ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.SignedMeasure.singularPart_mutuallySingular
theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
    s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
      s.toJordanDecomposition.negPart.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ s.toJordanDecomposition.negPart.singularPart μ
-- MeasureTheory.SignedMeasure.singularPart_totalVariation
theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
    (s.singularPart μ).totalVariation =
      s.toJordanDecomposition.posPart.singularPart μ +
        s.toJordanDecomposition.negPart.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: (s.singularPart μ).totalVariation =
  s.toJordanDecomposition.posPart.singularPart μ + s.toJordanDecomposition.negPart.singularPart μ
-- MeasureTheory.SignedMeasure.mutuallySingular_singularPart
nonrec theorem mutuallySingular_singularPart (s : SignedMeasure α) (μ : Measure α) :
    singularPart s μ ⟂ᵥ μ.toENNRealVectorMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: s.singularPart μ ⟂ᵥ μ.toENNRealVectorMeasure
-- MeasureTheory.SignedMeasure.rnDeriv_def
theorem rnDeriv_def (s : SignedMeasure α) (μ : Measure α) : rnDeriv s μ = fun x =>
    (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal -
      (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: s.rnDeriv μ = fun x =>
  (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal - (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal
-- MeasureTheory.SignedMeasure.measurable_rnDeriv
@[measurability]
theorem measurable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Measurable (rnDeriv s μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: Measurable (s.rnDeriv μ)
-- MeasureTheory.SignedMeasure.integrable_rnDeriv
theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Integrable (s.rnDeriv μ) μ
-- MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq
theorem singularPart_add_withDensity_rnDeriv_eq [s.HaveLebesgueDecomposition μ] :
    s.singularPart μ + μ.withDensityᵥ (s.rnDeriv μ) = s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : MeasureTheory.SignedMeasure α) [s.HaveLebesgueDecomposition μ]
Goal: s.singularPart μ + μ.withDensityᵥ (s.rnDeriv μ) = s
-- MeasureTheory.SignedMeasure.jordanDecomposition_add_withDensity_mutuallySingular
theorem jordanDecomposition_add_withDensity_mutuallySingular {f : α → ℝ} (hf : Measurable f)
    (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) :
    (t.toJordanDecomposition.posPart + μ.withDensity fun x : α => ENNReal.ofReal (f x)) ⟂ₘ
      t.toJordanDecomposition.negPart + μ.withDensity fun x : α => ENNReal.ofReal (-f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : MeasureTheory.SignedMeasure α} {f : α → ℝ} (hf : Measurable f) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure)
Goal: (t.toJordanDecomposition.posPart + μ.withDensity fun x => ENNReal.ofReal (f x)) ⟂ₘ
  t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x)
-- MeasureTheory.SignedMeasure.toJordanDecomposition_eq_of_eq_add_withDensity
theorem toJordanDecomposition_eq_of_eq_add_withDensity {f : α → ℝ} (hf : Measurable f)
    (hfi : Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
    s.toJordanDecomposition =
      @JordanDecomposition.mk α _
        (t.toJordanDecomposition.posPart + μ.withDensity fun x => ENNReal.ofReal (f x))
        (t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x))
        (by haveI := isFiniteMeasure_withDensity_ofReal hfi.2; infer_instance)
        (by haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2; infer_instance)
        (jordanDecomposition_add_withDensity_mutuallySingular hf htμ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : MeasureTheory.SignedMeasure α} {f : α → ℝ} (hf : Measurable f) (hfi : MeasureTheory.Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f)
Goal: s.toJordanDecomposition =
  MeasureTheory.JordanDecomposition.mk (t.toJordanDecomposition.posPart + μ.withDensity fun x => ENNReal.ofReal (f x))
    (t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x)) ⋯
-- _private.Mathlib.MeasureTheory.Decomposition.SignedLebesgue.0.MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk'
