ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.le_hausdorffMeasure
theorem le_hausdorffMeasure (d : ℝ) (μ : Measure X) (ε : ℝ≥0∞) (h₀ : 0 < ε)
    (h : ∀ s : Set X, diam s ≤ ε → μ s ≤ diam s ^ d) : μ ≤ μH[d] 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (d : ℝ) (μ : MeasureTheory.Measure X) (ε : ℝ≥0∞) (h₀ : 0 < ε) (h : ∀ (s : Set X), EMetric.diam s ≤ ε → μ s ≤ EMetric.diam s ^ d)
Goal: μ ≤ μH[d]
-- MeasureTheory.Measure.hausdorffMeasure_apply
theorem hausdorffMeasure_apply (d : ℝ) (s : Set X) :
    μH[d] s =
      ⨆ (r : ℝ≥0∞) (_ : 0 < r),
        ⨅ (t : ℕ → Set X) (_ : s ⊆ ⋃ n, t n) (_ : ∀ n, diam (t n) ≤ r),
          ∑' n, ⨆ _ : (t n).Nonempty, diam (t n) ^ d 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (d : ℝ) (s : Set X)
Goal: μH[d] s =
  ⨆ r,
    ⨆ (_ : 0 < r),
      ⨅ t,
        ⨅ (_ : s ⊆ ⋃ n, t n),
          ⨅ (_ : ∀ (n : ℕ), EMetric.diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), EMetric.diam (t n) ^ d
-- MeasureTheory.Measure.hausdorffMeasure_le_liminf_tsum
theorem hausdorffMeasure_le_liminf_tsum {β : Type*} {ι : β → Type*} [∀ n, Countable (ι n)]
    (d : ℝ) (s : Set X) {l : Filter β} (r : β → ℝ≥0∞) (hr : Tendsto r l (𝓝 0))
    (t : ∀ n : β, ι n → Set X) (ht : ∀ᶠ n in l, ∀ i, diam (t n i) ≤ r n)
    (hst : ∀ᶠ n in l, s ⊆ ⋃ i, t n i) : μH[d] s ≤ liminf (fun n => ∑' i, diam (t n i) ^ d) l 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {β : Type u_4} {ι : β → Type u_5} [∀ (n : β), Countable (ι n)] (d : ℝ) (s : Set X) {l : Filter β} (r : β → ℝ≥0∞) (hr : Filter.Tendsto r l (𝓝 0)) (t : (n : β) → ι n → Set X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), EMetric.diam (t n i) ≤ r n) (hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ i, t n i)
Goal: μH[d] s ≤ Filter.liminf (fun n => ∑' (i : ι n), EMetric.diam (t n i) ^ d) l
-- MeasureTheory.Measure.hausdorffMeasure_le_liminf_sum
theorem hausdorffMeasure_le_liminf_sum {β : Type*} {ι : β → Type*} [∀ n, Fintype (ι n)]
    (d : ℝ) (s : Set X) {l : Filter β} (r : β → ℝ≥0∞) (hr : Tendsto r l (𝓝 0))
    (t : ∀ n : β, ι n → Set X) (ht : ∀ᶠ n in l, ∀ i, diam (t n i) ≤ r n)
    (hst : ∀ᶠ n in l, s ⊆ ⋃ i, t n i) : μH[d] s ≤ liminf (fun n => ∑ i, diam (t n i) ^ d) l 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {β : Type u_4} {ι : β → Type u_5} [(n : β) → Fintype (ι n)] (d : ℝ) (s : Set X) {l : Filter β} (r : β → ℝ≥0∞) (hr : Filter.Tendsto r l (𝓝 0)) (t : (n : β) → ι n → Set X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), EMetric.diam (t n i) ≤ r n) (hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ i, t n i)
Goal: μH[d] s ≤ Filter.liminf (fun n => ∑ i : ι n, EMetric.diam (t n i) ^ d) l
-- MeasureTheory.Measure.hausdorffMeasure_zero_or_top
theorem hausdorffMeasure_zero_or_top {d₁ d₂ : ℝ} (h : d₁ < d₂) (s : Set X) :
    μH[d₂] s = 0 ∨ μH[d₁] s = ∞ 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {d₁ d₂ : ℝ} (h : d₁ < d₂) (s : Set X)
Goal: μH[d₂] s = 0 ∨ μH[d₁] s = ⊤
-- MeasureTheory.Measure.hausdorffMeasure_mono
theorem hausdorffMeasure_mono {d₁ d₂ : ℝ} (h : d₁ ≤ d₂) (s : Set X) : μH[d₂] s ≤ μH[d₁] s 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {d₁ d₂ : ℝ} (h : d₁ ≤ d₂) (s : Set X)
