ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IsSetRing.isSetSemiring
lemma isSetSemiring (hC : IsSetRing C) : IsSetSemiring C 
Arguments: {α : Type u_1} {C : Set (Set α)} (hC : MeasureTheory.IsSetRing C)
Goal: MeasureTheory.IsSetSemiring C
-- MeasureTheory.IsSetRing.biUnion_mem
lemma biUnion_mem {ι : Type*} (hC : IsSetRing C) {s : ι → Set α}
    (S : Finset ι) (hs : ∀ n ∈ S, s n ∈ C) :
    ⋃ i ∈ S, s i ∈ C 
Arguments: {α : Type u_1} {C : Set (Set α)} {ι : Type u_2} (hC : MeasureTheory.IsSetRing C) {s : ι → Set α} (S : Finset ι) (hs : ∀ n ∈ S, s n ∈ C)
Goal: ⋃ i ∈ S, s i ∈ C
-- MeasureTheory.IsSetRing.biInter_mem
lemma biInter_mem {ι : Type*} (hC : IsSetRing C) {s : ι → Set α}
    (S : Finset ι) (hS : S.Nonempty) (hs : ∀ n ∈ S, s n ∈ C) :
    ⋂ i ∈ S, s i ∈ C 
Arguments: {α : Type u_1} {C : Set (Set α)} {ι : Type u_2} (hC : MeasureTheory.IsSetRing C) {s : ι → Set α} (S : Finset ι) (hS : S.Nonempty) (hs : ∀ n ∈ S, s n ∈ C)
Goal: ⋂ i ∈ S, s i ∈ C
-- BoundedContinuousFunction.apply_le_nndist_zero
lemma apply_le_nndist_zero {X : Type*} [TopologicalSpace X] (f : X →ᵇ ℝ≥0) (x : X) :
    f x ≤ nndist 0 f 
Arguments: {X : Type u_1} [TopologicalSpace X] (f : X →ᵇ ℝ≥0) (x : X)
Goal: f x ≤ nndist 0 f
-- BoundedContinuousFunction.lintegral_le_edist_mul
lemma lintegral_le_edist_mul (f : X →ᵇ ℝ≥0) (μ : Measure X) :
    (∫⁻ x, f x ∂μ) ≤ edist 0 f * (μ Set.univ) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] (f : X →ᵇ ℝ≥0) (μ : MeasureTheory.Measure X)
Goal: ∫⁻ (x : X), ↑(f x) ∂μ ≤ edist 0 f * μ Set.univ
-- BoundedContinuousFunction.measurable_coe_ennreal_comp
theorem measurable_coe_ennreal_comp (f : X →ᵇ ℝ≥0) :
    Measurable fun x ↦ (f x : ℝ≥0∞) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] (f : X →ᵇ ℝ≥0)
Goal: Measurable fun x => ↑(f x)
-- BoundedContinuousFunction.lintegral_lt_top_of_nnreal
theorem lintegral_lt_top_of_nnreal (f : X →ᵇ ℝ≥0) : ∫⁻ x, f x ∂μ < ∞ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ ℝ≥0)
Goal: ∫⁻ (x : X), ↑(f x) ∂μ < ⊤
-- BoundedContinuousFunction.integrable_of_nnreal
theorem integrable_of_nnreal (f : X →ᵇ ℝ≥0) : Integrable (((↑) : ℝ≥0 → ℝ) ∘ ⇑f) μ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ ℝ≥0)
Goal: MeasureTheory.Integrable (NNReal.toReal ∘ ⇑f) μ
-- BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub
theorem integral_eq_integral_nnrealPart_sub (f : X →ᵇ ℝ) :
    ∫ x, f x ∂μ = (∫ x, (f.nnrealPart x : ℝ) ∂μ) - ∫ x, ((-f).nnrealPart x : ℝ) ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ ℝ)
Goal: ∫ (x : X), f x ∂μ = ∫ (x : X), ↑(f.nnrealPart x) ∂μ - ∫ (x : X), ↑((-f).nnrealPart x) ∂μ
-- BoundedContinuousFunction.lintegral_of_real_lt_top
theorem lintegral_of_real_lt_top (f : X →ᵇ ℝ) :
    ∫⁻ x, ENNReal.ofReal (f x) ∂μ < ∞ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ ℝ)
Goal: ∫⁻ (x : X), ENNReal.ofReal (f x) ∂μ < ⊤
-- BoundedContinuousFunction.toReal_lintegral_coe_eq_integral
theorem toReal_lintegral_coe_eq_integral (f : X →ᵇ ℝ≥0) (μ : Measure X) :
    (∫⁻ x, (f x : ℝ≥0∞) ∂μ).toReal = ∫ x, (f x : ℝ) ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] (f : X →ᵇ ℝ≥0) (μ : MeasureTheory.Measure X)
Goal: (∫⁻ (x : X), ↑(f x) ∂μ).toReal = ∫ (x : X), ↑(f x) ∂μ
-- BoundedContinuousFunction.lintegral_nnnorm_le
lemma lintegral_nnnorm_le (f : X →ᵇ E) :