private theorem haveLebesgueDecomposition_mk' (μ : Measure α) {f : α → ℝ} (hf : Measurable f)
    (hfi : Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
    s.HaveLebesgueDecomposition μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : MeasureTheory.SignedMeasure α} (μ : MeasureTheory.Measure α) {f : α → ℝ} (hf : Measurable f) (hfi : MeasureTheory.Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f)
Goal: s.HaveLebesgueDecomposition μ
-- MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk
theorem haveLebesgueDecomposition_mk (μ : Measure α) {f : α → ℝ} (hf : Measurable f)
    (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
    s.HaveLebesgueDecomposition μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : MeasureTheory.SignedMeasure α} (μ : MeasureTheory.Measure α) {f : α → ℝ} (hf : Measurable f) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f)
Goal: s.HaveLebesgueDecomposition μ
-- _private.Mathlib.MeasureTheory.Decomposition.SignedLebesgue.0.MeasureTheory.SignedMeasure.eq_singularPart'
private theorem eq_singularPart' (t : SignedMeasure α) {f : α → ℝ} (hf : Measurable f)
    (hfi : Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
    t = s.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : MeasureTheory.SignedMeasure α} (t : MeasureTheory.SignedMeasure α) {f : α → ℝ} (hf : Measurable f) (hfi : MeasureTheory.Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f)
Goal: t = s.singularPart μ
-- MeasureTheory.SignedMeasure.eq_singularPart
theorem eq_singularPart (t : SignedMeasure α) (f : α → ℝ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure)
    (hadd : s = t + μ.withDensityᵥ f) : t = s.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : MeasureTheory.SignedMeasure α} (t : MeasureTheory.SignedMeasure α) (f : α → ℝ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f)
Goal: t = s.singularPart μ
-- MeasureTheory.SignedMeasure.singularPart_zero
theorem singularPart_zero (μ : Measure α) : (0 : SignedMeasure α).singularPart μ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.SignedMeasure.singularPart 0 μ = 0
-- MeasureTheory.SignedMeasure.singularPart_neg
theorem singularPart_neg (s : SignedMeasure α) (μ : Measure α) :
    (-s).singularPart μ = -s.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
Goal: (-s).singularPart μ = -s.singularPart μ
-- MeasureTheory.SignedMeasure.singularPart_smul_nnreal
theorem singularPart_smul_nnreal (s : SignedMeasure α) (μ : Measure α) (r : ℝ≥0) :
    (r • s).singularPart μ = r • s.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) (r : ℝ≥0)
Goal: (r • s).singularPart μ = r • s.singularPart μ
-- MeasureTheory.SignedMeasure.singularPart_smul
nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
    (r • s).singularPart μ = r • s.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) (r : ℝ)
Goal: (r • s).singularPart μ = r • s.singularPart μ
-- MeasureTheory.SignedMeasure.singularPart_add
theorem singularPart_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
    [t.HaveLebesgueDecomposition μ] :
    (s + t).singularPart μ = s.singularPart μ + t.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ]
Goal: (s + t).singularPart μ = s.singularPart μ + t.singularPart μ
-- MeasureTheory.SignedMeasure.singularPart_sub
theorem singularPart_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
    [t.HaveLebesgueDecomposition μ] :
    (s - t).singularPart μ = s.singularPart μ - t.singularPart μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ]
Goal: (s - t).singularPart μ = s.singularPart μ - t.singularPart μ
-- MeasureTheory.SignedMeasure.eq_rnDeriv
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f μ)
    (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
    f =ᵐ[μ] s.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : MeasureTheory.SignedMeasure α} (t : MeasureTheory.SignedMeasure α) (f : α → ℝ) (hfi : MeasureTheory.Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f)
Goal: f =ᵐ[μ] s.rnDeriv μ
-- MeasureTheory.SignedMeasure.rnDeriv_neg
theorem rnDeriv_neg (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] :