Goal: μH[d₂] s ≤ μH[d₁] s
-- MeasureTheory.Measure.noAtoms_hausdorff
theorem noAtoms_hausdorff {d : ℝ} (hd : 0 < d) : NoAtoms (hausdorffMeasure d : Measure X) 
Arguments: (X : Type u_2) [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {d : ℝ} (hd : 0 < d)
Goal: MeasureTheory.NoAtoms μH[d]
-- MeasureTheory.Measure.hausdorffMeasure_zero_singleton
@[simp]
theorem hausdorffMeasure_zero_singleton (x : X) : μH[0] ({x} : Set X) = 1 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (x : X)
Goal: μH[0] {x} = 1
-- MeasureTheory.Measure.one_le_hausdorffMeasure_zero_of_nonempty
theorem one_le_hausdorffMeasure_zero_of_nonempty {s : Set X} (h : s.Nonempty) : 1 ≤ μH[0] s 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} (h : s.Nonempty)
Goal: 1 ≤ μH[0] s
-- MeasureTheory.Measure.hausdorffMeasure_le_one_of_subsingleton
theorem hausdorffMeasure_le_one_of_subsingleton {s : Set X} (hs : s.Subsingleton) {d : ℝ}
    (hd : 0 ≤ d) : μH[d] s ≤ 1 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} (hs : s.Subsingleton) {d : ℝ} (hd : 0 ≤ d)
Goal: μH[d] s ≤ 1
-- HolderOnWith.hausdorffMeasure_image_le
theorem hausdorffMeasure_image_le (h : HolderOnWith C r f s) (hr : 0 < r) {d : ℝ} (hd : 0 ≤ d) :
    μH[d] (f '' s) ≤ (C : ℝ≥0∞) ^ d * μH[r * d] s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (h : HolderOnWith C r f s) (hr : 0 < r) {d : ℝ} (hd : 0 ≤ d)
Goal: μH[d] (f '' s) ≤ ↑C ^ d * μH[↑r * d] s
-- LipschitzOnWith.hausdorffMeasure_image_le
theorem hausdorffMeasure_image_le (h : LipschitzOnWith K f s) {d : ℝ} (hd : 0 ≤ d) :
    μH[d] (f '' s) ≤ (K : ℝ≥0∞) ^ d * μH[d] s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {K : ℝ≥0} {f : X → Y} {s : Set X} (h : LipschitzOnWith K f s) {d : ℝ} (hd : 0 ≤ d)
Goal: μH[d] (f '' s) ≤ ↑K ^ d * μH[d] s
-- LipschitzWith.hausdorffMeasure_image_le
theorem hausdorffMeasure_image_le (h : LipschitzWith K f) {d : ℝ} (hd : 0 ≤ d) (s : Set X) :
    μH[d] (f '' s) ≤ (K : ℝ≥0∞) ^ d * μH[d] s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {K : ℝ≥0} {f : X → Y} (h : LipschitzWith K f) {d : ℝ} (hd : 0 ≤ d) (s : Set X)
Goal: μH[d] (f '' s) ≤ ↑K ^ d * μH[d] s
-- MeasureTheory.Measure.hausdorffMeasure_smul₀
theorem MeasureTheory.Measure.hausdorffMeasure_smul₀ {𝕜 E : Type*} [NormedAddCommGroup E]
    [NormedField 𝕜] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {d : ℝ} (hd : 0 ≤ d)
    {r : 𝕜} (hr : r ≠ 0) (s : Set E) : μH[d] (r • s) = ‖r‖₊ ^ d • μH[d] s 
Arguments: {𝕜 : Type u_4} {E : Type u_5} [NormedAddCommGroup E] [NormedField 𝕜] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {d : ℝ} (hd : 0 ≤ d) {r : 𝕜} (hr : r ≠ 0) (s : Set E)
Goal: μH[d] (r • s) = ‖r‖₊ ^ d • μH[d] s
-- AntilipschitzWith.hausdorffMeasure_preimage_le
theorem hausdorffMeasure_preimage_le (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set Y) :
    μH[d] (f ⁻¹' s) ≤ (K : ℝ≥0∞) ^ d * μH[d] s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {K : ℝ≥0} {d : ℝ} (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set Y)
Goal: μH[d] (f ⁻¹' s) ≤ ↑K ^ d * μH[d] s
-- AntilipschitzWith.le_hausdorffMeasure_image