    ∫⁻ x, ‖f x‖₊ ∂μ ≤ ‖f‖₊ * (μ Set.univ) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] (μ : MeasureTheory.Measure X) {E : Type u_2} [NormedAddCommGroup E] (f : X →ᵇ E)
Goal: ∫⁻ (x : X), ↑‖f x‖₊ ∂μ ≤ ↑‖f‖₊ * μ Set.univ
-- BoundedContinuousFunction.integrable
lemma integrable [IsFiniteMeasure μ] (f : X →ᵇ E) :
    Integrable f μ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] (μ : MeasureTheory.Measure X) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ E)
Goal: MeasureTheory.Integrable (⇑f) μ
-- BoundedContinuousFunction.norm_integral_le_mul_norm
lemma norm_integral_le_mul_norm [IsFiniteMeasure μ] (f : X →ᵇ E) :
    ‖∫ x, f x ∂μ‖ ≤ ENNReal.toReal (μ Set.univ) * ‖f‖ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] (μ : MeasureTheory.Measure X) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] [NormedSpace ℝ E] [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ E)
Goal: ‖∫ (x : X), f x ∂μ‖ ≤ (μ Set.univ).toReal * ‖f‖
-- BoundedContinuousFunction.norm_integral_le_norm
lemma norm_integral_le_norm [IsProbabilityMeasure μ] (f : X →ᵇ E) :
    ‖∫ x, f x ∂μ‖ ≤ ‖f‖ 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] (μ : MeasureTheory.Measure X) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] [NormedSpace ℝ E] [MeasureTheory.IsProbabilityMeasure μ] (f : X →ᵇ E)
Goal: ‖∫ (x : X), f x ∂μ‖ ≤ ‖f‖
-- BoundedContinuousFunction.isBounded_range_integral
lemma isBounded_range_integral
    {ι : Type*} (μs : ι → Measure X) [∀ i, IsProbabilityMeasure (μs i)] (f : X →ᵇ E) :
    Bornology.IsBounded (Set.range (fun i ↦ ∫ x, f x ∂ (μs i))) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] [NormedSpace ℝ E] {ι : Type u_3} (μs : ι → MeasureTheory.Measure X) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] (f : X →ᵇ E)
Goal: Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)
-- BoundedContinuousFunction.integral_add_const
lemma integral_add_const (f : X →ᵇ ℝ) (c : ℝ) :
    ∫ x, (f + const X c) x ∂μ = ∫ x, f x ∂μ + ENNReal.toReal (μ (Set.univ)) • c 
Arguments: {X : Type u_1} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ ℝ) (c : ℝ)
Goal: ∫ (x : X), (f + BoundedContinuousFunction.const X c) x ∂μ = ∫ (x : X), f x ∂μ + (μ Set.univ).toReal • c
-- BoundedContinuousFunction.integral_const_sub
lemma integral_const_sub (f : X →ᵇ ℝ) (c : ℝ) :
    ∫ x, (const X c - f) x ∂μ = ENNReal.toReal (μ (Set.univ)) • c - ∫ x, f x ∂μ 
Arguments: {X : Type u_1} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] (f : X →ᵇ ℝ) (c : ℝ)
Goal: ∫ (x : X), (BoundedContinuousFunction.const X c - f) x ∂μ = (μ Set.univ).toReal • c - ∫ (x : X), f x ∂μ
-- BoundedContinuousFunction.tendsto_integral_of_forall_limsup_integral_le_integral
lemma tendsto_integral_of_forall_limsup_integral_le_integral {ι : Type*} {L : Filter ι}
    {μ : Measure X} [IsProbabilityMeasure μ] {μs : ι → Measure X} [∀ i, IsProbabilityMeasure (μs i)]
    (h : ∀ f : X →ᵇ ℝ, 0 ≤ f → L.limsup (fun i ↦ ∫ x, f x ∂ (μs i)) ≤ ∫ x, f x ∂μ)
    (f : X →ᵇ ℝ) :
    Tendsto (fun i ↦ ∫ x, f x ∂ (μs i)) L (𝓝 (∫ x, f x ∂μ)) 
Arguments: {X : Type u_1} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {μs : ι → MeasureTheory.Measure X} [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] (h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → Filter.limsup (fun i => ∫ (x : X), f x ∂μs i) L ≤ ∫ (x : X), f x ∂μ) (f : X →ᵇ ℝ)
