ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.FiniteMeasure.restrict_measure_eq
theorem restrict_measure_eq (μ : FiniteMeasure Ω) (A : Set Ω) :
    (μ.restrict A : Measure Ω) = (μ : Measure Ω).restrict A 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (A : Set Ω)
Goal: ↑(μ.restrict A) = (↑μ).restrict A
-- MeasureTheory.FiniteMeasure.restrict_apply_measure
theorem restrict_apply_measure (μ : FiniteMeasure Ω) (A : Set Ω) {s : Set Ω}
    (s_mble : MeasurableSet s) : (μ.restrict A : Measure Ω) s = (μ : Measure Ω) (s ∩ A) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (A : Set Ω) {s : Set Ω} (s_mble : MeasurableSet s)
Goal: ↑(μ.restrict A) s = ↑μ (s ∩ A)
-- MeasureTheory.FiniteMeasure.restrict_apply
theorem restrict_apply (μ : FiniteMeasure Ω) (A : Set Ω) {s : Set Ω} (s_mble : MeasurableSet s) :
    (μ.restrict A) s = μ (s ∩ A) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (A : Set Ω) {s : Set Ω} (s_mble : MeasurableSet s)
Goal: (μ.restrict A) s = μ (s ∩ A)
-- MeasureTheory.FiniteMeasure.restrict_mass
theorem restrict_mass (μ : FiniteMeasure Ω) (A : Set Ω) : (μ.restrict A).mass = μ A 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (A : Set Ω)
Goal: (μ.restrict A).mass = μ A
-- MeasureTheory.FiniteMeasure.restrict_eq_zero_iff
theorem restrict_eq_zero_iff (μ : FiniteMeasure Ω) (A : Set Ω) : μ.restrict A = 0 ↔ μ A = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (A : Set Ω)
Goal: μ.restrict A = 0 ↔ μ A = 0
-- MeasureTheory.FiniteMeasure.restrict_nonzero_iff
theorem restrict_nonzero_iff (μ : FiniteMeasure Ω) (A : Set Ω) : μ.restrict A ≠ 0 ↔ μ A ≠ 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (A : Set Ω)
Goal: μ.restrict A ≠ 0 ↔ μ A ≠ 0
-- MeasureTheory.FiniteMeasure.ext_of_forall_lintegral_eq
theorem ext_of_forall_lintegral_eq [HasOuterApproxClosed Ω] [BorelSpace Ω]
    {μ ν : FiniteMeasure Ω} (h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ x, f x ∂μ = ∫⁻ x, f x ∂ν) :
    μ = ν 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω] [BorelSpace Ω] {μ ν : MeasureTheory.FiniteMeasure Ω} (h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂↑μ = ∫⁻ (x : Ω), ↑(f x) ∂↑ν)
Goal: μ = ν
-- MeasureTheory.FiniteMeasure.testAgainstNN_coe_eq
@[simp]
theorem testAgainstNN_coe_eq {μ : FiniteMeasure Ω} {f : Ω →ᵇ ℝ≥0} :
    (μ.testAgainstNN f : ℝ≥0∞) = ∫⁻ ω, f ω ∂(μ : Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] {μ : MeasureTheory.FiniteMeasure Ω} {f : Ω →ᵇ ℝ≥0}
Goal: ↑(μ.testAgainstNN f) = ∫⁻ (ω : Ω), ↑(f ω) ∂↑μ
-- MeasureTheory.FiniteMeasure.testAgainstNN_const
theorem testAgainstNN_const (μ : FiniteMeasure Ω) (c : ℝ≥0) :
    μ.testAgainstNN (BoundedContinuousFunction.const Ω c) = c * μ.mass 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (c : ℝ≥0)
Goal: μ.testAgainstNN (BoundedContinuousFunction.const Ω c) = c * μ.mass
-- MeasureTheory.FiniteMeasure.testAgainstNN_mono
theorem testAgainstNN_mono (μ : FiniteMeasure Ω) {f g : Ω →ᵇ ℝ≥0} (f_le_g : (f : Ω → ℝ≥0) ≤ g) :
    μ.testAgainstNN f ≤ μ.testAgainstNN g 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) {f g : Ω →ᵇ ℝ≥0} (f_le_g : ⇑f ≤ ⇑g)
Goal: μ.testAgainstNN f ≤ μ.testAgainstNN g
-- MeasureTheory.FiniteMeasure.testAgainstNN_zero
@[simp]
theorem testAgainstNN_zero (μ : FiniteMeasure Ω) : μ.testAgainstNN 0 = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω)
Goal: μ.testAgainstNN 0 = 0
-- MeasureTheory.FiniteMeasure.testAgainstNN_one
