ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.SeparableSpace.exists_measurable_partition_diam_le
lemma SeparableSpace.exists_measurable_partition_diam_le {ε : ℝ} (ε_pos : 0 < ε) :
    ∃ (As : ℕ → Set Ω), (∀ n, MeasurableSet (As n)) ∧ (∀ n, Bornology.IsBounded (As n)) ∧
        (∀ n, diam (As n) ≤ ε) ∧ (⋃ n, As n = univ) ∧
        (Pairwise (fun (n m : ℕ) ↦ Disjoint (As n) (As m))) 
Arguments: (Ω : Type u_2) [PseudoMetricSpace Ω] [TopologicalSpace.SeparableSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] {ε : ℝ} (ε_pos : 0 < ε)
Goal: ∃ As,
  (∀ (n : ℕ), MeasurableSet (As n)) ∧
    (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
      (∀ (n : ℕ), Metric.diam (As n) ≤ ε) ∧ ⋃ n, As n = Set.univ ∧ Pairwise fun n m => Disjoint (As n) (As m)
-- MeasureTheory.ProbabilityMeasure.toMeasure_add_pos_gt_mem_nhds
lemma ProbabilityMeasure.toMeasure_add_pos_gt_mem_nhds (P : ProbabilityMeasure Ω)
    {G : Set Ω} (G_open : IsOpen G) {ε : ℝ≥0∞} (ε_pos : 0 < ε) :
    {Q | P.toMeasure G < Q.toMeasure G + ε} ∈ 𝓝 P 
Arguments: {Ω : Type u_2} [PseudoMetricSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] (P : MeasureTheory.ProbabilityMeasure Ω) {G : Set Ω} (G_open : IsOpen G) {ε : ℝ≥0∞} (ε_pos : 0 < ε)
Goal: {Q | ↑P G < ↑Q G + ε} ∈ 𝓝 P
-- MeasureTheory.ProbabilityMeasure.continuous_toLevyProkhorov
lemma ProbabilityMeasure.continuous_toLevyProkhorov :
    Continuous (ProbabilityMeasure.toLevyProkhorov (Ω := Ω)) 
Arguments: {Ω : Type u_2} [PseudoMetricSpace Ω] [TopologicalSpace.SeparableSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω]
Goal: Continuous MeasureTheory.ProbabilityMeasure.toLevyProkhorov
-- MeasureTheory.levyProkhorov_eq_convergenceInDistribution
theorem levyProkhorov_eq_convergenceInDistribution :
    (inferInstance : TopologicalSpace (ProbabilityMeasure Ω))
      = TopologicalSpace.coinduced (LevyProkhorov.toProbabilityMeasure (Ω := Ω)) inferInstance 
Arguments: {Ω : Type u_2} [PseudoMetricSpace Ω] [TopologicalSpace.SeparableSpace Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω]
Goal: inferInstance = TopologicalSpace.coinduced MeasureTheory.LevyProkhorov.toProbabilityMeasure inferInstance
-- AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux
theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f : E → F} {U : Set E}
    (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
    EqOn f 0 U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0)
Goal: Set.EqOn f 0 U
-- AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero
theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero {f : E → F} {U : Set E}
    (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
    EqOn f 0 U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0)
Goal: Set.EqOn f 0 U
-- AnalyticOn.eqOn_of_preconnected_of_eventuallyEq
theorem eqOn_of_preconnected_of_eventuallyEq {f g : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U)
    (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[𝓝 z₀] g) :
    EqOn f g U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[𝓝 z₀] g)
Goal: Set.EqOn f g U
-- AnalyticOn.eq_of_eventuallyEq
theorem eq_of_eventuallyEq {f g : E → F} [PreconnectedSpace E] (hf : AnalyticOn 𝕜 f univ)
    (hg : AnalyticOn 𝕜 g univ) {z₀ : E} (hfg : f =ᶠ[𝓝 z₀] g) : f = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} [PreconnectedSpace E] (hf : AnalyticOn 𝕜 f Set.univ) (hg : AnalyticOn 𝕜 g Set.univ) {z₀ : E} (hfg : f =ᶠ[𝓝 z₀] g)
Goal: f = g
-- MeasureTheory.JordanDecomposition.mutuallySingular
@[ext]
structure JordanDecomposition (α : Type*) [MeasurableSpace α] where
  (posPart negPart : Measure α)
