ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.IdentDistrib.const_sub
@[to_additive]
theorem const_div [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => c / f x) (fun x => c / g x) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Sub γ] [MeasurableSub γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => c - f x) (fun x => c - g x) μ ν
-- ProbabilityTheory.IdentDistrib.inv
@[to_additive]
lemma inv [Inv γ] [MeasurableInv γ] (h : IdentDistrib f g μ ν) :
    IdentDistrib f⁻¹ g⁻¹ μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Inv γ] [MeasurableInv γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.IdentDistrib f⁻¹ g⁻¹ μ ν
-- ProbabilityTheory.IdentDistrib.neg
@[to_additive]
lemma inv [Inv γ] [MeasurableInv γ] (h : IdentDistrib f g μ ν) :
    IdentDistrib f⁻¹ g⁻¹ μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Neg γ] [MeasurableNeg γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.IdentDistrib (-f) (-g) μ ν
-- ProbabilityTheory.IdentDistrib.evariance_eq
theorem evariance_eq {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν) :
    evariance f μ = evariance g ν 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → ℝ} {g : β → ℝ} (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.evariance f μ = ProbabilityTheory.evariance g ν
-- ProbabilityTheory.IdentDistrib.variance_eq
theorem variance_eq {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν) :
    variance f μ = variance g ν 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → ℝ} {g : β → ℝ} (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.variance f μ = ProbabilityTheory.variance g ν
-- ProbabilityTheory.Memℒp.uniformIntegrable_of_identDistrib_aux
theorem Memℒp.uniformIntegrable_of_identDistrib_aux {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
    (hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : Memℒp (f j) p μ) (hfmeas : ∀ i, StronglyMeasurable (f i))
    (hf : ∀ i, IdentDistrib (f i) (f j) μ μ) : UniformIntegrable f p μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {E : Type u_5} [MeasurableSpace E] [NormedAddCommGroup E] [BorelSpace E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_6} {f : ι → α → E} {j : ι} {p : ℝ≥0∞} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hℒp : MeasureTheory.Memℒp (f j) p μ) (hfmeas : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (hf : ∀ (i : ι), ProbabilityTheory.IdentDistrib (f i) (f j) μ μ)
Goal: MeasureTheory.UniformIntegrable f p μ
-- ProbabilityTheory.Memℒp.uniformIntegrable_of_identDistrib
theorem Memℒp.uniformIntegrable_of_identDistrib {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
    (hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : Memℒp (f j) p μ) (hf : ∀ i, IdentDistrib (f i) (f j) μ μ) :
    UniformIntegrable f p μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {E : Type u_5} [MeasurableSpace E] [NormedAddCommGroup E] [BorelSpace E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_6} {f : ι → α → E} {j : ι} {p : ℝ≥0∞} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hℒp : MeasureTheory.Memℒp (f j) p μ) (hf : ∀ (i : ι), ProbabilityTheory.IdentDistrib (f i) (f j) μ μ)
Goal: MeasureTheory.UniformIntegrable f p μ
-- ProbabilityTheory.indepFun_of_identDistrib_pair
lemma indepFun_of_identDistrib_pair
    {μ : Measure γ} {μ' : Measure δ} [IsFiniteMeasure μ] [IsFiniteMeasure μ']
    {X : γ → α} {X' : δ → α} {Y : γ → β} {Y' : δ → β} (h_indep : IndepFun X Y μ)
    (h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ') :
    IndepFun X' Y' μ' 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure γ} {μ' : MeasureTheory.Measure δ} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'] {X : γ → α} {X' : δ → α} {Y : γ → β} {Y' : δ → β} (h_indep : ProbabilityTheory.IndepFun X Y μ) (h_ident : ProbabilityTheory.IdentDistrib (fun ω => (X ω, Y ω)) (fun ω => (X' ω, Y' ω)) μ μ')
Goal: ProbabilityTheory.IndepFun X' Y' μ'
-- ProbabilityTheory.exponentialPDF_eq
lemma exponentialPDF_eq (r x : ℝ) :