theorem le_hausdorffMeasure_image (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set X) :
    μH[d] s ≤ (K : ℝ≥0∞) ^ d * μH[d] (f '' s) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {K : ℝ≥0} {d : ℝ} (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set X)
Goal: μH[d] s ≤ ↑K ^ d * μH[d] (f '' s)
-- Isometry.hausdorffMeasure_image
theorem hausdorffMeasure_image (hf : Isometry f) (hd : 0 ≤ d ∨ Surjective f) (s : Set X) :
    μH[d] (f '' s) = μH[d] s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℝ} (hf : Isometry f) (hd : 0 ≤ d ∨ Function.Surjective f) (s : Set X)
Goal: μH[d] (f '' s) = μH[d] s
-- Isometry.hausdorffMeasure_preimage
theorem hausdorffMeasure_preimage (hf : Isometry f) (hd : 0 ≤ d ∨ Surjective f) (s : Set Y) :
    μH[d] (f ⁻¹' s) = μH[d] (s ∩ range f) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℝ} (hf : Isometry f) (hd : 0 ≤ d ∨ Function.Surjective f) (s : Set Y)
Goal: μH[d] (f ⁻¹' s) = μH[d] (s ∩ Set.range f)
-- Isometry.map_hausdorffMeasure
theorem map_hausdorffMeasure (hf : Isometry f) (hd : 0 ≤ d ∨ Surjective f) :
    Measure.map f μH[d] = μH[d].restrict (range f) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℝ} (hf : Isometry f) (hd : 0 ≤ d ∨ Function.Surjective f)
Goal: MeasureTheory.Measure.map f μH[d] = μH[d].restrict (Set.range f)
-- IsometryEquiv.hausdorffMeasure_image
@[simp]
theorem hausdorffMeasure_image (e : X ≃ᵢ Y) (d : ℝ) (s : Set X) : μH[d] (e '' s) = μH[d] s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] (e : X ≃ᵢ Y) (d : ℝ) (s : Set X)
Goal: μH[d] (⇑e '' s) = μH[d] s
-- IsometryEquiv.hausdorffMeasure_preimage
@[simp]
theorem hausdorffMeasure_preimage (e : X ≃ᵢ Y) (d : ℝ) (s : Set Y) : μH[d] (e ⁻¹' s) = μH[d] s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] (e : X ≃ᵢ Y) (d : ℝ) (s : Set Y)
Goal: μH[d] (⇑e ⁻¹' s) = μH[d] s
-- IsometryEquiv.map_hausdorffMeasure
@[simp]
theorem map_hausdorffMeasure (e : X ≃ᵢ Y) (d : ℝ) : Measure.map e μH[d] = μH[d] 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] (e : X ≃ᵢ Y) (d : ℝ)
Goal: MeasureTheory.Measure.map ⇑e μH[d] = μH[d]
-- IsometryEquiv.measurePreserving_hausdorffMeasure
theorem measurePreserving_hausdorffMeasure (e : X ≃ᵢ Y) (d : ℝ) : MeasurePreserving e μH[d] μH[d] 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] (e : X ≃ᵢ Y) (d : ℝ)
Goal: MeasureTheory.MeasurePreserving ⇑e μH[d] μH[d]
-- MeasureTheory.hausdorffMeasure_smul
@[to_additive]
theorem hausdorffMeasure_smul {α : Type*} [SMul α X] [IsometricSMul α X] {d : ℝ} (c : α)
    (h : 0 ≤ d ∨ Surjective (c • · : X → X)) (s : Set X) : μH[d] (c • s) = μH[d] s 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {α : Type u_4} [SMul α X] [IsometricSMul α X] {d : ℝ} (c : α) (h : 0 ≤ d ∨ Function.Surjective fun x => c • x) (s : Set X)
Goal: μH[d] (c • s) = μH[d] s
-- MeasureTheory.hausdorffMeasure_vadd
@[to_additive]
theorem hausdorffMeasure_smul {α : Type*} [SMul α X] [IsometricSMul α X] {d : ℝ} (c : α)
    (h : 0 ≤ d ∨ Surjective (c • · : X → X)) (s : Set X) : μH[d] (c • s) = μH[d] s 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {α : Type u_4} [VAdd α X] [IsometricVAdd α X] {d : ℝ} (c : α) (h : 0 ≤ d ∨ Function.Surjective fun x => c +ᵥ x) (s : Set X)
