ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Measurable.im
@[measurability]
theorem Measurable.im (hf : Measurable f) : Measurable fun x => RCLike.im (f x) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {m : MeasurableSpace α} {f : α → 𝕜} (hf : Measurable f)
Goal: Measurable fun x => RCLike.im (f x)
-- AEMeasurable.im
@[measurability]
theorem AEMeasurable.im (hf : AEMeasurable f μ) : AEMeasurable (fun x => RCLike.im (f x)) μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {m : MeasurableSpace α} {f : α → 𝕜} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => RCLike.im (f x)) μ
-- RCLike.measurable_ofReal
@[measurability]
theorem RCLike.measurable_ofReal : Measurable ((↑) : ℝ → 𝕜) 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜]
Goal: Measurable RCLike.ofReal
-- measurable_of_re_im
theorem measurable_of_re_im (hre : Measurable fun x => RCLike.re (f x))
    (him : Measurable fun x => RCLike.im (f x)) : Measurable f 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] [MeasurableSpace α] {f : α → 𝕜} (hre : Measurable fun x => RCLike.re (f x)) (him : Measurable fun x => RCLike.im (f x))
Goal: Measurable f
-- aemeasurable_of_re_im
theorem aemeasurable_of_re_im (hre : AEMeasurable (fun x => RCLike.re (f x)) μ)
    (him : AEMeasurable (fun x => RCLike.im (f x)) μ) : AEMeasurable f μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] [MeasurableSpace α] {f : α → 𝕜} {μ : MeasureTheory.Measure α} (hre : AEMeasurable (fun x => RCLike.re (f x)) μ) (him : AEMeasurable (fun x => RCLike.im (f x)) μ)
Goal: AEMeasurable f μ
-- MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {β : Type u_2} [CompleteLinearOrder β] [DenselyOrdered β] [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β] [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β) (h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0)
Goal: AEMeasurable f μ
-- ENNReal.aemeasurable_of_exist_almost_disjoint_supersets
theorem ENNReal.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*} {m : MeasurableSpace α}
    (μ : Measure α) (f : α → ℝ≥0∞)
    (h : ∀ (p : ℝ≥0) (q : ℝ≥0), p < q →
      ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
        { x | f x < p } ⊆ u ∧ { x | (q : ℝ≥0∞) < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞) (h :
  ∀ (p q : ℝ≥0),
    p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < ↑p} ⊆ u ∧ {x | ↑q < f x} ⊆ v ∧ μ (u ∩ v) = 0)
Goal: AEMeasurable f μ
-- MeasureTheory.AEEqFun.stronglyMeasurable
protected theorem stronglyMeasurable (f : α →ₘ[μ] β) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α →ₘ[μ] β)
Goal: MeasureTheory.StronglyMeasurable ↑f
-- MeasureTheory.AEEqFun.aestronglyMeasurable
protected theorem aestronglyMeasurable (f : α →ₘ[μ] β) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α →ₘ[μ] β)
Goal: MeasureTheory.AEStronglyMeasurable (↑f) μ
-- MeasureTheory.AEEqFun.measurable
protected theorem measurable [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
    (f : α →ₘ[μ] β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] (f : α →ₘ[μ] β)
Goal: Measurable ↑f
-- MeasureTheory.AEEqFun.aemeasurable