    exponentialPDF r x = ENNReal.ofReal (if 0 ≤ x then r * exp (-(r * x)) else 0) 
Arguments: (r x : ℝ)
Goal: ProbabilityTheory.exponentialPDF r x = ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)
-- ProbabilityTheory.exponentialPDF_of_neg
lemma exponentialPDF_of_neg {r x : ℝ} (hx : x < 0) : exponentialPDF r x = 0 
Arguments: {r x : ℝ} (hx : x < 0)
Goal: ProbabilityTheory.exponentialPDF r x = 0
-- ProbabilityTheory.exponentialPDF_of_nonneg
lemma exponentialPDF_of_nonneg {r x : ℝ} (hx : 0 ≤ x) :
    exponentialPDF r x = ENNReal.ofReal (r * rexp (-(r * x))) 
Arguments: {r x : ℝ} (hx : 0 ≤ x)
Goal: ProbabilityTheory.exponentialPDF r x = ENNReal.ofReal (r * rexp (-(r * x)))
-- ProbabilityTheory.lintegral_exponentialPDF_of_nonpos
lemma lintegral_exponentialPDF_of_nonpos {x r : ℝ} (hx : x ≤ 0) :
    ∫⁻ y in Iio x, exponentialPDF r y = 0 
Arguments: {x r : ℝ} (hx : x ≤ 0)
Goal: ∫⁻ (y : ℝ) in Set.Iio x, ProbabilityTheory.exponentialPDF r y = 0
-- ProbabilityTheory.measurable_exponentialPDFReal
@[measurability]
lemma measurable_exponentialPDFReal (r : ℝ) : Measurable (exponentialPDFReal r) 
Arguments: (r : ℝ)
Goal: Measurable (ProbabilityTheory.exponentialPDFReal r)
-- ProbabilityTheory.stronglyMeasurable_exponentialPDFReal
@[measurability]
 lemma stronglyMeasurable_exponentialPDFReal (r : ℝ) :
     StronglyMeasurable (exponentialPDFReal r) 
Arguments: (r : ℝ)
Goal: MeasureTheory.StronglyMeasurable (ProbabilityTheory.exponentialPDFReal r)
-- ProbabilityTheory.exponentialPDFReal_pos
lemma exponentialPDFReal_pos {x r : ℝ} (hr : 0 < r) (hx : 0 < x) :
    0 < exponentialPDFReal r x 
Arguments: {x r : ℝ} (hr : 0 < r) (hx : 0 < x)
Goal: 0 < ProbabilityTheory.exponentialPDFReal r x
-- ProbabilityTheory.exponentialPDFReal_nonneg
lemma exponentialPDFReal_nonneg {r : ℝ} (hr : 0 < r) (x : ℝ) :
    0 ≤ exponentialPDFReal r x 
Arguments: {r : ℝ} (hr : 0 < r) (x : ℝ)
Goal: 0 ≤ ProbabilityTheory.exponentialPDFReal r x
-- ProbabilityTheory.lintegral_exponentialPDF_eq_one
@[simp]
lemma lintegral_exponentialPDF_eq_one {r : ℝ} (hr : 0 < r) : ∫⁻ x, exponentialPDF r x = 1 
Arguments: {r : ℝ} (hr : 0 < r)
Goal: ∫⁻ (x : ℝ), ProbabilityTheory.exponentialPDF r x = 1
-- ProbabilityTheory.isProbabilityMeasureExponential
lemma isProbabilityMeasureExponential {r : ℝ} (hr : 0 < r) :
    IsProbabilityMeasure (expMeasure r) 
Arguments: {r : ℝ} (hr : 0 < r)
Goal: MeasureTheory.IsProbabilityMeasure (ProbabilityTheory.expMeasure r)
-- ProbabilityTheory.exponentialCDFReal_eq_integral
lemma exponentialCDFReal_eq_integral {r : ℝ} (hr : 0 < r) (x : ℝ) :
    exponentialCDFReal r x = ∫ x in Iic x, exponentialPDFReal r x 
Arguments: {r : ℝ} (hr : 0 < r) (x : ℝ)
Goal: ↑(ProbabilityTheory.exponentialCDFReal r) x = ∫ (x : ℝ) in Set.Iic x, ProbabilityTheory.exponentialPDFReal r x
-- ProbabilityTheory.exponentialCDFReal_eq_lintegral
lemma exponentialCDFReal_eq_lintegral {r : ℝ} (hr : 0 < r) (x : ℝ) :
    exponentialCDFReal r x = ENNReal.toReal (∫⁻ x in Iic x, exponentialPDF r x) 
Arguments: {r : ℝ} (hr : 0 < r) (x : ℝ)
Goal: ↑(ProbabilityTheory.exponentialCDFReal r) x = (∫⁻ (x : ℝ) in Set.Iic x, ProbabilityTheory.exponentialPDF r x).toReal
-- ProbabilityTheory.hasDerivAt_neg_exp_mul_exp
lemma hasDerivAt_neg_exp_mul_exp {r x : ℝ} :
    HasDerivAt (fun a ↦ -exp (-(r * a))) (r * exp (-(r * x))) x 
Arguments: {r x : ℝ}
Goal: HasDerivAt (fun a => -rexp (-(r * a))) (r * rexp (-(r * x))) x
-- ProbabilityTheory.exp_neg_integrableOn_Ioc
lemma exp_neg_integrableOn_Ioc {b x : ℝ} (hb : 0 < b) :
    IntegrableOn (fun x ↦ rexp (-(b * x))) (Ioc 0 x) 
Arguments: {b x : ℝ} (hb : 0 < b)
Goal: MeasureTheory.IntegrableOn (fun x => rexp (-(b * x))) (Set.Ioc 0 x) MeasureTheory.volume
-- ProbabilityTheory.lintegral_exponentialPDF_eq_antiDeriv
lemma lintegral_exponentialPDF_eq_antiDeriv {r : ℝ} (hr : 0 < r) (x : ℝ) :
    ∫⁻ y in Iic x, exponentialPDF r y
    = ENNReal.ofReal (if 0 ≤ x then 1 - exp (-(r * x)) else 0) 
Arguments: {r : ℝ} (hr : 0 < r) (x : ℝ)
Goal: ∫⁻ (y : ℝ) in Set.Iic x, ProbabilityTheory.exponentialPDF r y =