Goal: μH[d] (c +ᵥ s) = μH[d] s
-- MeasureTheory.hausdorffMeasure_pi_real
@[simp]
theorem hausdorffMeasure_pi_real {ι : Type*} [Fintype ι] :
    (μH[Fintype.card ι] : Measure (ι → ℝ)) = volume 
Arguments: {ι : Type u_4} [Fintype ι]
Goal: μH[↑(Fintype.card ι)] = MeasureTheory.volume
-- MeasureTheory.hausdorffMeasure_measurePreserving_funUnique
theorem hausdorffMeasure_measurePreserving_funUnique [Unique ι]
    [SecondCountableTopology X] (d : ℝ) :
    MeasurePreserving (MeasurableEquiv.funUnique ι X) μH[d] μH[d] 
Arguments: (ι : Type u_1) (X : Type u_2) [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] [Unique ι] [SecondCountableTopology X] (d : ℝ)
Goal: MeasureTheory.MeasurePreserving ⇑(MeasurableEquiv.funUnique ι X) μH[d] μH[d]
-- MeasureTheory.hausdorffMeasure_measurePreserving_piFinTwo
theorem hausdorffMeasure_measurePreserving_piFinTwo (α : Fin 2 → Type*)
    [∀ i, MeasurableSpace (α i)] [∀ i, EMetricSpace (α i)] [∀ i, BorelSpace (α i)]
    [∀ i, SecondCountableTopology (α i)] (d : ℝ) :
    MeasurePreserving (MeasurableEquiv.piFinTwo α) μH[d] μH[d] 
Arguments: (α : Fin 2 → Type u_4) [(i : Fin 2) → MeasurableSpace (α i)] [(i : Fin 2) → EMetricSpace (α i)] [∀ (i : Fin 2), BorelSpace (α i)] [∀ (i : Fin 2), SecondCountableTopology (α i)] (d : ℝ)
Goal: MeasureTheory.MeasurePreserving ⇑(MeasurableEquiv.piFinTwo α) μH[d] μH[d]
-- MeasureTheory.hausdorffMeasure_real
@[simp]
theorem hausdorffMeasure_real : (μH[1] : Measure ℝ) = volume 
Goal: μH[1] = MeasureTheory.volume
-- MeasureTheory.hausdorffMeasure_prod_real
@[simp]
theorem hausdorffMeasure_prod_real : (μH[2] : Measure (ℝ × ℝ)) = volume 
Goal: μH[2] = MeasureTheory.volume
-- MeasureTheory.hausdorffMeasure_smul_right_image
theorem hausdorffMeasure_smul_right_image [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] (v : E) (s : Set ℝ) :
    μH[1] ((fun r => r • v) '' s) = ‖v‖₊ • μH[1] s 
Arguments: {E : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (v : E) (s : Set ℝ)
Goal: μH[1] ((fun r => r • v) '' s) = ‖v‖₊ • μH[1] s
-- MeasureTheory.hausdorffMeasure_homothety_image
theorem hausdorffMeasure_homothety_image {d : ℝ} (hd : 0 ≤ d) (x : P) {c : 𝕜} (hc : c ≠ 0)
    (s : Set P) : μH[d] (AffineMap.homothety x c '' s) = ‖c‖₊ ^ d • μH[d] s 
Arguments: {𝕜 : Type u_4} {E : Type u_5} {P : Type u_6} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace P] [MetricSpace P] [NormedAddTorsor E P] [BorelSpace P] {d : ℝ} (hd : 0 ≤ d) (x : P) {c : 𝕜} (hc : c ≠ 0) (s : Set P)
Goal: μH[d] (⇑(AffineMap.homothety x c) '' s) = ‖c‖₊ ^ d • μH[d] s
-- MeasureTheory.hausdorffMeasure_homothety_preimage
theorem hausdorffMeasure_homothety_preimage {d : ℝ} (hd : 0 ≤ d) (x : P) {c : 𝕜} (hc : c ≠ 0)
    (s : Set P) : μH[d] (AffineMap.homothety x c ⁻¹' s) = ‖c‖₊⁻¹ ^ d • μH[d] s 
Arguments: {𝕜 : Type u_4} {E : Type u_5} {P : Type u_6} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace P] [MetricSpace P] [NormedAddTorsor E P] [BorelSpace P] {d : ℝ} (hd : 0 ≤ d) (x : P) {c : 𝕜} (hc : c ≠ 0) (s : Set P)