  [posPart_finite : IsFiniteMeasure posPart]
  [negPart_finite : IsFiniteMeasure negPart]
  mutuallySingular : posPart ⟂ₘ negPart
Arguments: {α : Type u_3} [MeasurableSpace α] (self : MeasureTheory.JordanDecomposition α)
Goal: self.posPart ⟂ₘ self.negPart
-- MeasureTheory.JordanDecomposition.negPart_finite
@[ext]
structure JordanDecomposition (α : Type*) [MeasurableSpace α] where
  (posPart negPart : Measure α)
  [posPart_finite : IsFiniteMeasure posPart]
  [negPart_finite : IsFiniteMeasure negPart]
  mutuallySingular : posPart ⟂ₘ negPart
Arguments: {α : Type u_3} [MeasurableSpace α] (self : MeasureTheory.JordanDecomposition α)
Goal: MeasureTheory.IsFiniteMeasure self.negPart
-- MeasureTheory.JordanDecomposition.posPart_finite
@[ext]
structure JordanDecomposition (α : Type*) [MeasurableSpace α] where
  (posPart negPart : Measure α)
  [posPart_finite : IsFiniteMeasure posPart]
  [negPart_finite : IsFiniteMeasure negPart]
  mutuallySingular : posPart ⟂ₘ negPart
Arguments: {α : Type u_3} [MeasurableSpace α] (self : MeasureTheory.JordanDecomposition α)
Goal: MeasureTheory.IsFiniteMeasure self.posPart
-- MeasureTheory.JordanDecomposition.zero_posPart
@[simp]
theorem zero_posPart : (0 : JordanDecomposition α).posPart = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.JordanDecomposition.posPart 0 = 0
-- MeasureTheory.JordanDecomposition.zero_negPart
@[simp]
theorem zero_negPart : (0 : JordanDecomposition α).negPart = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.JordanDecomposition.negPart 0 = 0
-- MeasureTheory.JordanDecomposition.neg_posPart
@[simp]
theorem neg_posPart : (-j).posPart = j.negPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α)
Goal: (-j).posPart = j.negPart
-- MeasureTheory.JordanDecomposition.neg_negPart
@[simp]
theorem neg_negPart : (-j).negPart = j.posPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α)
Goal: (-j).negPart = j.posPart
-- MeasureTheory.JordanDecomposition.smul_posPart
@[simp]
theorem smul_posPart (r : ℝ≥0) : (r • j).posPart = r • j.posPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ≥0)
Goal: (r • j).posPart = r • j.posPart
-- MeasureTheory.JordanDecomposition.smul_negPart
@[simp]
theorem smul_negPart (r : ℝ≥0) : (r • j).negPart = r • j.negPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ≥0)
Goal: (r • j).negPart = r • j.negPart
-- MeasureTheory.JordanDecomposition.real_smul_def
theorem real_smul_def (r : ℝ) (j : JordanDecomposition α) :
    r • j = if 0 ≤ r then r.toNNReal • j else -((-r).toNNReal • j) 
Arguments: {α : Type u_1} [MeasurableSpace α] (r : ℝ) (j : MeasureTheory.JordanDecomposition α)
Goal: r • j = if 0 ≤ r then r.toNNReal • j else -((-r).toNNReal • j)
-- MeasureTheory.JordanDecomposition.coe_smul
@[simp]
theorem coe_smul (r : ℝ≥0) : (r : ℝ) • j = r • j 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ≥0)
Goal: ↑r • j = r • j
-- MeasureTheory.JordanDecomposition.real_smul_nonneg
theorem real_smul_nonneg (r : ℝ) (hr : 0 ≤ r) : r • j = r.toNNReal • j 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ) (hr : 0 ≤ r)
Goal: r • j = r.toNNReal • j
-- MeasureTheory.JordanDecomposition.real_smul_neg
theorem real_smul_neg (r : ℝ) (hr : r < 0) : r • j = -((-r).toNNReal • j) 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ) (hr : r < 0)
Goal: r • j = -((-r).toNNReal • j)
-- MeasureTheory.JordanDecomposition.real_smul_posPart_nonneg
theorem real_smul_posPart_nonneg (r : ℝ) (hr : 0 ≤ r) :
    (r • j).posPart = r.toNNReal • j.posPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ) (hr : 0 ≤ r)
Goal: (r • j).posPart = r.toNNReal • j.posPart