@[simp]
theorem testAgainstNN_one (μ : FiniteMeasure Ω) : μ.testAgainstNN 1 = μ.mass 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω)
Goal: μ.testAgainstNN 1 = μ.mass
-- MeasureTheory.FiniteMeasure.zero_testAgainstNN_apply
@[simp]
theorem zero_testAgainstNN_apply (f : Ω →ᵇ ℝ≥0) : (0 : FiniteMeasure Ω).testAgainstNN f = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] (f : Ω →ᵇ ℝ≥0)
Goal: MeasureTheory.FiniteMeasure.testAgainstNN 0 f = 0
-- MeasureTheory.FiniteMeasure.zero_testAgainstNN
theorem zero_testAgainstNN : (0 : FiniteMeasure Ω).testAgainstNN = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω]
Goal: MeasureTheory.FiniteMeasure.testAgainstNN 0 = 0
-- MeasureTheory.FiniteMeasure.smul_testAgainstNN_apply
@[simp]
theorem smul_testAgainstNN_apply (c : ℝ≥0) (μ : FiniteMeasure Ω) (f : Ω →ᵇ ℝ≥0) :
    (c • μ).testAgainstNN f = c • μ.testAgainstNN f 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] (c : ℝ≥0) (μ : MeasureTheory.FiniteMeasure Ω) (f : Ω →ᵇ ℝ≥0)
Goal: (c • μ).testAgainstNN f = c • μ.testAgainstNN f
-- MeasureTheory.FiniteMeasure.testAgainstNN_add
theorem testAgainstNN_add (μ : FiniteMeasure Ω) (f₁ f₂ : Ω →ᵇ ℝ≥0) :
    μ.testAgainstNN (f₁ + f₂) = μ.testAgainstNN f₁ + μ.testAgainstNN f₂ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (f₁ f₂ : Ω →ᵇ ℝ≥0)
Goal: μ.testAgainstNN (f₁ + f₂) = μ.testAgainstNN f₁ + μ.testAgainstNN f₂
-- MeasureTheory.FiniteMeasure.testAgainstNN_smul
theorem testAgainstNN_smul [IsScalarTower R ℝ≥0 ℝ≥0] [PseudoMetricSpace R] [Zero R]
    [BoundedSMul R ℝ≥0] (μ : FiniteMeasure Ω) (c : R) (f : Ω →ᵇ ℝ≥0) :
    μ.testAgainstNN (c • f) = c • μ.testAgainstNN f 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {R : Type u_2} [SMul R ℝ≥0] [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0 ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [IsScalarTower R ℝ≥0 ℝ≥0] [PseudoMetricSpace R] [Zero R] [BoundedSMul R ℝ≥0] (μ : MeasureTheory.FiniteMeasure Ω) (c : R) (f : Ω →ᵇ ℝ≥0)
Goal: μ.testAgainstNN (c • f) = c • μ.testAgainstNN f
-- MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz_estimate
theorem testAgainstNN_lipschitz_estimate (μ : FiniteMeasure Ω) (f g : Ω →ᵇ ℝ≥0) :
    μ.testAgainstNN f ≤ μ.testAgainstNN g + nndist f g * μ.mass 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (f g : Ω →ᵇ ℝ≥0)
Goal: μ.testAgainstNN f ≤ μ.testAgainstNN g + nndist f g * μ.mass
-- MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz
theorem testAgainstNN_lipschitz (μ : FiniteMeasure Ω) :
    LipschitzWith μ.mass fun f : Ω →ᵇ ℝ≥0 => μ.testAgainstNN f 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω)
Goal: LipschitzWith μ.mass fun f => μ.testAgainstNN f
-- MeasureTheory.FiniteMeasure.coe_toWeakDualBCNN
@[simp]
theorem coe_toWeakDualBCNN (μ : FiniteMeasure Ω) : ⇑μ.toWeakDualBCNN = μ.testAgainstNN 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω)
Goal: ⇑μ.toWeakDualBCNN = μ.testAgainstNN
-- MeasureTheory.FiniteMeasure.toWeakDualBCNN_apply
@[simp]
theorem toWeakDualBCNN_apply (μ : FiniteMeasure Ω) (f : Ω →ᵇ ℝ≥0) :
    μ.toWeakDualBCNN f = (∫⁻ x, f x ∂(μ : Measure Ω)).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (f : Ω →ᵇ ℝ≥0)
Goal: μ.toWeakDualBCNN f = (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toNNReal
-- MeasureTheory.FiniteMeasure.toWeakDualBCNN_continuous
theorem toWeakDualBCNN_continuous : Continuous (@toWeakDualBCNN Ω _ _ _) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω]