Goal: μH[d] (⇑(AffineMap.homothety x c) ⁻¹' s) = ‖c‖₊⁻¹ ^ d • μH[d] s
-- MeasureTheory.hausdorffMeasure_lineMap_image
theorem hausdorffMeasure_lineMap_image (x y : P) (s : Set ℝ) :
    μH[1] (AffineMap.lineMap x y '' s) = nndist x y • μH[1] s 
Arguments: {E : Type u_5} {P : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace P] [MetricSpace P] [NormedAddTorsor E P] [BorelSpace P] (x y : P) (s : Set ℝ)
Goal: μH[1] (⇑(AffineMap.lineMap x y) '' s) = nndist x y • μH[1] s
-- MeasureTheory.hausdorffMeasure_affineSegment
@[simp]
theorem hausdorffMeasure_affineSegment (x y : P) : μH[1] (affineSegment ℝ x y) = edist x y 
Arguments: {E : Type u_5} {P : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace P] [MetricSpace P] [NormedAddTorsor E P] [BorelSpace P] (x y : P)
Goal: μH[1] (affineSegment ℝ x y) = edist x y
-- MeasureTheory.hausdorffMeasure_segment
@[simp]
theorem hausdorffMeasure_segment {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] (x y : E) : μH[1] (segment ℝ x y) = edist x y 
Arguments: {E : Type u_7} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (x y : E)
Goal: μH[1] (segment ℝ x y) = edist x y
-- MeasureTheory.ProbabilityMeasure.coe_mk
@[simp, norm_cast] lemma coe_mk (μ : Measure Ω) (hμ) : toMeasure ⟨μ, hμ⟩ = μ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (hμ : MeasureTheory.IsProbabilityMeasure μ)
Goal: ↑⟨μ, hμ⟩ = μ
-- MeasureTheory.ProbabilityMeasure.val_eq_to_measure
@[simp]
theorem val_eq_to_measure (ν : ProbabilityMeasure Ω) : ν.val = (ν : Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.ProbabilityMeasure Ω)
Goal: ↑ν = ↑ν
-- MeasureTheory.ProbabilityMeasure.toMeasure_injective
theorem toMeasure_injective : Function.Injective ((↑) : ProbabilityMeasure Ω → Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω]
Goal: Function.Injective MeasureTheory.ProbabilityMeasure.toMeasure
-- MeasureTheory.ProbabilityMeasure.coeFn_def
lemma coeFn_def (μ : ProbabilityMeasure Ω) : μ = fun s ↦ ((μ : Measure Ω) s).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: ⇑μ = fun s => (↑μ s).toNNReal
-- MeasureTheory.ProbabilityMeasure.coeFn_mk
lemma coeFn_mk (μ : Measure Ω) (hμ) :
    DFunLike.coe (F := ProbabilityMeasure Ω) ⟨μ, hμ⟩ = fun s ↦ (μ s).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (hμ : MeasureTheory.IsProbabilityMeasure μ)
Goal: ⇑⟨μ, hμ⟩ = fun s => (μ s).toNNReal
-- MeasureTheory.ProbabilityMeasure.mk_apply
@[simp, norm_cast]
lemma mk_apply (μ : Measure Ω) (hμ) (s : Set Ω) :
    DFunLike.coe (F := ProbabilityMeasure Ω) ⟨μ, hμ⟩ s = (μ s).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (hμ : MeasureTheory.IsProbabilityMeasure μ) (s : Set Ω)
Goal: ⟨μ, hμ⟩ s = (μ s).toNNReal
-- MeasureTheory.ProbabilityMeasure.coeFn_univ
@[simp, norm_cast]
theorem coeFn_univ (ν : ProbabilityMeasure Ω) : ν univ = 1 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.ProbabilityMeasure Ω)
Goal: ν Set.univ = 1
-- MeasureTheory.ProbabilityMeasure.coeFn_univ_ne_zero
theorem coeFn_univ_ne_zero (ν : ProbabilityMeasure Ω) : ν univ ≠ 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.ProbabilityMeasure Ω)
Goal: ν Set.univ ≠ 0