-- MeasureTheory.JordanDecomposition.real_smul_negPart_nonneg
theorem real_smul_negPart_nonneg (r : ℝ) (hr : 0 ≤ r) :
    (r • j).negPart = r.toNNReal • j.negPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ) (hr : 0 ≤ r)
Goal: (r • j).negPart = r.toNNReal • j.negPart
-- MeasureTheory.JordanDecomposition.real_smul_posPart_neg
theorem real_smul_posPart_neg (r : ℝ) (hr : r < 0) :
    (r • j).posPart = (-r).toNNReal • j.negPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ) (hr : r < 0)
Goal: (r • j).posPart = (-r).toNNReal • j.negPart
-- MeasureTheory.JordanDecomposition.real_smul_negPart_neg
theorem real_smul_negPart_neg (r : ℝ) (hr : r < 0) :
    (r • j).negPart = (-r).toNNReal • j.posPart 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ) (hr : r < 0)
Goal: (r • j).negPart = (-r).toNNReal • j.posPart
-- MeasureTheory.JordanDecomposition.toSignedMeasure_zero
theorem toSignedMeasure_zero : (0 : JordanDecomposition α).toSignedMeasure = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.JordanDecomposition.toSignedMeasure 0 = 0
-- MeasureTheory.JordanDecomposition.toSignedMeasure_neg
theorem toSignedMeasure_neg : (-j).toSignedMeasure = -j.toSignedMeasure 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α)
Goal: (-j).toSignedMeasure = -j.toSignedMeasure
-- MeasureTheory.JordanDecomposition.toSignedMeasure_smul
theorem toSignedMeasure_smul (r : ℝ≥0) : (r • j).toSignedMeasure = r • j.toSignedMeasure 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α) (r : ℝ≥0)
Goal: (r • j).toSignedMeasure = r • j.toSignedMeasure
-- MeasureTheory.JordanDecomposition.exists_compl_positive_negative
theorem exists_compl_positive_negative :
    ∃ S : Set α,
      MeasurableSet S ∧
        j.toSignedMeasure ≤[S] 0 ∧
          0 ≤[Sᶜ] j.toSignedMeasure ∧ j.posPart S = 0 ∧ j.negPart Sᶜ = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α)
Goal: ∃ S,
  MeasurableSet S ∧
    MeasureTheory.VectorMeasure.restrict j.toSignedMeasure S ≤ MeasureTheory.VectorMeasure.restrict 0 S ∧
      MeasureTheory.VectorMeasure.restrict 0 Sᶜ ≤ MeasureTheory.VectorMeasure.restrict j.toSignedMeasure Sᶜ ∧
        j.posPart S = 0 ∧ j.negPart Sᶜ = 0
-- MeasureTheory.SignedMeasure.toJordanDecomposition_spec
theorem toJordanDecomposition_spec (s : SignedMeasure α) :
    ∃ (i : Set α) (hi₁ : MeasurableSet i) (hi₂ : 0 ≤[i] s) (hi₃ : s ≤[iᶜ] 0),
      s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂ ∧
        s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ hi₁.compl hi₃ 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α)
Goal: ∃ i,
  ∃ (hi₁ : MeasurableSet i) (hi₂ : MeasureTheory.VectorMeasure.restrict 0 i ≤ MeasureTheory.VectorMeasure.restrict s i)
    (hi₃ : MeasureTheory.VectorMeasure.restrict s iᶜ ≤ MeasureTheory.VectorMeasure.restrict 0 iᶜ),
    s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂ ∧
      s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃
-- MeasureTheory.SignedMeasure.toSignedMeasure_toJordanDecomposition
@[simp]
theorem toSignedMeasure_toJordanDecomposition (s : SignedMeasure α) :
    s.toJordanDecomposition.toSignedMeasure = s 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α)
Goal: s.toJordanDecomposition.toSignedMeasure = s
-- MeasureTheory.SignedMeasure.subset_positive_null_set
theorem subset_positive_null_set (hu : MeasurableSet u) (hv : MeasurableSet v)
    (hw : MeasurableSet w) (hsu : 0 ≤[u] s) (hw₁ : s w = 0) (hw₂ : w ⊆ u) (hwt : v ⊆ w) :
    s v = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {u v w : Set α} (hu : MeasurableSet u) (hv : MeasurableSet v) (hw : MeasurableSet w) (hsu : MeasureTheory.VectorMeasure.restrict 0 u ≤ MeasureTheory.VectorMeasure.restrict s u) (hw₁ : ↑s w = 0) (hw₂ : w ⊆ u) (hwt : v ⊆ w)