Goal: Continuous MeasureTheory.FiniteMeasure.toWeakDualBCNN
-- MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval
theorem continuous_testAgainstNN_eval (f : Ω →ᵇ ℝ≥0) :
    Continuous fun μ : FiniteMeasure Ω => μ.testAgainstNN f 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (f : Ω →ᵇ ℝ≥0)
Goal: Continuous fun μ => μ.testAgainstNN f
-- MeasureTheory.FiniteMeasure.continuous_mass
theorem continuous_mass : Continuous fun μ : FiniteMeasure Ω => μ.mass 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω]
Goal: Continuous fun μ => μ.mass
-- Filter.Tendsto.mass
theorem _root_.Filter.Tendsto.mass {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    {μ : FiniteMeasure Ω} (h : Tendsto μs F (𝓝 μ)) : Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_3} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω} (h : Filter.Tendsto μs F (𝓝 μ))
Goal: Filter.Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass)
-- MeasureTheory.FiniteMeasure.tendsto_iff_weak_star_tendsto
theorem tendsto_iff_weak_star_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    {μ : FiniteMeasure Ω} :
    Tendsto μs F (𝓝 μ) ↔ Tendsto (fun i => (μs i).toWeakDualBCNN) F (𝓝 μ.toWeakDualBCNN) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_3} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ) ↔ Filter.Tendsto (fun i => (μs i).toWeakDualBCNN) F (𝓝 μ.toWeakDualBCNN)
-- MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto
theorem tendsto_iff_forall_toWeakDualBCNN_tendsto {γ : Type*} {F : Filter γ}
    {μs : γ → FiniteMeasure Ω} {μ : FiniteMeasure Ω} :
    Tendsto μs F (𝓝 μ) ↔
      ∀ f : Ω →ᵇ ℝ≥0, Tendsto (fun i => (μs i).toWeakDualBCNN f) F (𝓝 (μ.toWeakDualBCNN f)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_3} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ) ↔
  ∀ (f : Ω →ᵇ ℝ≥0), Filter.Tendsto (fun i => (μs i).toWeakDualBCNN f) F (𝓝 (μ.toWeakDualBCNN f))
-- MeasureTheory.FiniteMeasure.tendsto_iff_forall_testAgainstNN_tendsto
theorem tendsto_iff_forall_testAgainstNN_tendsto {γ : Type*} {F : Filter γ}
    {μs : γ → FiniteMeasure Ω} {μ : FiniteMeasure Ω} :
    Tendsto μs F (𝓝 μ) ↔
      ∀ f : Ω →ᵇ ℝ≥0, Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_3} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Filter.Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f))
-- MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass
theorem tendsto_zero_testAgainstNN_of_tendsto_zero_mass {γ : Type*} {F : Filter γ}
    {μs : γ → FiniteMeasure Ω} (mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)) (f : Ω →ᵇ ℝ≥0) :
    Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 0) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_3} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} (mass_lim : Filter.Tendsto (fun i => (μs i).mass) F (𝓝 0)) (f : Ω →ᵇ ℝ≥0)
Goal: Filter.Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 0)
-- MeasureTheory.FiniteMeasure.tendsto_zero_of_tendsto_zero_mass
theorem tendsto_zero_of_tendsto_zero_mass {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    (mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)) : Tendsto μs F (𝓝 0) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_3} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} (mass_lim : Filter.Tendsto (fun i => (μs i).mass) F (𝓝 0))
Goal: Filter.Tendsto μs F (𝓝 0)
-- MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto
theorem tendsto_iff_forall_lintegral_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    {μ : FiniteMeasure Ω} :
    Tendsto μs F (𝓝 μ) ↔
      ∀ f : Ω →ᵇ ℝ≥0,
        Tendsto (fun i => ∫⁻ x, f x ∂(μs i : Measure Ω)) F (𝓝 (∫⁻ x, f x ∂(μ : Measure Ω))) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_3} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ) ↔
  ∀ (f : Ω →ᵇ ℝ≥0), Filter.Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
-- MeasureTheory.FiniteMeasure.injective_toWeakDualBCNN
lemma injective_toWeakDualBCNN :
    Injective (toWeakDualBCNN : FiniteMeasure Ω → WeakDual ℝ≥0 (Ω →ᵇ ℝ≥0)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω] [BorelSpace Ω]
Goal: Function.Injective MeasureTheory.FiniteMeasure.toWeakDualBCNN
-- MeasureTheory.FiniteMeasure.embedding_toWeakDualBCNN
lemma embedding_toWeakDualBCNN :
    Embedding (toWeakDualBCNN : FiniteMeasure Ω → WeakDual ℝ≥0 (Ω →ᵇ ℝ≥0)) 
Arguments: (Ω : Type u_1) [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω] [BorelSpace Ω]
Goal: Embedding MeasureTheory.FiniteMeasure.toWeakDualBCNN
-- MeasureTheory.FiniteMeasure.tendsto_lintegral_nn_of_le_const
theorem tendsto_lintegral_nn_of_le_const (μ : FiniteMeasure Ω) {fs : ℕ → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
    (fs_le_const : ∀ n ω, fs n ω ≤ c) {f : Ω → ℝ≥0}
    (fs_lim : ∀ ω, Tendsto (fun n => fs n ω) atTop (𝓝 (f ω))) :
    Tendsto (fun n => ∫⁻ ω, fs n ω ∂(μ : Measure Ω)) atTop (𝓝 (∫⁻ ω, f ω ∂(μ : Measure Ω))) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) {fs : ℕ → Ω →ᵇ ℝ≥0} {c : ℝ≥0} (fs_le_const : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c) {f : Ω → ℝ≥0} (fs_lim : ∀ (ω : Ω), Filter.Tendsto (fun n => (fs n) ω) Filter.atTop (𝓝 (f ω)))
Goal: Filter.Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) Filter.atTop (𝓝 (∫⁻ (ω : Ω), ↑(f ω) ∂↑μ))
-- MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_filter_of_le_const
theorem tendsto_testAgainstNN_filter_of_le_const {ι : Type*} {L : Filter ι}
    [L.IsCountablyGenerated] {μ : FiniteMeasure Ω} {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
    (fs_le_const : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂(μ : Measure Ω), fs i ω ≤ c) {f : Ω →ᵇ ℝ≥0}
    (fs_lim : ∀ᵐ ω : Ω ∂(μ : Measure Ω), Tendsto (fun i => fs i ω) L (𝓝 (f ω))) :
    Tendsto (fun i => μ.testAgainstNN (fs i)) L (𝓝 (μ.testAgainstNN f)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {ι : Type u_2} {L : Filter ι} [L.IsCountablyGenerated] {μ : MeasureTheory.FiniteMeasure Ω} {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0} (fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂↑μ, (fs i) ω ≤ c) {f : Ω →ᵇ ℝ≥0} (fs_lim : ∀ᵐ (ω : Ω) ∂↑μ, Filter.Tendsto (fun i => (fs i) ω) L (𝓝 (f ω)))
Goal: Filter.Tendsto (fun i => μ.testAgainstNN (fs i)) L (𝓝 (μ.testAgainstNN f))
-- MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_le_const
theorem tendsto_testAgainstNN_of_le_const {μ : FiniteMeasure Ω} {fs : ℕ → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
    (fs_le_const : ∀ n ω, fs n ω ≤ c) {f : Ω →ᵇ ℝ≥0}
    (fs_lim : ∀ ω, Tendsto (fun n => fs n ω) atTop (𝓝 (f ω))) :
    Tendsto (fun n => μ.testAgainstNN (fs n)) atTop (𝓝 (μ.testAgainstNN f)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {μ : MeasureTheory.FiniteMeasure Ω} {fs : ℕ → Ω →ᵇ ℝ≥0} {c : ℝ≥0} (fs_le_const : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c) {f : Ω →ᵇ ℝ≥0} (fs_lim : ∀ (ω : Ω), Filter.Tendsto (fun n => (fs n) ω) Filter.atTop (𝓝 (f ω)))