-- MeasureTheory.ProbabilityMeasure.coeFn_toFiniteMeasure
@[simp] lemma coeFn_toFiniteMeasure (μ : ProbabilityMeasure Ω) : ⇑μ.toFiniteMeasure = μ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: ⇑μ.toFiniteMeasure = ⇑μ
-- MeasureTheory.ProbabilityMeasure.toFiniteMeasure_apply
lemma toFiniteMeasure_apply (μ : ProbabilityMeasure Ω) (s : Set Ω) :
    μ.toFiniteMeasure s = μ s 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω) (s : Set Ω)
Goal: μ.toFiniteMeasure s = μ s
-- MeasureTheory.ProbabilityMeasure.toMeasure_comp_toFiniteMeasure_eq_toMeasure
@[simp]
theorem toMeasure_comp_toFiniteMeasure_eq_toMeasure (ν : ProbabilityMeasure Ω) :
    (ν.toFiniteMeasure : Measure Ω) = (ν : Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.ProbabilityMeasure Ω)
Goal: ↑ν.toFiniteMeasure = ↑ν
-- MeasureTheory.ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn
@[simp]
theorem coeFn_comp_toFiniteMeasure_eq_coeFn (ν : ProbabilityMeasure Ω) :
    (ν.toFiniteMeasure : Set Ω → ℝ≥0) = (ν : Set Ω → ℝ≥0) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.ProbabilityMeasure Ω)
Goal: ⇑ν.toFiniteMeasure = ⇑ν
-- MeasureTheory.ProbabilityMeasure.toFiniteMeasure_apply_eq_apply
@[simp]
theorem toFiniteMeasure_apply_eq_apply (ν : ProbabilityMeasure Ω) (s : Set Ω) :
    ν.toFiniteMeasure s = ν s 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.ProbabilityMeasure Ω) (s : Set Ω)
Goal: ν.toFiniteMeasure s = ν s
-- MeasureTheory.ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure
@[simp]
theorem ennreal_coeFn_eq_coeFn_toMeasure (ν : ProbabilityMeasure Ω) (s : Set Ω) :
    (ν s : ℝ≥0∞) = (ν : Measure Ω) s 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.ProbabilityMeasure Ω) (s : Set Ω)
Goal: ↑(ν s) = ↑ν s
-- MeasureTheory.ProbabilityMeasure.apply_mono
theorem apply_mono (μ : ProbabilityMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂)
Goal: μ s₁ ≤ μ s₂
-- MeasureTheory.ProbabilityMeasure.apply_le_one
@[simp] theorem apply_le_one (μ : ProbabilityMeasure Ω) (s : Set Ω) : μ s ≤ 1 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω) (s : Set Ω)
Goal: μ s ≤ 1
-- MeasureTheory.ProbabilityMeasure.nonempty
theorem nonempty (μ : ProbabilityMeasure Ω) : Nonempty Ω 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: Nonempty Ω
-- MeasureTheory.ProbabilityMeasure.eq_of_forall_toMeasure_apply_eq
@[ext]
theorem eq_of_forall_toMeasure_apply_eq (μ ν : ProbabilityMeasure Ω)
    (h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ ν : MeasureTheory.ProbabilityMeasure Ω) (h : ∀ (s : Set Ω), MeasurableSet s → ↑μ s = ↑ν s)
Goal: μ = ν
-- MeasureTheory.ProbabilityMeasure.eq_of_forall_apply_eq
theorem eq_of_forall_apply_eq (μ ν : ProbabilityMeasure Ω)
    (h : ∀ s : Set Ω, MeasurableSet s → μ s = ν s) : μ = ν 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ ν : MeasureTheory.ProbabilityMeasure Ω) (h : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s)
Goal: μ = ν
-- MeasureTheory.ProbabilityMeasure.mass_toFiniteMeasure
@[simp]
theorem mass_toFiniteMeasure (μ : ProbabilityMeasure Ω) : μ.toFiniteMeasure.mass = 1 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: μ.toFiniteMeasure.mass = 1