Goal: ↑s v = 0
-- MeasureTheory.SignedMeasure.subset_negative_null_set
theorem subset_negative_null_set (hu : MeasurableSet u) (hv : MeasurableSet v)
    (hw : MeasurableSet w) (hsu : s ≤[u] 0) (hw₁ : s w = 0) (hw₂ : w ⊆ u) (hwt : v ⊆ w) :
    s v = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {u v w : Set α} (hu : MeasurableSet u) (hv : MeasurableSet v) (hw : MeasurableSet w) (hsu : MeasureTheory.VectorMeasure.restrict s u ≤ MeasureTheory.VectorMeasure.restrict 0 u) (hw₁ : ↑s w = 0) (hw₂ : w ⊆ u) (hwt : v ⊆ w)
Goal: ↑s v = 0
-- MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive
theorem of_diff_eq_zero_of_symmDiff_eq_zero_positive (hu : MeasurableSet u) (hv : MeasurableSet v)
    (hsu : 0 ≤[u] s) (hsv : 0 ≤[v] s) (hs : s (u ∆ v) = 0) : s (u \ v) = 0 ∧ s (v \ u) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {u v : Set α} (hu : MeasurableSet u) (hv : MeasurableSet v) (hsu : MeasureTheory.VectorMeasure.restrict 0 u ≤ MeasureTheory.VectorMeasure.restrict s u) (hsv : MeasureTheory.VectorMeasure.restrict 0 v ≤ MeasureTheory.VectorMeasure.restrict s v) (hs : ↑s (symmDiff u v) = 0)
Goal: ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
-- MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_negative
theorem of_diff_eq_zero_of_symmDiff_eq_zero_negative (hu : MeasurableSet u) (hv : MeasurableSet v)
    (hsu : s ≤[u] 0) (hsv : s ≤[v] 0) (hs : s (u ∆ v) = 0) : s (u \ v) = 0 ∧ s (v \ u) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {u v : Set α} (hu : MeasurableSet u) (hv : MeasurableSet v) (hsu : MeasureTheory.VectorMeasure.restrict s u ≤ MeasureTheory.VectorMeasure.restrict 0 u) (hsv : MeasureTheory.VectorMeasure.restrict s v ≤ MeasureTheory.VectorMeasure.restrict 0 v) (hs : ↑s (symmDiff u v) = 0)
Goal: ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
-- MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive
theorem of_inter_eq_of_symmDiff_eq_zero_positive (hu : MeasurableSet u) (hv : MeasurableSet v)
    (hw : MeasurableSet w) (hsu : 0 ≤[u] s) (hsv : 0 ≤[v] s) (hs : s (u ∆ v) = 0) :
    s (w ∩ u) = s (w ∩ v) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {u v w : Set α} (hu : MeasurableSet u) (hv : MeasurableSet v) (hw : MeasurableSet w) (hsu : MeasureTheory.VectorMeasure.restrict 0 u ≤ MeasureTheory.VectorMeasure.restrict s u) (hsv : MeasureTheory.VectorMeasure.restrict 0 v ≤ MeasureTheory.VectorMeasure.restrict s v) (hs : ↑s (symmDiff u v) = 0)
Goal: ↑s (w ∩ u) = ↑s (w ∩ v)
-- MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_negative
theorem of_inter_eq_of_symmDiff_eq_zero_negative (hu : MeasurableSet u) (hv : MeasurableSet v)
    (hw : MeasurableSet w) (hsu : s ≤[u] 0) (hsv : s ≤[v] 0) (hs : s (u ∆ v) = 0) :
    s (w ∩ u) = s (w ∩ v) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {u v w : Set α} (hu : MeasurableSet u) (hv : MeasurableSet v) (hw : MeasurableSet w) (hsu : MeasureTheory.VectorMeasure.restrict s u ≤ MeasureTheory.VectorMeasure.restrict 0 u) (hsv : MeasureTheory.VectorMeasure.restrict s v ≤ MeasureTheory.VectorMeasure.restrict 0 v) (hs : ↑s (symmDiff u v) = 0)
Goal: ↑s (w ∩ u) = ↑s (w ∩ v)
-- _private.Mathlib.MeasureTheory.Decomposition.Jordan.0.MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart
private theorem eq_of_posPart_eq_posPart {j₁ j₂ : JordanDecomposition α}
    (hj : j₁.posPart = j₂.posPart) (hj' : j₁.toSignedMeasure = j₂.toSignedMeasure) : j₁ = j₂ 
Arguments: {α : Type u_1} [MeasurableSpace α] {j₁ j₂ : MeasureTheory.JordanDecomposition α} (hj : j₁.posPart = j₂.posPart) (hj' : j₁.toSignedMeasure = j₂.toSignedMeasure)