Goal: Filter.Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))
-- BoundedContinuousFunction.tendsto_integral_of_forall_integral_le_liminf_integral
lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
    {μ : Measure X} [IsProbabilityMeasure μ] {μs : ι → Measure X} [∀ i, IsProbabilityMeasure (μs i)]
    (h : ∀ f : X →ᵇ ℝ, 0 ≤ f → ∫ x, f x ∂μ ≤ L.liminf (fun i ↦ ∫ x, f x ∂ (μs i)))
    (f : X →ᵇ ℝ) :
    Tendsto (fun i ↦ ∫ x, f x ∂ (μs i)) L (𝓝 (∫ x, f x ∂μ)) 
Arguments: {X : Type u_1} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] {ι : Type u_2} {L : Filter ι} {μ : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {μs : ι → MeasureTheory.Measure X} [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] (h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ Filter.liminf (fun i => ∫ (x : X), f x ∂μs i) L) (f : X →ᵇ ℝ)
Goal: Filter.Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))
-- ContinuousLinearMap.measurable
@[fun_prop, measurability]
protected theorem measurable (L : E →L[𝕜] F) : Measurable L 
Arguments: {𝕜 : Type u_2} [NormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [MeasurableSpace F] [BorelSpace F] (L : E →L[𝕜] F)
Goal: Measurable ⇑L
-- ContinuousLinearMap.measurable_comp
@[fun_prop]
theorem measurable_comp (L : E →L[𝕜] F) {φ : α → E} (φ_meas : Measurable φ) :
    Measurable fun a : α => L (φ a) 
Arguments: {α : Type u_1} [MeasurableSpace α] {𝕜 : Type u_2} [NormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [MeasurableSpace F] [BorelSpace F] (L : E →L[𝕜] F) {φ : α → E} (φ_meas : Measurable φ)
Goal: Measurable fun a => L (φ a)
-- ContinuousLinearMap.measurable_apply
@[fun_prop, measurability]
theorem measurable_apply [MeasurableSpace F] [BorelSpace F] (x : E) :
    Measurable fun f : E →L[𝕜] F => f x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [MeasurableSpace F] [BorelSpace F] (x : E)
Goal: Measurable fun f => f x
-- ContinuousLinearMap.measurable_apply'
@[measurability]
theorem measurable_apply' [MeasurableSpace E] [OpensMeasurableSpace E] [MeasurableSpace F]
    [BorelSpace F] : Measurable fun (x : E) (f : E →L[𝕜] F) => f x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [MeasurableSpace E] [OpensMeasurableSpace E] [MeasurableSpace F] [BorelSpace F]
Goal: Measurable fun x f => f x
-- ContinuousLinearMap.measurable_coe
@[measurability]
theorem measurable_coe [MeasurableSpace F] [BorelSpace F] :
    Measurable fun (f : E →L[𝕜] F) (x : E) => f x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [MeasurableSpace F] [BorelSpace F]
Goal: Measurable fun f x => f x
-- Measurable.apply_continuousLinearMap
@[fun_prop, measurability]
theorem Measurable.apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : Measurable φ) (v : F) :
    Measurable fun a => φ a v 
Arguments: {α : Type u_1} [MeasurableSpace α] {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ : α → F →L[𝕜] E} (hφ : Measurable φ) (v : F)
Goal: Measurable fun a => (φ a) v
-- AEMeasurable.apply_continuousLinearMap
@[measurability]
theorem AEMeasurable.apply_continuousLinearMap {φ : α → F →L[𝕜] E} {μ : Measure α}
    (hφ : AEMeasurable φ μ) (v : F) : AEMeasurable (fun a => φ a v) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ : α → F →L[𝕜] E} {μ : MeasureTheory.Measure α} (hφ : AEMeasurable φ μ) (v : F)
Goal: AEMeasurable (fun a => (φ a) v) μ
-- measurable_smul_const