protected theorem aemeasurable [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
    (f : α →ₘ[μ] β) : AEMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] (f : α →ₘ[μ] β)
Goal: AEMeasurable (↑f) μ
-- MeasureTheory.AEEqFun.quot_mk_eq_mk
@[simp]
theorem quot_mk_eq_mk (f : α → β) (hf) :
    (Quot.mk (@Setoid.r _ <| μ.aeEqSetoid β) ⟨f, hf⟩ : α →ₘ[μ] β) = mk f hf 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: Quot.mk Setoid.r ⟨f, hf⟩ = MeasureTheory.AEEqFun.mk f hf
-- MeasureTheory.AEEqFun.mk_eq_mk
@[simp]
theorem mk_eq_mk {f g : α → β} {hf hg} : (mk f hf : α →ₘ[μ] β) = mk g hg ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} {hf : MeasureTheory.AEStronglyMeasurable f μ} {hg : MeasureTheory.AEStronglyMeasurable g μ}
Goal: MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk g hg ↔ f =ᵐ[μ] g
-- MeasureTheory.AEEqFun.mk_coeFn
@[simp]
theorem mk_coeFn (f : α →ₘ[μ] β) : mk f f.aestronglyMeasurable = f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α →ₘ[μ] β)
Goal: MeasureTheory.AEEqFun.mk ↑f ⋯ = f
-- MeasureTheory.AEEqFun.ext
@[ext]
theorem ext {f g : α →ₘ[μ] β} (h : f =ᵐ[μ] g) : f = g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α →ₘ[μ] β} (h : ↑f =ᵐ[μ] ↑g)
Goal: f = g
-- MeasureTheory.AEEqFun.ext_iff
theorem ext_iff {f g : α →ₘ[μ] β} : f = g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α →ₘ[μ] β}
Goal: f = g ↔ ↑f =ᵐ[μ] ↑g
-- MeasureTheory.AEEqFun.coeFn_mk
theorem coeFn_mk (f : α → β) (hf) : (mk f hf : α →ₘ[μ] β) =ᵐ[μ] f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: ↑(MeasureTheory.AEEqFun.mk f hf) =ᵐ[μ] f
-- MeasureTheory.AEEqFun.induction_on
@[elab_as_elim]
theorem induction_on (f : α →ₘ[μ] β) {p : (α →ₘ[μ] β) → Prop} (H : ∀ f hf, p (mk f hf)) : p f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α →ₘ[μ] β) {p : (α →ₘ[μ] β) → Prop} (H : ∀ (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ), p (MeasureTheory.AEEqFun.mk f hf))
Goal: p f
-- MeasureTheory.AEEqFun.induction_on₂
@[elab_as_elim]
theorem induction_on₂ {α' β' : Type*} [MeasurableSpace α'] [TopologicalSpace β'] {μ' : Measure α'}
    (f : α →ₘ[μ] β) (f' : α' →ₘ[μ'] β') {p : (α →ₘ[μ] β) → (α' →ₘ[μ'] β') → Prop}
    (H : ∀ f hf f' hf', p (mk f hf) (mk f' hf')) : p f f' 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] {α' : Type u_5} {β' : Type u_6} [MeasurableSpace α'] [TopologicalSpace β'] {μ' : MeasureTheory.Measure α'} (f : α →ₘ[μ] β) (f' : α' →ₘ[μ'] β') {p : (α →ₘ[μ] β) → (α' →ₘ[μ'] β') → Prop} (H :
  ∀ (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ) (f' : α' → β')
    (hf' : MeasureTheory.AEStronglyMeasurable f' μ'),
    p (MeasureTheory.AEEqFun.mk f hf) (MeasureTheory.AEEqFun.mk f' hf'))
Goal: p f f'
-- MeasureTheory.AEEqFun.induction_on₃
@[elab_as_elim]
theorem induction_on₃ {α' β' : Type*} [MeasurableSpace α'] [TopologicalSpace β'] {μ' : Measure α'}
    {α'' β'' : Type*} [MeasurableSpace α''] [TopologicalSpace β''] {μ'' : Measure α''}
    (f : α →ₘ[μ] β) (f' : α' →ₘ[μ'] β') (f'' : α'' →ₘ[μ''] β'')
    {p : (α →ₘ[μ] β) → (α' →ₘ[μ'] β') → (α'' →ₘ[μ''] β'') → Prop}