    (-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [s.HaveLebesgueDecomposition μ]
Goal: (-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ
-- MeasureTheory.SignedMeasure.rnDeriv_smul
theorem rnDeriv_smul (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] (r : ℝ) :
    (r • s).rnDeriv μ =ᵐ[μ] r • s.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [s.HaveLebesgueDecomposition μ] (r : ℝ)
Goal: (r • s).rnDeriv μ =ᵐ[μ] r • s.rnDeriv μ
-- MeasureTheory.SignedMeasure.rnDeriv_add
theorem rnDeriv_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
    [t.HaveLebesgueDecomposition μ] [(s + t).HaveLebesgueDecomposition μ] :
    (s + t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + t.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ] [(s + t).HaveLebesgueDecomposition μ]
Goal: (s + t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + t.rnDeriv μ
-- MeasureTheory.SignedMeasure.rnDeriv_sub
theorem rnDeriv_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
    [t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
    (s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ]
Goal: (s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ
-- MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.imPart
class HaveLebesgueDecomposition (c : ComplexMeasure α) (μ : Measure α) : Prop where
  rePart : c.re.HaveLebesgueDecomposition μ
  imPart : c.im.HaveLebesgueDecomposition μ
Arguments: {α : Type u_1} {m : MeasurableSpace α} {c : MeasureTheory.ComplexMeasure α} {μ : MeasureTheory.Measure α} [self : c.HaveLebesgueDecomposition μ]
Goal: (MeasureTheory.ComplexMeasure.im c).HaveLebesgueDecomposition μ
-- MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.rePart
class HaveLebesgueDecomposition (c : ComplexMeasure α) (μ : Measure α) : Prop where
  rePart : c.re.HaveLebesgueDecomposition μ
  imPart : c.im.HaveLebesgueDecomposition μ
Arguments: {α : Type u_1} {m : MeasurableSpace α} {c : MeasureTheory.ComplexMeasure α} {μ : MeasureTheory.Measure α} [self : c.HaveLebesgueDecomposition μ]
Goal: (MeasureTheory.ComplexMeasure.re c).HaveLebesgueDecomposition μ
-- MeasureTheory.ComplexMeasure.integrable_rnDeriv
theorem integrable_rnDeriv (c : ComplexMeasure α) (μ : Measure α) : Integrable (c.rnDeriv μ) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Integrable (c.rnDeriv μ) μ
-- MeasureTheory.ComplexMeasure.singularPart_add_withDensity_rnDeriv_eq
theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
    c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {c : MeasureTheory.ComplexMeasure α} [c.HaveLebesgueDecomposition μ]
Goal: c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c
-- MeasureTheory.VectorMeasure.m_iUnion'
structure VectorMeasure (α : Type*) [MeasurableSpace α] (M : Type*) [AddCommMonoid M]
    [TopologicalSpace M] where
  measureOf' : Set α → M
  empty' : measureOf' ∅ = 0
  not_measurable' ⦃i : Set α⦄ : ¬MeasurableSet i → measureOf' i = 0
  m_iUnion' ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    HasSum (fun i => measureOf' (f i)) (measureOf' (⋃ i, f i))
Arguments: {α : Type u_3} [MeasurableSpace α] {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] (self : MeasureTheory.VectorMeasure α M) ⦃f : ℕ → Set α⦄
Goal: (∀ (i : ℕ), MeasurableSet (f i)) → Pairwise (Disjoint on f) → HasSum (fun i => ↑self (f i)) (↑self (⋃ i, f i))
-- MeasureTheory.VectorMeasure.not_measurable'
structure VectorMeasure (α : Type*) [MeasurableSpace α] (M : Type*) [AddCommMonoid M]
    [TopologicalSpace M] where
  measureOf' : Set α → M
  empty' : measureOf' ∅ = 0
  not_measurable' ⦃i : Set α⦄ : ¬MeasurableSet i → measureOf' i = 0
  m_iUnion' ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    HasSum (fun i => measureOf' (f i)) (measureOf' (⋃ i, f i))
Arguments: {α : Type u_3} [MeasurableSpace α] {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] (self : MeasureTheory.VectorMeasure α M) ⦃i : Set α⦄
Goal: ¬MeasurableSet i → ↑self i = 0
-- MeasureTheory.VectorMeasure.empty'
structure VectorMeasure (α : Type*) [MeasurableSpace α] (M : Type*) [AddCommMonoid M]
    [TopologicalSpace M] where
  measureOf' : Set α → M
  empty' : measureOf' ∅ = 0
  not_measurable' ⦃i : Set α⦄ : ¬MeasurableSet i → measureOf' i = 0