-- MeasureTheory.ProbabilityMeasure.toFiniteMeasure_nonzero
theorem toFiniteMeasure_nonzero (μ : ProbabilityMeasure Ω) : μ.toFiniteMeasure ≠ 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: μ.toFiniteMeasure ≠ 0
-- MeasureTheory.ProbabilityMeasure.testAgainstNN_lipschitz
theorem testAgainstNN_lipschitz (μ : ProbabilityMeasure Ω) :
    LipschitzWith 1 fun f : Ω →ᵇ ℝ≥0 => μ.toFiniteMeasure.testAgainstNN f 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: LipschitzWith 1 fun f => μ.toFiniteMeasure.testAgainstNN f
-- MeasureTheory.ProbabilityMeasure.toFiniteMeasure_continuous
theorem toFiniteMeasure_continuous :
    Continuous (toFiniteMeasure : ProbabilityMeasure Ω → FiniteMeasure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω]
Goal: Continuous MeasureTheory.ProbabilityMeasure.toFiniteMeasure
-- MeasureTheory.ProbabilityMeasure.coe_toWeakDualBCNN
@[simp]
theorem coe_toWeakDualBCNN (μ : ProbabilityMeasure Ω) :
    ⇑μ.toWeakDualBCNN = μ.toFiniteMeasure.testAgainstNN 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: ⇑μ.toWeakDualBCNN = μ.toFiniteMeasure.testAgainstNN
-- MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_apply
@[simp]
theorem toWeakDualBCNN_apply (μ : ProbabilityMeasure Ω) (f : Ω →ᵇ ℝ≥0) :
    μ.toWeakDualBCNN f = (∫⁻ ω, f ω ∂(μ : Measure Ω)).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.ProbabilityMeasure Ω) (f : Ω →ᵇ ℝ≥0)
Goal: μ.toWeakDualBCNN f = (∫⁻ (ω : Ω), ↑(f ω) ∂↑μ).toNNReal
-- MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_continuous
theorem toWeakDualBCNN_continuous : Continuous fun μ : ProbabilityMeasure Ω => μ.toWeakDualBCNN 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω]
Goal: Continuous fun μ => μ.toWeakDualBCNN
-- MeasureTheory.ProbabilityMeasure.continuous_testAgainstNN_eval
theorem continuous_testAgainstNN_eval (f : Ω →ᵇ ℝ≥0) :
    Continuous fun μ : ProbabilityMeasure Ω => μ.toFiniteMeasure.testAgainstNN f 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (f : Ω →ᵇ ℝ≥0)
Goal: Continuous fun μ => μ.toFiniteMeasure.testAgainstNN f
-- MeasureTheory.ProbabilityMeasure.toFiniteMeasure_embedding
theorem toFiniteMeasure_embedding (Ω : Type*) [MeasurableSpace Ω] [TopologicalSpace Ω]
    [OpensMeasurableSpace Ω] :
    Embedding (toFiniteMeasure : ProbabilityMeasure Ω → FiniteMeasure Ω) 
Arguments: (Ω : Type u_2) [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω]
Goal: Embedding MeasureTheory.ProbabilityMeasure.toFiniteMeasure
-- MeasureTheory.ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds
theorem tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds {δ : Type*} (F : Filter δ)
    {μs : δ → ProbabilityMeasure Ω} {μ₀ : ProbabilityMeasure Ω} :
    Tendsto μs F (𝓝 μ₀) ↔ Tendsto (toFiniteMeasure ∘ μs) F (𝓝 μ₀.toFiniteMeasure) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {δ : Type u_2} (F : Filter δ) {μs : δ → MeasureTheory.ProbabilityMeasure Ω} {μ₀ : MeasureTheory.ProbabilityMeasure Ω}
Goal: Filter.Tendsto μs F (𝓝 μ₀) ↔
  Filter.Tendsto (MeasureTheory.ProbabilityMeasure.toFiniteMeasure ∘ μs) F (𝓝 μ₀.toFiniteMeasure)