    (H : ∀ f hf f' hf' f'' hf'', p (mk f hf) (mk f' hf') (mk f'' hf'')) : p f f' f'' 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] {α' : Type u_5} {β' : Type u_6} [MeasurableSpace α'] [TopologicalSpace β'] {μ' : MeasureTheory.Measure α'} {α'' : Type u_7} {β'' : Type u_8} [MeasurableSpace α''] [TopologicalSpace β''] {μ'' : MeasureTheory.Measure α''} (f : α →ₘ[μ] β) (f' : α' →ₘ[μ'] β') (f'' : α'' →ₘ[μ''] β'') {p : (α →ₘ[μ] β) → (α' →ₘ[μ'] β') → (α'' →ₘ[μ''] β'') → Prop} (H :
  ∀ (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ) (f' : α' → β')
    (hf' : MeasureTheory.AEStronglyMeasurable f' μ') (f'' : α'' → β'')
    (hf'' : MeasureTheory.AEStronglyMeasurable f'' μ''),
    p (MeasureTheory.AEEqFun.mk f hf) (MeasureTheory.AEEqFun.mk f' hf') (MeasureTheory.AEEqFun.mk f'' hf''))
Goal: p f f' f''
-- MeasureTheory.AEEqFun.compQuasiMeasurePreserving_mk
@[simp]
theorem compQuasiMeasurePreserving_mk {g : β → γ} (hg : AEStronglyMeasurable g ν)
    (hf : QuasiMeasurePreserving f μ ν) :
    (mk g hg).compQuasiMeasurePreserving f hf = mk (g ∘ f) (hg.comp_quasiMeasurePreserving hf) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → β} {g : β → γ} (hg : MeasureTheory.AEStronglyMeasurable g ν) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν)
Goal: (MeasureTheory.AEEqFun.mk g hg).compQuasiMeasurePreserving f hf = MeasureTheory.AEEqFun.mk (g ∘ f) ⋯
-- MeasureTheory.AEEqFun.compQuasiMeasurePreserving_eq_mk
theorem compQuasiMeasurePreserving_eq_mk (g : β →ₘ[ν] γ) (hf : QuasiMeasurePreserving f μ ν) :
    g.compQuasiMeasurePreserving f hf =
      mk (g ∘ f) (g.aestronglyMeasurable.comp_quasiMeasurePreserving hf) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → β} (g : β →ₘ[ν] γ) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν)
Goal: g.compQuasiMeasurePreserving f hf = MeasureTheory.AEEqFun.mk (↑g ∘ f) ⋯
-- MeasureTheory.AEEqFun.coeFn_compQuasiMeasurePreserving
theorem coeFn_compQuasiMeasurePreserving (g : β →ₘ[ν] γ) (hf : QuasiMeasurePreserving f μ ν) :
    g.compQuasiMeasurePreserving f hf =ᵐ[μ] g ∘ f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → β} (g : β →ₘ[ν] γ) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν)
Goal: ↑(g.compQuasiMeasurePreserving f hf) =ᵐ[μ] ↑g ∘ f
-- MeasureTheory.AEEqFun.compMeasurePreserving_mk
@[simp]
theorem compMeasurePreserving_mk (hg : AEStronglyMeasurable g ν) (hf : MeasurePreserving f μ ν) :
    (mk g hg).compMeasurePreserving f hf =
      mk (g ∘ f) (hg.comp_quasiMeasurePreserving hf.quasiMeasurePreserving) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → β} {g : β → γ} (hg : MeasureTheory.AEStronglyMeasurable g ν) (hf : MeasureTheory.MeasurePreserving f μ ν)
Goal: (MeasureTheory.AEEqFun.mk g hg).compMeasurePreserving f hf = MeasureTheory.AEEqFun.mk (g ∘ f) ⋯
-- MeasureTheory.AEEqFun.compMeasurePreserving_eq_mk
theorem compMeasurePreserving_eq_mk (g : β →ₘ[ν] γ) (hf : MeasurePreserving f μ ν) :
    g.compMeasurePreserving f hf =
      mk (g ∘ f) (g.aestronglyMeasurable.comp_quasiMeasurePreserving hf.quasiMeasurePreserving) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → β} (g : β →ₘ[ν] γ) (hf : MeasureTheory.MeasurePreserving f μ ν)