  m_iUnion' ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    HasSum (fun i => measureOf' (f i)) (measureOf' (⋃ i, f i))
Arguments: {α : Type u_3} [MeasurableSpace α] {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] (self : MeasureTheory.VectorMeasure α M)
Goal: ↑self ∅ = 0
-- MeasureTheory.VectorMeasure.empty
@[simp]
theorem empty (v : VectorMeasure α M) : v ∅ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M)
Goal: ↑v ∅ = 0
-- MeasureTheory.VectorMeasure.not_measurable
theorem not_measurable (v : VectorMeasure α M) {i : Set α} (hi : ¬MeasurableSet i) : v i = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : ¬MeasurableSet i)
Goal: ↑v i = 0
-- MeasureTheory.VectorMeasure.m_iUnion
theorem m_iUnion (v : VectorMeasure α M) {f : ℕ → Set α} (hf₁ : ∀ i, MeasurableSet (f i))
    (hf₂ : Pairwise (Disjoint on f)) : HasSum (fun i => v (f i)) (v (⋃ i, f i)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {f : ℕ → Set α} (hf₁ : ∀ (i : ℕ), MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f))
Goal: HasSum (fun i => ↑v (f i)) (↑v (⋃ i, f i))
-- MeasureTheory.VectorMeasure.of_disjoint_iUnion_nat
theorem of_disjoint_iUnion_nat [T2Space M] (v : VectorMeasure α M) {f : ℕ → Set α}
    (hf₁ : ∀ i, MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f)) :
    v (⋃ i, f i) = ∑' i, v (f i) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [T2Space M] (v : MeasureTheory.VectorMeasure α M) {f : ℕ → Set α} (hf₁ : ∀ (i : ℕ), MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f))
Goal: ↑v (⋃ i, f i) = ∑' (i : ℕ), ↑v (f i)
-- MeasureTheory.VectorMeasure.coe_injective
theorem coe_injective : @Function.Injective (VectorMeasure α M) (Set α → M) (⇑) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M]
Goal: Function.Injective MeasureTheory.VectorMeasure.measureOf'
-- MeasureTheory.VectorMeasure.ext_iff'
theorem ext_iff' (v w : VectorMeasure α M) : v = w ↔ ∀ i : Set α, v i = w i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v w : MeasureTheory.VectorMeasure α M)
Goal: v = w ↔ ∀ (i : Set α), ↑v i = ↑w i
-- MeasureTheory.VectorMeasure.ext_iff
theorem ext_iff (v w : VectorMeasure α M) : v = w ↔ ∀ i : Set α, MeasurableSet i → v i = w i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v w : MeasureTheory.VectorMeasure α M)
Goal: v = w ↔ ∀ (i : Set α), MeasurableSet i → ↑v i = ↑w i
-- MeasureTheory.VectorMeasure.ext
@[ext]
theorem ext {s t : VectorMeasure α M} (h : ∀ i : Set α, MeasurableSet i → s i = t i) : s = t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {s t : MeasureTheory.VectorMeasure α M} (h : ∀ (i : Set α), MeasurableSet i → ↑s i = ↑t i)
Goal: s = t
-- MeasureTheory.VectorMeasure.hasSum_of_disjoint_iUnion
theorem hasSum_of_disjoint_iUnion [Countable β] {f : β → Set α} (hf₁ : ∀ i, MeasurableSet (f i))
    (hf₂ : Pairwise (Disjoint on f)) : HasSum (fun i => v (f i)) (v (⋃ i, f i)) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [T2Space M] {v : MeasureTheory.VectorMeasure α M} [Countable β] {f : β → Set α} (hf₁ : ∀ (i : β), MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f))
Goal: HasSum (fun i => ↑v (f i)) (↑v (⋃ i, f i))
-- MeasureTheory.VectorMeasure.of_disjoint_iUnion
theorem of_disjoint_iUnion [Countable β] {f : β → Set α} (hf₁ : ∀ i, MeasurableSet (f i))
    (hf₂ : Pairwise (Disjoint on f)) : v (⋃ i, f i) = ∑' i, v (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [T2Space M] {v : MeasureTheory.VectorMeasure α M} [Countable β] {f : β → Set α} (hf₁ : ∀ (i : β), MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f))
Goal: ↑v (⋃ i, f i) = ∑' (i : β), ↑v (f i)
-- MeasureTheory.VectorMeasure.of_union
theorem of_union {A B : Set α} (h : Disjoint A B) (hA : MeasurableSet A) (hB : MeasurableSet B) :
    v (A ∪ B) = v A + v B 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [T2Space M] {v : MeasureTheory.VectorMeasure α M} {A B : Set α} (h : Disjoint A B) (hA : MeasurableSet A) (hB : MeasurableSet B)
Goal: ↑v (A ∪ B) = ↑v A + ↑v B
-- MeasureTheory.VectorMeasure.of_add_of_diff
theorem of_add_of_diff {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B) (h : A ⊆ B) :