theorem measurable_smul_const {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
    (Measurable fun x => f x • c) ↔ Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [MeasurableSpace 𝕜] [BorelSpace 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {f : α → 𝕜} {c : E} (hc : c ≠ 0)
Goal: (Measurable fun x => f x • c) ↔ Measurable f
-- aemeasurable_smul_const
theorem aemeasurable_smul_const {f : α → 𝕜} {μ : Measure α} {c : E} (hc : c ≠ 0) :
    AEMeasurable (fun x => f x • c) μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [MeasurableSpace 𝕜] [BorelSpace 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {f : α → 𝕜} {μ : MeasureTheory.Measure α} {c : E} (hc : c ≠ 0)
Goal: AEMeasurable (fun x => f x • c) μ ↔ AEMeasurable f μ
-- _private.Mathlib.MeasureTheory.Order.UpperLower.0.aux₀
private lemma aux₀
    (h : ∀ δ, 0 < δ →
      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s) :
    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)
        (𝓝 0) 
Arguments: {ι : Type u_1} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ} (h :
  ∀ (δ : ℝ),
    0 < δ → ∃ y, Metric.closedBall y (δ / 4) ⊆ Metric.closedBall x δ ∧ Metric.closedBall y (δ / 4) ⊆ interior s)
Goal: ¬Filter.Tendsto
    (fun r => MeasureTheory.volume (closure s ∩ Metric.closedBall x r) / MeasureTheory.volume (Metric.closedBall x r))
    (𝓝[>] 0) (𝓝 0)
-- _private.Mathlib.MeasureTheory.Order.UpperLower.0.aux₁
private lemma aux₁
    (h : ∀ δ, 0 < δ →
      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ) :
    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)
        (𝓝 1) 
Arguments: {ι : Type u_1} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ} (h :
  ∀ (δ : ℝ),
    0 < δ → ∃ y, Metric.closedBall y (δ / 4) ⊆ Metric.closedBall x δ ∧ Metric.closedBall y (δ / 4) ⊆ interior sᶜ)
Goal: ¬Filter.Tendsto
    (fun r => MeasureTheory.volume (closure s ∩ Metric.closedBall x r) / MeasureTheory.volume (Metric.closedBall x r))
    (𝓝[>] 0) (𝓝 1)
-- IsUpperSet.null_frontier
theorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0 
Arguments: {ι : Type u_1} [Fintype ι] {s : Set (ι → ℝ)} (hs : IsUpperSet s)
Goal: MeasureTheory.volume (frontier s) = 0
-- IsLowerSet.null_frontier
theorem IsLowerSet.null_frontier (hs : IsLowerSet s) : volume (frontier s) = 0 
Arguments: {ι : Type u_1} [Fintype ι] {s : Set (ι → ℝ)} (hs : IsLowerSet s)
Goal: MeasureTheory.volume (frontier s) = 0
-- Set.OrdConnected.null_frontier
theorem Set.OrdConnected.null_frontier (hs : s.OrdConnected) : volume (frontier s) = 0 
Arguments: {ι : Type u_1} [Fintype ι] {s : Set (ι → ℝ)} (hs : s.OrdConnected)
Goal: MeasureTheory.volume (frontier s) = 0
-- Set.OrdConnected.nullMeasurableSet
protected theorem Set.OrdConnected.nullMeasurableSet (hs : s.OrdConnected) : NullMeasurableSet s 
Arguments: {ι : Type u_1} [Fintype ι] {s : Set (ι → ℝ)} (hs : s.OrdConnected)
Goal: MeasureTheory.NullMeasurableSet s MeasureTheory.volume
-- IsAntichain.volume_eq_zero
theorem IsAntichain.volume_eq_zero [Nonempty ι] (hs : IsAntichain (· ≤ ·) s) : volume s = 0 
Arguments: {ι : Type u_1} [Fintype ι] {s : Set (ι → ℝ)} [Nonempty ι] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: MeasureTheory.volume s = 0
-- measurable_of_tendsto_metrizable'
theorem measurable_of_tendsto_metrizable' {ι} {f : ι → α → β} {g : α → β} (u : Filter ι) [NeBot u]