Goal: g.compMeasurePreserving f hf = MeasureTheory.AEEqFun.mk (↑g ∘ f) ⋯
-- MeasureTheory.AEEqFun.coeFn_compMeasurePreserving
theorem coeFn_compMeasurePreserving (g : β →ₘ[ν] γ) (hf : MeasurePreserving f μ ν) :
    g.compMeasurePreserving f hf =ᵐ[μ] g ∘ f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → β} (g : β →ₘ[ν] γ) (hf : MeasureTheory.MeasurePreserving f μ ν)
Goal: ↑(g.compMeasurePreserving f hf) =ᵐ[μ] ↑g ∘ f
-- MeasureTheory.AEEqFun.comp_mk
@[simp]
theorem comp_mk (g : β → γ) (hg : Continuous g) (f : α → β) (hf) :
    comp g hg (mk f hf : α →ₘ[μ] β) = mk (g ∘ f) (hg.comp_aestronglyMeasurable hf) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] (g : β → γ) (hg : Continuous g) (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEEqFun.comp g hg (MeasureTheory.AEEqFun.mk f hf) = MeasureTheory.AEEqFun.mk (g ∘ f) ⋯
-- MeasureTheory.AEEqFun.comp_eq_mk
theorem comp_eq_mk (g : β → γ) (hg : Continuous g) (f : α →ₘ[μ] β) :
    comp g hg f = mk (g ∘ f) (hg.comp_aestronglyMeasurable f.aestronglyMeasurable) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] (g : β → γ) (hg : Continuous g) (f : α →ₘ[μ] β)
Goal: MeasureTheory.AEEqFun.comp g hg f = MeasureTheory.AEEqFun.mk (g ∘ ↑f) ⋯
-- MeasureTheory.AEEqFun.coeFn_comp
theorem coeFn_comp (g : β → γ) (hg : Continuous g) (f : α →ₘ[μ] β) : comp g hg f =ᵐ[μ] g ∘ f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] (g : β → γ) (hg : Continuous g) (f : α →ₘ[μ] β)
Goal: ↑(MeasureTheory.AEEqFun.comp g hg f) =ᵐ[μ] g ∘ ↑f
-- MeasureTheory.AEEqFun.comp_compQuasiMeasurePreserving
theorem comp_compQuasiMeasurePreserving [MeasurableSpace β] {ν} (g : γ → δ) (hg : Continuous g)
    (f : β →ₘ[ν] γ) {φ : α → β} (hφ : Measure.QuasiMeasurePreserving φ μ ν) :
    (comp g hg f).compQuasiMeasurePreserving φ hφ =
      comp g hg (f.compQuasiMeasurePreserving φ hφ) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [TopologicalSpace δ] [MeasurableSpace β] {ν : MeasureTheory.Measure β} (g : γ → δ) (hg : Continuous g) (f : β →ₘ[ν] γ) {φ : α → β} (hφ : MeasureTheory.Measure.QuasiMeasurePreserving φ μ ν)
Goal: (MeasureTheory.AEEqFun.comp g hg f).compQuasiMeasurePreserving φ hφ =
  MeasureTheory.AEEqFun.comp g hg (f.compQuasiMeasurePreserving φ hφ)
-- MeasureTheory.AEEqFun.compMeasurable_mk
@[simp]
theorem compMeasurable_mk (g : β → γ) (hg : Measurable g) (f : α → β)
    (hf : AEStronglyMeasurable f μ) :
    compMeasurable g hg (mk f hf : α →ₘ[μ] β) =
      mk (g ∘ f) (hg.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [MeasurableSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [OpensMeasurableSpace γ] [SecondCountableTopology γ] (g : β → γ) (hg : Measurable g) (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEEqFun.compMeasurable g hg (MeasureTheory.AEEqFun.mk f hf) = MeasureTheory.AEEqFun.mk (g ∘ f) ⋯
-- MeasureTheory.AEEqFun.compMeasurable_eq_mk
theorem compMeasurable_eq_mk (g : β → γ) (hg : Measurable g) (f : α →ₘ[μ] β) :
    compMeasurable g hg f =
    mk (g ∘ f) (hg.comp_aemeasurable f.aemeasurable).aestronglyMeasurable 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [MeasurableSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [OpensMeasurableSpace γ] [SecondCountableTopology γ] (g : β → γ) (hg : Measurable g) (f : α →ₘ[μ] β)