Goal: j₁ = j₂
-- MeasureTheory.JordanDecomposition.toSignedMeasure_injective
theorem toSignedMeasure_injective : Injective <| @JordanDecomposition.toSignedMeasure α _ 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: Function.Injective MeasureTheory.JordanDecomposition.toSignedMeasure
-- MeasureTheory.JordanDecomposition.toJordanDecomposition_toSignedMeasure
@[simp]
theorem toJordanDecomposition_toSignedMeasure (j : JordanDecomposition α) :
    j.toSignedMeasure.toJordanDecomposition = j 
Arguments: {α : Type u_1} [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α)
Goal: j.toSignedMeasure.toJordanDecomposition = j
-- MeasureTheory.SignedMeasure.toJordanDecompositionEquiv_symm_apply
@[simps apply symm_apply]
def toJordanDecompositionEquiv (α : Type*) [MeasurableSpace α] :
    SignedMeasure α ≃ JordanDecomposition α 
Arguments: (α : Type u_3) [MeasurableSpace α] (j : MeasureTheory.JordanDecomposition α)
Goal: (MeasureTheory.SignedMeasure.toJordanDecompositionEquiv α).symm j = j.toSignedMeasure
-- MeasureTheory.SignedMeasure.toJordanDecompositionEquiv_apply
@[simps apply symm_apply]
def toJordanDecompositionEquiv (α : Type*) [MeasurableSpace α] :
    SignedMeasure α ≃ JordanDecomposition α 
Arguments: (α : Type u_3) [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α)
Goal: (MeasureTheory.SignedMeasure.toJordanDecompositionEquiv α) s = s.toJordanDecomposition
-- MeasureTheory.SignedMeasure.toJordanDecomposition_zero
theorem toJordanDecomposition_zero : (0 : SignedMeasure α).toJordanDecomposition = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.SignedMeasure.toJordanDecomposition 0 = 0
-- MeasureTheory.SignedMeasure.toJordanDecomposition_neg
theorem toJordanDecomposition_neg (s : SignedMeasure α) :
    (-s).toJordanDecomposition = -s.toJordanDecomposition 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α)
Goal: (-s).toJordanDecomposition = -s.toJordanDecomposition
-- MeasureTheory.SignedMeasure.toJordanDecomposition_smul
theorem toJordanDecomposition_smul (s : SignedMeasure α) (r : ℝ≥0) :
    (r • s).toJordanDecomposition = r • s.toJordanDecomposition 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) (r : ℝ≥0)
Goal: (r • s).toJordanDecomposition = r • s.toJordanDecomposition
-- _private.Mathlib.MeasureTheory.Decomposition.Jordan.0.MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real_nonneg
private theorem toJordanDecomposition_smul_real_nonneg (s : SignedMeasure α) (r : ℝ)
    (hr : 0 ≤ r) : (r • s).toJordanDecomposition = r • s.toJordanDecomposition 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) (r : ℝ) (hr : 0 ≤ r)
Goal: (r • s).toJordanDecomposition = r • s.toJordanDecomposition
-- MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real
theorem toJordanDecomposition_smul_real (s : SignedMeasure α) (r : ℝ) :
    (r • s).toJordanDecomposition = r • s.toJordanDecomposition 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) (r : ℝ)
Goal: (r • s).toJordanDecomposition = r • s.toJordanDecomposition
-- MeasureTheory.SignedMeasure.toJordanDecomposition_eq
theorem toJordanDecomposition_eq {s : SignedMeasure α} {j : JordanDecomposition α}
    (h : s = j.toSignedMeasure) : s.toJordanDecomposition = j 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {j : MeasureTheory.JordanDecomposition α} (h : s = j.toSignedMeasure)
Goal: s.toJordanDecomposition = j
-- MeasureTheory.SignedMeasure.totalVariation_zero
theorem totalVariation_zero : (0 : SignedMeasure α).totalVariation = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.SignedMeasure.totalVariation 0 = 0