    [IsCountablyGenerated u] (hf : ∀ i, Measurable (f i)) (lim : Tendsto f u (𝓝 g)) :
    Measurable g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {ι : Type u_3} {f : ι → α → β} {g : α → β} (u : Filter ι) [u.NeBot] [u.IsCountablyGenerated] (hf : ∀ (i : ι), Measurable (f i)) (lim : Filter.Tendsto f u (𝓝 g))
Goal: Measurable g
-- measurable_of_tendsto_metrizable
theorem measurable_of_tendsto_metrizable {f : ℕ → α → β} {g : α → β} (hf : ∀ i, Measurable (f i))
    (lim : Tendsto f atTop (𝓝 g)) : Measurable g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {f : ℕ → α → β} {g : α → β} (hf : ∀ (i : ℕ), Measurable (f i)) (lim : Filter.Tendsto f Filter.atTop (𝓝 g))
Goal: Measurable g
-- aemeasurable_of_tendsto_metrizable_ae
theorem aemeasurable_of_tendsto_metrizable_ae {ι} {μ : Measure α} {f : ι → α → β} {g : α → β}
    (u : Filter ι) [hu : NeBot u] [IsCountablyGenerated u] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) : AEMeasurable g μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {ι : Type u_3} {μ : MeasureTheory.Measure α} {f : ι → α → β} {g : α → β} (u : Filter ι) [hu : u.NeBot] [u.IsCountablyGenerated] (hf : ∀ (n : ι), AEMeasurable (f n) μ) (h_tendsto : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) u (𝓝 (g x)))
Goal: AEMeasurable g μ
-- aemeasurable_of_tendsto_metrizable_ae'
theorem aemeasurable_of_tendsto_metrizable_ae' {μ : Measure α} {f : ℕ → α → β} {g : α → β}
    (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : AEMeasurable g μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : MeasureTheory.Measure α} {f : ℕ → α → β} {g : α → β} (hf : ∀ (n : ℕ), AEMeasurable (f n) μ) (h_ae_tendsto : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: AEMeasurable g μ
-- aemeasurable_of_unif_approx
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_3} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β] {μ : MeasureTheory.Measure α} {g : α → β} (hf : ∀ ε > 0, ∃ f, AEMeasurable f μ ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε)
Goal: AEMeasurable g μ
-- measurable_of_tendsto_metrizable_ae
theorem measurable_of_tendsto_metrizable_ae {μ : Measure α} [μ.IsComplete] {f : ℕ → α → β}
    {g : α → β} (hf : ∀ n, Measurable (f n))
    (h_ae_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : Measurable g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : MeasureTheory.Measure α} [μ.IsComplete] {f : ℕ → α → β} {g : α → β} (hf : ∀ (n : ℕ), Measurable (f n)) (h_ae_tendsto : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: Measurable g
-- measurable_limit_of_tendsto_metrizable_ae
theorem measurable_limit_of_tendsto_metrizable_ae {ι} [Countable ι] [Nonempty ι] {μ : Measure α}
    {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) L (𝓝 l)) :
    ∃ f_lim : α → β, Measurable f_lim ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {ι : Type u_3} [Countable ι] [Nonempty ι] {μ : MeasureTheory.Measure α} {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ (n : ι), AEMeasurable (f n) μ) (h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Filter.Tendsto (fun n => f n x) L (𝓝 l))
Goal: ∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) L (𝓝 (f_lim x))
-- measurableSet_of_tendsto_indicator
lemma measurableSet_of_tendsto_indicator [NeBot L] (As_mble : ∀ i, MeasurableSet (As i))
