective f)
Goal: (MeasureTheory.Measure.map f μ).IsOpenPosMeasure
-- MeasureTheory.Measure.measure_Ioi_pos
theorem measure_Ioi_pos [NoMaxOrder X] (a : X) : 0 < μ (Ioi a) 
Arguments: {X : Type u_1} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [NoMaxOrder X] (a : X)
Goal: 0 < μ (Set.Ioi a)
-- MeasureTheory.Measure.measure_Iio_pos
theorem measure_Iio_pos [NoMinOrder X] (a : X) : 0 < μ (Iio a) 
Arguments: {X : Type u_1} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [NoMinOrder X] (a : X)
Goal: 0 < μ (Set.Iio a)
-- MeasureTheory.Measure.measure_Ioo_pos
theorem measure_Ioo_pos [DenselyOrdered X] {a b : X} : 0 < μ (Ioo a b) ↔ a < b 
Arguments: {X : Type u_1} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X}
Goal: 0 < μ (Set.Ioo a b) ↔ a < b
-- MeasureTheory.Measure.measure_Ioo_eq_zero
theorem measure_Ioo_eq_zero [DenselyOrdered X] {a b : X} : μ (Ioo a b) = 0 ↔ b ≤ a 
Arguments: {X : Type u_1} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X}
Goal: μ (Set.Ioo a b) = 0 ↔ b ≤ a
-- MeasureTheory.Measure.eqOn_Ioo_of_ae_eq
theorem eqOn_Ioo_of_ae_eq {a b : X} {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Ioo a b)] g)
    (hf : ContinuousOn f (Ioo a b)) (hg : ContinuousOn g (Ioo a b)) : EqOn f g (Ioo a b) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {a b : X} {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Ioo a b)] g) (hf : ContinuousOn f (Set.Ioo a b)) (hg : ContinuousOn g (Set.Ioo a b))
Goal: Set.EqOn f g (Set.Ioo a b)
-- MeasureTheory.Measure.eqOn_Ioc_of_ae_eq
theorem eqOn_Ioc_of_ae_eq [DenselyOrdered X] {a b : X} {f g : X → Y}
    (hfg : f =ᵐ[μ.restrict (Ioc a b)] g) (hf : ContinuousOn f (Ioc a b))
    (hg : ContinuousOn g (Ioc a b)) : EqOn f g (Ioc a b) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X} {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Ioc a b)] g) (hf : ContinuousOn f (Set.Ioc a b)) (hg : ContinuousOn g (Set.Ioc a b))
Goal: Set.EqOn f g (Set.Ioc a b)
-- MeasureTheory.Measure.eqOn_Ico_of_ae_eq
theorem eqOn_Ico_of_ae_eq [DenselyOrdered X] {a b : X} {f g : X → Y}
    (hfg : f =ᵐ[μ.restrict (Ico a b)] g) (hf : ContinuousOn f (Ico a b))
    (hg : ContinuousOn g (Ico a b)) : EqOn f g (Ico a b) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X} {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Ico a b)] g) (hf : ContinuousOn f (Set.Ico a b)) (hg : ContinuousOn g (Set.Ico a b))
Goal: Set.EqOn f g (Set.Ico a b)
-- MeasureTheory.Measure.eqOn_Icc_of_ae_eq
theorem eqOn_Icc_of_ae_eq [DenselyOrdered X] {a b : X} (hne : a ≠ b) {f g : X → Y}
    (hfg : f =ᵐ[μ.restrict (Icc a b)] g) (hf : ContinuousOn f (Icc a b))
    (hg : ContinuousOn g (Icc a b)) : EqOn f g (Icc a b) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X} (hne : a ≠ b) {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Icc a b)] g) (hf : ContinuousOn f (Set.Icc a b)) (hg : ContinuousOn g (Set.Icc a b))
Goal: Set.EqOn f g (Set.Icc a b)
-- Metric.measure_ball_pos
theorem measure_ball_pos (x : X) {r : ℝ} (hr : 0 < r) : 0 < μ (ball x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] (x : X) {r : ℝ} (hr : 0 < r)
Goal: 0 < μ (Metric.ball x r)
-- Metric.measure_closedBall_pos
theorem measure_closedBall_pos (x : X) {r : ℝ} (hr : 0 < r) : 0 < μ (closedBall x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] (x : X) {r : ℝ} (hr : 0 < r)
Goal: 0 < μ (Metric.closedBall x r)
-- Metric.measure_closedBall_pos_iff
@[simp] lemma measure_closedBall_pos_iff {X : Type*} [MetricSpace X] {m : MeasurableSpace X}
    (μ : Measure X) [IsOpenPosMeasure μ] [NoAtoms μ] {x : X} {r : ℝ} :
    0 < μ (closedBall x r) ↔ 0 < r 
Arguments: {X : Type u_2} [MetricSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [MeasureTheory.NoAtoms μ] {x : X} {r : ℝ}
Goal: 0 < μ (Metric.closedBall x r) ↔ 0 < r
-- EMetric.measure_ball_pos
theorem measure_ball_pos (x : X) {r : ℝ≥0∞} (hr : r ≠ 0) : 0 < μ (ball x r) 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] (x : X) {r : ℝ≥0∞} (hr : r ≠ 0)
Goal: 0 < μ (EMetric.ball x r)
-- EMetric.measure_closedBall_pos
theorem measure_closedBall_pos (x : X) {r : ℝ≥0∞} (hr : r ≠ 0) : 0 < μ (closedBall x r) 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] (x : X) {r : ℝ≥0∞} (hr : r ≠ 0)
Goal: 0 < μ (EMetric.closedBall x r)
-- IsNowhereDense.of_isClosed_null
lemma IsNowhereDense.of_isClosed_null (h₁s : IsClosed s) (h₂s : μ s = 0) :