-- MeasureTheory.SignedMeasure.totalVariation_neg
theorem totalVariation_neg (s : SignedMeasure α) : (-s).totalVariation = s.totalVariation 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α)
Goal: (-s).totalVariation = s.totalVariation
-- MeasureTheory.SignedMeasure.null_of_totalVariation_zero
theorem null_of_totalVariation_zero (s : SignedMeasure α) {i : Set α}
    (hs : s.totalVariation i = 0) : s i = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) {i : Set α} (hs : s.totalVariation i = 0)
Goal: ↑s i = 0
-- MeasureTheory.SignedMeasure.absolutelyContinuous_ennreal_iff
theorem absolutelyContinuous_ennreal_iff (s : SignedMeasure α) (μ : VectorMeasure α ℝ≥0∞) :
    s ≪ᵥ μ ↔ s.totalVariation ≪ μ.ennrealToMeasure 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.VectorMeasure α ℝ≥0∞)
Goal: s ≪ᵥ μ ↔ s.totalVariation ≪ μ.ennrealToMeasure
-- MeasureTheory.SignedMeasure.totalVariation_absolutelyContinuous_iff
theorem totalVariation_absolutelyContinuous_iff (s : SignedMeasure α) (μ : Measure α) :
    s.totalVariation ≪ μ ↔
      s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: s.totalVariation ≪ μ ↔ s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ
-- MeasureTheory.SignedMeasure.mutuallySingular_iff
theorem mutuallySingular_iff (s t : SignedMeasure α) :
    s ⟂ᵥ t ↔ s.totalVariation ⟂ₘ t.totalVariation 
Arguments: {α : Type u_1} [MeasurableSpace α] (s t : MeasureTheory.SignedMeasure α)
Goal: s ⟂ᵥ t ↔ s.totalVariation ⟂ₘ t.totalVariation
-- MeasureTheory.SignedMeasure.mutuallySingular_ennreal_iff
theorem mutuallySingular_ennreal_iff (s : SignedMeasure α) (μ : VectorMeasure α ℝ≥0∞) :
    s ⟂ᵥ μ ↔ s.totalVariation ⟂ₘ μ.ennrealToMeasure 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.VectorMeasure α ℝ≥0∞)
Goal: s ⟂ᵥ μ ↔ s.totalVariation ⟂ₘ μ.ennrealToMeasure
-- MeasureTheory.SignedMeasure.totalVariation_mutuallySingular_iff
theorem totalVariation_mutuallySingular_iff (s : SignedMeasure α) (μ : Measure α) :
    s.totalVariation ⟂ₘ μ ↔
      s.toJordanDecomposition.posPart ⟂ₘ μ ∧ s.toJordanDecomposition.negPart ⟂ₘ μ 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: s.totalVariation ⟂ₘ μ ↔ s.toJordanDecomposition.posPart ⟂ₘ μ ∧ s.toJordanDecomposition.negPart ⟂ₘ μ
-- MeasureTheory.SignedMeasure.HaveLebesgueDecomposition.posPart
class HaveLebesgueDecomposition (s : SignedMeasure α) (μ : Measure α) : Prop where
  posPart : s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ
  negPart : s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : MeasureTheory.SignedMeasure α} {μ : MeasureTheory.Measure α} [self : s.HaveLebesgueDecomposition μ]
Goal: s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ
-- MeasureTheory.SignedMeasure.HaveLebesgueDecomposition.negPart
class HaveLebesgueDecomposition (s : SignedMeasure α) (μ : Measure α) : Prop where
  posPart : s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ
  negPart : s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : MeasureTheory.SignedMeasure α} {μ : MeasureTheory.Measure α} [self : s.HaveLebesgueDecomposition μ]
Goal: s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ
-- MeasureTheory.SignedMeasure.not_haveLebesgueDecomposition_iff
theorem not_haveLebesgueDecomposition_iff (s : SignedMeasure α) (μ : Measure α) :
    ¬s.HaveLebesgueDecomposition μ ↔
      ¬s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ ∨
        ¬s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: ¬s.HaveLebesgueDecomposition μ ↔
  ¬s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ ∨
    ¬s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