Goal: Filter.Tendsto (fun n => μ.testAgainstNN (fs n)) Filter.atTop (𝓝 (μ.testAgainstNN f))
-- MeasureTheory.FiniteMeasure.tendsto_of_forall_integral_tendsto
theorem tendsto_of_forall_integral_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    {μ : FiniteMeasure Ω}
    (h :
      ∀ f : Ω →ᵇ ℝ,
        Tendsto (fun i => ∫ x, f x ∂(μs i : Measure Ω)) F (𝓝 (∫ x, f x ∂(μ : Measure Ω)))) :
    Tendsto μs F (𝓝 μ) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω} (h : ∀ (f : Ω →ᵇ ℝ), Filter.Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ)))
Goal: Filter.Tendsto μs F (𝓝 μ)
-- MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendsto
theorem tendsto_iff_forall_integral_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
    {μ : FiniteMeasure Ω} :
    Tendsto μs F (𝓝 μ) ↔
      ∀ f : Ω →ᵇ ℝ,
        Tendsto (fun i => ∫ x, f x ∂(μs i : Measure Ω)) F (𝓝 (∫ x, f x ∂(μ : Measure Ω))) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ), Filter.Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
-- MeasureTheory.FiniteMeasure.toMeasure_map
@[simp] lemma toMeasure_map (ν : FiniteMeasure Ω) (f : Ω → Ω') :
    (ν.map f).toMeasure = ν.toMeasure.map f 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.FiniteMeasure Ω) (f : Ω → Ω')
Goal: ↑(ν.map f) = MeasureTheory.Measure.map f ↑ν
-- MeasureTheory.FiniteMeasure.map_apply'
lemma map_apply' (ν : FiniteMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)
    {A : Set Ω'} (A_mble : MeasurableSet A) :
    (ν.map f : Measure Ω') A = (ν : Measure Ω) (f ⁻¹' A) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.FiniteMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ↑ν) {A : Set Ω'} (A_mble : MeasurableSet A)
Goal: ↑(ν.map f) A = ↑ν (f ⁻¹' A)
-- MeasureTheory.FiniteMeasure.map_apply_of_aemeasurable
lemma map_apply_of_aemeasurable (ν : FiniteMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)
    {A : Set Ω'} (A_mble : MeasurableSet A) :
    ν.map f A = ν (f ⁻¹' A) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.FiniteMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ↑ν) {A : Set Ω'} (A_mble : MeasurableSet A)
Goal: (ν.map f) A = ν (f ⁻¹' A)
-- MeasureTheory.FiniteMeasure.map_apply
lemma map_apply (ν : FiniteMeasure Ω) {f : Ω → Ω'} (f_mble : Measurable f)
    {A : Set Ω'} (A_mble : MeasurableSet A) :
    ν.map f A = ν (f ⁻¹' A) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (ν : MeasureTheory.FiniteMeasure Ω) {f : Ω → Ω'} (f_mble : Measurable f) {A : Set Ω'} (A_mble : MeasurableSet A)
Goal: (ν.map f) A = ν (f ⁻¹' A)
-- MeasureTheory.FiniteMeasure.map_add
@[simp] lemma map_add {f : Ω → Ω'} (f_mble : Measurable f) (ν₁ ν₂ : FiniteMeasure Ω) :
    (ν₁ + ν₂).map f = ν₁.map f + ν₂.map f 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] {f : Ω → Ω'} (f_mble : Measurable f) (ν₁ ν₂ : MeasureTheory.FiniteMeasure Ω)
Goal: (ν₁ + ν₂).map f = ν₁.map f + ν₂.map f
-- MeasureTheory.FiniteMeasure.map_smul
@[simp] lemma map_smul {f : Ω → Ω'} (c : ℝ≥0) (ν : FiniteMeasure Ω) :
    (c • ν).map f = c • (ν.map f) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] {f : Ω → Ω'} (c : ℝ≥0) (ν : MeasureTheory.FiniteMeasure Ω)
Goal: (c • ν).map f = c • ν.map f
-- MeasureTheory.FiniteMeasure.tendsto_map_of_tendsto_of_continuous
lemma tendsto_map_of_tendsto_of_continuous {ι : Type*} {L : Filter ι}