Goal: MeasureTheory.AEEqFun.compMeasurable g hg f = MeasureTheory.AEEqFun.mk (g ∘ ↑f) ⋯
-- MeasureTheory.AEEqFun.coeFn_compMeasurable
theorem coeFn_compMeasurable (g : β → γ) (hg : Measurable g) (f : α →ₘ[μ] β) :
    compMeasurable g hg f =ᵐ[μ] g ∘ f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [MeasurableSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [OpensMeasurableSpace γ] [SecondCountableTopology γ] (g : β → γ) (hg : Measurable g) (f : α →ₘ[μ] β)
Goal: ↑(MeasureTheory.AEEqFun.compMeasurable g hg f) =ᵐ[μ] g ∘ ↑f
-- MeasureTheory.AEEqFun.pair_mk_mk
@[simp]
theorem pair_mk_mk (f : α → β) (hf) (g : α → γ) (hg) :
    (mk f hf : α →ₘ[μ] β).pair (mk g hg) = mk (fun x => (f x, g x)) (hf.prod_mk hg) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ) (g : α → γ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: (MeasureTheory.AEEqFun.mk f hf).pair (MeasureTheory.AEEqFun.mk g hg) = MeasureTheory.AEEqFun.mk (fun x => (f x, g x)) ⋯
-- MeasureTheory.AEEqFun.pair_eq_mk
theorem pair_eq_mk (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) :
    f.pair g =
      mk (fun x => (f x, g x)) (f.aestronglyMeasurable.prod_mk g.aestronglyMeasurable) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ)
Goal: f.pair g = MeasureTheory.AEEqFun.mk (fun x => (↑f x, ↑g x)) ⋯
-- MeasureTheory.AEEqFun.coeFn_pair
theorem coeFn_pair (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) : f.pair g =ᵐ[μ] fun x => (f x, g x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ)
Goal: ↑(f.pair g) =ᵐ[μ] fun x => (↑f x, ↑g x)
-- MeasureTheory.AEEqFun.comp₂_mk_mk
@[simp]
theorem comp₂_mk_mk (g : β → γ → δ) (hg : Continuous (uncurry g)) (f₁ : α → β) (f₂ : α → γ)
    (hf₁ hf₂) :
    comp₂ g hg (mk f₁ hf₁ : α →ₘ[μ] β) (mk f₂ hf₂) =
      mk (fun a => g (f₁ a) (f₂ a)) (hg.comp_aestronglyMeasurable (hf₁.prod_mk hf₂)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (g : β → γ → δ) (hg : Continuous (Function.uncurry g)) (f₁ : α → β) (f₂ : α → γ) (hf₁ : MeasureTheory.AEStronglyMeasurable f₁ μ) (hf₂ : MeasureTheory.AEStronglyMeasurable f₂ μ)
Goal: MeasureTheory.AEEqFun.comp₂ g hg (MeasureTheory.AEEqFun.mk f₁ hf₁) (MeasureTheory.AEEqFun.mk f₂ hf₂) =
  MeasureTheory.AEEqFun.mk (fun a => g (f₁ a) (f₂ a)) ⋯
-- MeasureTheory.AEEqFun.comp₂_eq_pair
theorem comp₂_eq_pair (g : β → γ → δ) (hg : Continuous (uncurry g)) (f₁ : α →ₘ[μ] β)
    (f₂ : α →ₘ[μ] γ) : comp₂ g hg f₁ f₂ = comp _ hg (f₁.pair f₂) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (g : β → γ → δ) (hg : Continuous (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: MeasureTheory.AEEqFun.comp₂ g hg f₁ f₂ = MeasureTheory.AEEqFun.comp (Function.uncurry g) hg (f₁.pair f₂)
-- MeasureTheory.AEEqFun.comp₂_eq_mk
theorem comp₂_eq_mk (g : β → γ → δ) (hg : Continuous (uncurry g)) (f₁ : α →ₘ[μ] β)
    (f₂ : α →ₘ[μ] γ) : comp₂ g hg f₁ f₂ = mk (fun a => g (f₁ a) (f₂ a))