    v A + v (B \ A) = v B 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [T2Space M] {v : MeasureTheory.VectorMeasure α M} {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B) (h : A ⊆ B)
Goal: ↑v A + ↑v (B \ A) = ↑v B
-- MeasureTheory.VectorMeasure.of_diff
theorem of_diff {M : Type*} [AddCommGroup M] [TopologicalSpace M] [T2Space M]
    {v : VectorMeasure α M} {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B)
    (h : A ⊆ B) : v (B \ A) = v B - v A 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommGroup M] [TopologicalSpace M] [T2Space M] {v : MeasureTheory.VectorMeasure α M} {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B) (h : A ⊆ B)
Goal: ↑v (B \ A) = ↑v B - ↑v A
-- MeasureTheory.VectorMeasure.of_diff_of_diff_eq_zero
theorem of_diff_of_diff_eq_zero {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B)
    (h' : v (B \ A) = 0) : v (A \ B) + v B = v A 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [T2Space M] {v : MeasureTheory.VectorMeasure α M} {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B) (h' : ↑v (B \ A) = 0)
Goal: ↑v (A \ B) + ↑v B = ↑v A
-- MeasureTheory.VectorMeasure.of_iUnion_nonneg
theorem of_iUnion_nonneg {M : Type*} [TopologicalSpace M] [OrderedAddCommMonoid M]
    [OrderClosedTopology M] {v : VectorMeasure α M} (hf₁ : ∀ i, MeasurableSet (f i))
    (hf₂ : Pairwise (Disjoint on f)) (hf₃ : ∀ i, 0 ≤ v (f i)) : 0 ≤ v (⋃ i, f i) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : ℕ → Set α} {M : Type u_4} [TopologicalSpace M] [OrderedAddCommMonoid M] [OrderClosedTopology M] {v : MeasureTheory.VectorMeasure α M} (hf₁ : ∀ (i : ℕ), MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f)) (hf₃ : ∀ (i : ℕ), 0 ≤ ↑v (f i))
Goal: 0 ≤ ↑v (⋃ i, f i)
-- MeasureTheory.VectorMeasure.of_iUnion_nonpos
theorem of_iUnion_nonpos {M : Type*} [TopologicalSpace M] [OrderedAddCommMonoid M]
    [OrderClosedTopology M] {v : VectorMeasure α M} (hf₁ : ∀ i, MeasurableSet (f i))
    (hf₂ : Pairwise (Disjoint on f)) (hf₃ : ∀ i, v (f i) ≤ 0) : v (⋃ i, f i) ≤ 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : ℕ → Set α} {M : Type u_4} [TopologicalSpace M] [OrderedAddCommMonoid M] [OrderClosedTopology M] {v : MeasureTheory.VectorMeasure α M} (hf₁ : ∀ (i : ℕ), MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f)) (hf₃ : ∀ (i : ℕ), ↑v (f i) ≤ 0)
Goal: ↑v (⋃ i, f i) ≤ 0
-- MeasureTheory.VectorMeasure.of_nonneg_disjoint_union_eq_zero
theorem of_nonneg_disjoint_union_eq_zero {s : SignedMeasure α} {A B : Set α} (h : Disjoint A B)
    (hA₁ : MeasurableSet A) (hB₁ : MeasurableSet B) (hA₂ : 0 ≤ s A) (hB₂ : 0 ≤ s B)
    (hAB : s (A ∪ B) = 0) : s A = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : MeasureTheory.SignedMeasure α} {A B : Set α} (h : Disjoint A B) (hA₁ : MeasurableSet A) (hB₁ : MeasurableSet B) (hA₂ : 0 ≤ ↑s A) (hB₂ : 0 ≤ ↑s B) (hAB : ↑s (A ∪ B) = 0)
Goal: ↑s A = 0
-- MeasureTheory.VectorMeasure.of_nonpos_disjoint_union_eq_zero
theorem of_nonpos_disjoint_union_eq_zero {s : SignedMeasure α} {A B : Set α} (h : Disjoint A B)
    (hA₁ : MeasurableSet A) (hB₁ : MeasurableSet B) (hA₂ : s A ≤ 0) (hB₂ : s B ≤ 0)
    (hAB : s (A ∪ B) = 0) : s A = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : MeasureTheory.SignedMeasure α} {A B : Set α} (h : Disjoint A B) (hA₁ : MeasurableSet A) (hB₁ : MeasurableSet B) (hA₂ : ↑s A ≤ 0) (hB₂ : ↑s B ≤ 0) (hAB : ↑s (A ∪ B) = 0)
Goal: ↑s A = 0
-- MeasureTheory.VectorMeasure.coe_smul
@[simp]
theorem coe_smul (r : R) (v : VectorMeasure α M) : ⇑(r • v) = r • ⇑v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {R : Type u_4} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] (r : R) (v : MeasureTheory.VectorMeasure α M)
Goal: ↑(r • v) = r • ↑v
-- MeasureTheory.VectorMeasure.smul_apply
theorem smul_apply (r : R) (v : VectorMeasure α M) (i : Set α) : (r • v) i = r • v i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {R : Type u_4} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] (r : R) (v : MeasureTheory.VectorMeasure α M) (i : Set α)
Goal: ↑(r • v) i = r • ↑v i
-- MeasureTheory.VectorMeasure.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : VectorMeasure α M) = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M]
Goal: ↑0 = 0