    (νs : ι → FiniteMeasure Ω) (ν : FiniteMeasure Ω) (lim : Tendsto νs L (𝓝 ν))
    {f : Ω → Ω'} (f_cont : Continuous f) :
    Tendsto (fun i ↦ (νs i).map f) L (𝓝 (ν.map f)) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [TopologicalSpace Ω'] [BorelSpace Ω'] {ι : Type u_3} {L : Filter ι} (νs : ι → MeasureTheory.FiniteMeasure Ω) (ν : MeasureTheory.FiniteMeasure Ω) (lim : Filter.Tendsto νs L (𝓝 ν)) {f : Ω → Ω'} (f_cont : Continuous f)
Goal: Filter.Tendsto (fun i => (νs i).map f) L (𝓝 (ν.map f))
-- MeasureTheory.FiniteMeasure.continuous_map
lemma continuous_map {f : Ω → Ω'} (f_cont : Continuous f) :
    Continuous (fun ν ↦ FiniteMeasure.map ν f) 
Arguments: {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [TopologicalSpace Ω'] [BorelSpace Ω'] {f : Ω → Ω'} (f_cont : Continuous f)
Goal: Continuous fun ν => ν.map f
-- MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd
theorem exists_pair_mem_lattice_not_disjoint_vadd [AddCommGroup L] [Countable L] [AddAction L E]
    [MeasurableSpace L] [MeasurableVAdd L E] [VAddInvariantMeasure L E μ]
    (fund : IsAddFundamentalDomain L F μ) (hS : NullMeasurableSet s μ) (h : μ F < μ s) :
    ∃ x y : L, x ≠ y ∧ ¬Disjoint (x +ᵥ s) (y +ᵥ s) 
Arguments: {E : Type u_1} {L : Type u_2} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {F s : Set E} [AddCommGroup L] [Countable L] [AddAction L E] [MeasurableSpace L] [MeasurableVAdd L E] [MeasureTheory.VAddInvariantMeasure L E μ] (fund : MeasureTheory.IsAddFundamentalDomain L F μ) (hS : MeasureTheory.NullMeasurableSet s μ) (h : μ F < μ s)
Goal: ∃ x y, x ≠ y ∧ ¬Disjoint (x +ᵥ s) (y +ᵥ s)
-- MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {F s : Set E} [NormedAddCommGroup E] [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure] {L : AddSubgroup E} [Countable ↥L] (fund : MeasureTheory.IsAddFundamentalDomain (↥L) F μ) (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ FiniteDimensional.finrank ℝ E < μ s)
Goal: ∃ x, x ≠ 0 ∧ ↑x ∈ s
-- MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] [DiscreteTopology L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h_cpt : IsCompact s)
    (h : μ F * 2 ^ finrank ℝ E ≤ μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {F s : Set E} [NormedAddCommGroup E] [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] [μ.IsAddHaarMeasure] {L : AddSubgroup E} [Countable ↥L] [DiscreteTopology ↥L] (fund : MeasureTheory.IsAddFundamentalDomain (↥L) F μ) (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h_cpt : IsCompact s) (h : μ F * 2 ^ FiniteDimensional.finrank ℝ E ≤ μ s)
Goal: ∃ x, x ≠ 0 ∧ ↑x ∈ s
-- MeasureTheory.measurePreserving_prod_mul
@[to_additive measurePreserving_prod_add
" The shear mapping `(x, y) ↦ (x, x + y)` preserves the measure `μ × ν`. "]
theorem measurePreserving_prod_mul [IsMulLeftInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.1 * z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [ν.IsMulLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, z.1 * z.2)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_prod_add
@[to_additive measurePreserving_prod_add
" The shear mapping `(x, y) ↦ (x, x + y)` preserves the measure `μ × ν`. "]
theorem measurePreserving_prod_mul [IsMulLeftInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.1 * z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [ν.IsAddLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, z.1 + z.2)) (μ.prod ν) (μ.prod ν)