    (h_lim : ∀ x, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
    MeasurableSet A 
Arguments: {α : Type u_3} [MeasurableSpace α] {A : Set α} {ι : Type u_4} (L : Filter ι) [L.IsCountablyGenerated] {As : ι → Set α} [L.NeBot] (As_mble : ∀ (i : ι), MeasurableSet (As i)) (h_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A)
Goal: MeasurableSet A
-- nullMeasurableSet_of_tendsto_indicator
lemma nullMeasurableSet_of_tendsto_indicator [NeBot L] {μ : Measure α}
    (As_mble : ∀ i, NullMeasurableSet (As i) μ)
    (h_lim : ∀ᵐ x ∂μ, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
    NullMeasurableSet A μ 
Arguments: {α : Type u_3} [MeasurableSpace α] {A : Set α} {ι : Type u_4} (L : Filter ι) [L.IsCountablyGenerated] {As : ι → Set α} [L.NeBot] {μ : MeasureTheory.Measure α} (As_mble : ∀ (i : ι), MeasureTheory.NullMeasurableSet (As i) μ) (h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A)
Goal: MeasureTheory.NullMeasurableSet A μ
-- measurableSet_ball
@[measurability]
theorem measurableSet_ball : MeasurableSet (Metric.ball x ε) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} {ε : ℝ}
Goal: MeasurableSet (Metric.ball x ε)
-- measurableSet_closedBall
@[measurability]
theorem measurableSet_closedBall : MeasurableSet (Metric.closedBall x ε) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} {ε : ℝ}
Goal: MeasurableSet (Metric.closedBall x ε)
-- measurable_infDist
@[measurability]
theorem measurable_infDist {s : Set α} : Measurable fun x => infDist x s 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α}
Goal: Measurable fun x => Metric.infDist x s
-- Measurable.infDist
@[measurability, fun_prop]
theorem Measurable.infDist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infDist (f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α}
Goal: Measurable fun x => Metric.infDist (f x) s
-- measurable_infNndist
@[measurability]
theorem measurable_infNndist {s : Set α} : Measurable fun x => infNndist x s 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α}
Goal: Measurable fun x => Metric.infNndist x s
-- Measurable.infNndist
@[measurability, fun_prop]
theorem Measurable.infNndist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infNndist (f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α}
Goal: Measurable fun x => Metric.infNndist (f x) s
-- measurable_dist
@[measurability]
theorem measurable_dist : Measurable fun p : α × α => dist p.1 p.2 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α]
Goal: Measurable fun p => dist p.1 p.2
-- Measurable.dist
@[measurability, fun_prop]
theorem Measurable.dist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => dist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [SecondCountableTopology α] {f g : β → α} (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun b => dist (f b) (g b)
-- measurable_nndist
@[measurability]
theorem measurable_nndist : Measurable fun p : α × α => nndist p.1 p.2 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α]
Goal: Measurable fun p => nndist p.1 p.2
-- Measurable.nndist
@[measurability, fun_prop]
theorem Measurable.nndist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => nndist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [SecondCountableTopology α] {f g : β → α} (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun b => nndist (f b) (g b)