    IsNowhereDense s 
Arguments: {X : Type u_1} [TopologicalSpace X] [MeasurableSpace X] {s : Set X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] (h₁s : IsClosed s) (h₂s : μ s = 0)
Goal: IsNowhereDense s
-- IsMeagre.of_isSigmaCompact_null
lemma IsMeagre.of_isSigmaCompact_null [T2Space X] (h₁s : IsSigmaCompact s) (h₂s : μ s = 0) :
    IsMeagre s 
Arguments: {X : Type u_1} [TopologicalSpace X] [MeasurableSpace X] {s : Set X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] [T2Space X] (h₁s : IsSigmaCompact s) (h₂s : μ s = 0)
Goal: IsMeagre s
-- CompletelyRegularSpace.exists_BCNN
lemma CompletelyRegularSpace.exists_BCNN {X : Type*} [TopologicalSpace X] [CompletelyRegularSpace X]
    {K : Set X} (K_closed : IsClosed K) {x : X} (x_notin_K : x ∉ K) :
    ∃ (f : X →ᵇ ℝ≥0), f x = 1 ∧ (∀ y ∈ K, f y = 0) 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompletelyRegularSpace X] {K : Set X} (K_closed : IsClosed K) {x : X} (x_notin_K : x ∉ K)
Goal: ∃ f, f x = 1 ∧ ∀ y ∈ K, f y = 0
-- MeasureTheory.injective_diracProba
lemma injective_diracProba {X : Type*} [MeasurableSpace X] [MeasurableSpace.SeparatesPoints X] :
    Function.Injective (fun (x : X) ↦ diracProba x) 
Arguments: {X : Type u_2} [MeasurableSpace X] [MeasurableSpace.SeparatesPoints X]
Goal: Function.Injective fun x => MeasureTheory.diracProba x
-- MeasureTheory.diracProba_toMeasure_apply'
@[simp] lemma diracProba_toMeasure_apply' (x : X) {A : Set X} (A_mble : MeasurableSet A) :
    (diracProba x).toMeasure A = A.indicator 1 x 
Arguments: {X : Type u_1} [MeasurableSpace X] (x : X) {A : Set X} (A_mble : MeasurableSet A)
Goal: ↑(MeasureTheory.diracProba x) A = A.indicator 1 x
-- MeasureTheory.diracProba_toMeasure_apply_of_mem
@[simp] lemma diracProba_toMeasure_apply_of_mem {x : X} {A : Set X} (x_in_A : x ∈ A) :
    (diracProba x).toMeasure A = 1 
Arguments: {X : Type u_1} [MeasurableSpace X] {x : X} {A : Set X} (x_in_A : x ∈ A)
Goal: ↑(MeasureTheory.diracProba x) A = 1
-- MeasureTheory.diracProba_toMeasure_apply
@[simp] lemma diracProba_toMeasure_apply [MeasurableSingletonClass X] (x : X) (A : Set X) :
    (diracProba x).toMeasure A = A.indicator 1 x 
Arguments: {X : Type u_1} [MeasurableSpace X] [MeasurableSingletonClass X] (x : X) (A : Set X)
Goal: ↑(MeasureTheory.diracProba x) A = A.indicator 1 x
-- MeasureTheory.continuous_diracProba
lemma continuous_diracProba : Continuous (fun (x : X) ↦ diracProba x) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X]
Goal: Continuous fun x => MeasureTheory.diracProba x
-- MeasureTheory.injective_diracProba_of_T0
lemma injective_diracProba_of_T0 [T0Space X] :
    Function.Injective (fun (x : X) ↦ diracProba x) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [T0Space X]
Goal: Function.Injective fun x => MeasureTheory.diracProba x
-- MeasureTheory.not_tendsto_diracProba_of_not_tendsto
lemma not_tendsto_diracProba_of_not_tendsto [CompletelyRegularSpace X] {x : X} (L : Filter X)
    (h : ¬ Tendsto id L (𝓝 x)) :
    ¬ Tendsto diracProba L (𝓝 (diracProba x)) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [CompletelyRegularSpace X] {x : X} (L : Filter X) (h : ¬Filter.Tendsto id L (𝓝 x))
Goal: ¬Filter.Tendsto MeasureTheory.diracProba L (𝓝 (MeasureTheory.diracProba x))
-- MeasureTheory.tendsto_diracProba_iff_tendsto
lemma tendsto_diracProba_iff_tendsto [CompletelyRegularSpace X] {x : X} (L : Filter X) :
    Tendsto diracProba L (𝓝 (diracProba x)) ↔ Tendsto id L (𝓝 x) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [CompletelyRegularSpace X] {x : X} (L : Filter X)
Goal: Filter.Tendsto MeasureTheory.diracProba L (𝓝 (MeasureTheory.diracProba x)) ↔ Filter.Tendsto id L (𝓝 x)
-- MeasureTheory.diracProba_diracProbaInverse
@[simp] lemma diracProba_diracProbaInverse (μ : range (diracProba (X := X))) :
    diracProba (diracProbaInverse μ) = μ 
Arguments: {X : Type u_1} [MeasurableSpace X] (μ : ↑(Set.range MeasureTheory.diracProba))
Goal: MeasureTheory.diracProba (MeasureTheory.diracProbaInverse μ) = ↑μ
-- MeasureTheory.diracProbaInverse_eq
lemma diracProbaInverse_eq [T0Space X] {x : X} {μ : range (diracProba (X := X))}
    (h : μ = diracProba x) :
    diracProbaInverse μ = x 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [T0Space X] {x : X} {μ : ↑(Set.range MeasureTheory.diracProba)} (h : ↑μ = MeasureTheory.diracProba x)
Goal: MeasureTheory.diracProbaInverse μ = x