      (hg.comp_aestronglyMeasurable (f₁.aestronglyMeasurable.prod_mk f₂.aestronglyMeasurable)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (g : β → γ → δ) (hg : Continuous (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: MeasureTheory.AEEqFun.comp₂ g hg f₁ f₂ = MeasureTheory.AEEqFun.mk (fun a => g (↑f₁ a) (↑f₂ a)) ⋯
-- MeasureTheory.AEEqFun.coeFn_comp₂
theorem coeFn_comp₂ (g : β → γ → δ) (hg : Continuous (uncurry g)) (f₁ : α →ₘ[μ] β)
    (f₂ : α →ₘ[μ] γ) : comp₂ g hg f₁ f₂ =ᵐ[μ] fun a => g (f₁ a) (f₂ a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (g : β → γ → δ) (hg : Continuous (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: ↑(MeasureTheory.AEEqFun.comp₂ g hg f₁ f₂) =ᵐ[μ] fun a => g (↑f₁ a) (↑f₂ a)
-- MeasureTheory.AEEqFun.comp₂Measurable_mk_mk
@[simp]
theorem comp₂Measurable_mk_mk (g : β → γ → δ) (hg : Measurable (uncurry g)) (f₁ : α → β)
    (f₂ : α → γ) (hf₁ hf₂) :
    comp₂Measurable g hg (mk f₁ hf₁ : α →ₘ[μ] β) (mk f₂ hf₂) =
      mk (fun a => g (f₁ a) (f₂ a))
        (hg.comp_aemeasurable (hf₁.aemeasurable.prod_mk hf₂.aemeasurable)).aestronglyMeasurable 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [MeasurableSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] [SecondCountableTopologyEither β γ] [MeasurableSpace δ] [TopologicalSpace.PseudoMetrizableSpace δ] [OpensMeasurableSpace δ] [SecondCountableTopology δ] (g : β → γ → δ) (hg : Measurable (Function.uncurry g)) (f₁ : α → β) (f₂ : α → γ) (hf₁ : MeasureTheory.AEStronglyMeasurable f₁ μ) (hf₂ : MeasureTheory.AEStronglyMeasurable f₂ μ)
Goal: MeasureTheory.AEEqFun.comp₂Measurable g hg (MeasureTheory.AEEqFun.mk f₁ hf₁) (MeasureTheory.AEEqFun.mk f₂ hf₂) =
  MeasureTheory.AEEqFun.mk (fun a => g (f₁ a) (f₂ a)) ⋯
-- MeasureTheory.AEEqFun.comp₂Measurable_eq_pair
theorem comp₂Measurable_eq_pair (g : β → γ → δ) (hg : Measurable (uncurry g)) (f₁ : α →ₘ[μ] β)
    (f₂ : α →ₘ[μ] γ) : comp₂Measurable g hg f₁ f₂ = compMeasurable _ hg (f₁.pair f₂) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [MeasurableSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] [SecondCountableTopologyEither β γ] [MeasurableSpace δ] [TopologicalSpace.PseudoMetrizableSpace δ] [OpensMeasurableSpace δ] [SecondCountableTopology δ] (g : β → γ → δ) (hg : Measurable (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: MeasureTheory.AEEqFun.comp₂Measurable g hg f₁ f₂ =
  MeasureTheory.AEEqFun.compMeasurable (Function.uncurry g) hg (f₁.pair f₂)
-- MeasureTheory.AEEqFun.comp₂Measurable_eq_mk
theorem comp₂Measurable_eq_mk (g : β → γ → δ) (hg : Measurable (uncurry g)) (f₁ : α →ₘ[μ] β)
    (f₂ : α →ₘ[μ] γ) :
    comp₂Measurable g hg f₁ f₂ =
      mk (fun a => g (f₁ a) (f₂ a))
        (hg.comp_aemeasurable (f₁.aemeasurable.prod_mk f₂.aemeasurable)).aestronglyMeasurable 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [MeasurableSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] [SecondCountableTopologyEither β γ] [MeasurableSpace δ] [TopologicalSpace.PseudoMetrizableSpace δ] [OpensMeasurableSpace δ] [SecondCountableTopology δ] (g : β → γ → δ) (hg : Measurable (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: MeasureTheory.AEEqFun.comp₂Measurable g hg f₁ f₂ = MeasureTheory.AEEqFun.mk (fun a => g (↑f₁ a) (↑f₂ a)) ⋯