  ENNReal.ofReal (if 0 ≤ x then 1 - rexp (-(r * x)) else 0)
-- ProbabilityTheory.exponentialCDFReal_eq
lemma exponentialCDFReal_eq {r : ℝ} (hr : 0 < r) (x : ℝ) :
    exponentialCDFReal r x = if 0 ≤ x then 1 - exp (-(r * x)) else 0 
Arguments: {r : ℝ} (hr : 0 < r) (x : ℝ)
Goal: ↑(ProbabilityTheory.exponentialCDFReal r) x = if 0 ≤ x then 1 - rexp (-(r * x)) else 0
-- MeasureTheory.HasPDF.pdf'
class HasPDF {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E := by volume_tac) : Prop where
  pdf' : AEMeasurable X ℙ ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : autoParam (MeasureTheory.Measure E) _auto✝} [self : MeasureTheory.HasPDF X ℙ μ]
Goal: AEMeasurable X ℙ ∧ (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧ MeasureTheory.Measure.map X ℙ ≪ μ
-- MeasureTheory.hasPDF_iff
theorem hasPDF_iff {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} :
    HasPDF X ℙ μ ↔ AEMeasurable X ℙ ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E},
  MeasureTheory.HasPDF X ℙ μ ↔
    AEMeasurable X ℙ ∧ (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧ MeasureTheory.Measure.map X ℙ ≪ μ
-- MeasureTheory.hasPDF_iff_of_aemeasurable
theorem hasPDF_iff_of_aemeasurable {X : Ω → E} {ℙ : Measure Ω}
    {μ : Measure E} (hX : AEMeasurable X ℙ) :
    HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E},
  AEMeasurable X ℙ →
    (MeasureTheory.HasPDF X ℙ μ ↔
      (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧ MeasureTheory.Measure.map X ℙ ≪ μ)
-- MeasureTheory.HasPDF.aemeasurable
@[measurability]
theorem HasPDF.aemeasurable (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E) [hX : HasPDF X ℙ μ] : AEMeasurable X ℙ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} (X : Ω → E) (ℙ : MeasureTheory.Measure Ω) (μ : MeasureTheory.Measure E)
  [hX : MeasureTheory.HasPDF X ℙ μ], AEMeasurable X ℙ
-- MeasureTheory.HasPDF.absolutelyContinuous
theorem HasPDF.absolutelyContinuous {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
    [hX : HasPDF X ℙ μ] : map X ℙ ≪ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E}
  [hX : MeasureTheory.HasPDF X ℙ μ], MeasureTheory.Measure.map X ℙ ≪ μ
-- MeasureTheory.HasPDF.quasiMeasurePreserving_of_measurable
theorem HasPDF.quasiMeasurePreserving_of_measurable (X : Ω → E) (ℙ : Measure Ω) (μ : Measure E)
    [HasPDF X ℙ μ] (h : Measurable X) : QuasiMeasurePreserving X ℙ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} (X : Ω → E) (ℙ : MeasureTheory.Measure Ω) (μ : MeasureTheory.Measure E)
  [inst : MeasureTheory.HasPDF X ℙ μ], Measurable X → MeasureTheory.Measure.QuasiMeasurePreserving X ℙ μ
-- MeasureTheory.HasPDF.congr
theorem HasPDF.congr {X Y : Ω → E} {ℙ : Measure Ω} {μ : Measure E} (hXY : X =ᵐ[ℙ] Y)
    [hX : HasPDF X ℙ μ] : HasPDF Y ℙ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {X Y : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E},
  X =ᵐ[ℙ] Y → ∀ [hX : MeasureTheory.HasPDF X ℙ μ], MeasureTheory.HasPDF Y ℙ μ
-- MeasureTheory.HasPDF.congr'
theorem HasPDF.congr' {X Y : Ω → E} {ℙ : Measure Ω} {μ : Measure E} (hXY : X =ᵐ[ℙ] Y) :
    HasPDF X ℙ μ ↔ HasPDF Y ℙ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {X Y : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E},
  X =ᵐ[ℙ] Y → (MeasureTheory.HasPDF X ℙ μ ↔ MeasureTheory.HasPDF Y ℙ μ)
-- MeasureTheory.hasPDF_of_map_eq_withDensity
theorem hasPDF_of_map_eq_withDensity {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
    (hX : AEMeasurable X ℙ) (f : E → ℝ≥0∞) (hf : AEMeasurable f μ) (h : map X ℙ = μ.withDensity f) :
    HasPDF X ℙ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E},
  AEMeasurable X ℙ →
    ∀ (f : E → ℝ≥0∞), AEMeasurable f μ → MeasureTheory.Measure.map X ℙ = μ.withDensity f → MeasureTheory.HasPDF X ℙ μ
-- MeasureTheory.pdf_def
theorem pdf_def {_ : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E} :
    pdf X ℙ μ = (map X ℙ).rnDeriv μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {X : Ω → E},
  MeasureTheory.pdf X ℙ μ = (MeasureTheory.Measure.map X ℙ).rnDeriv μ
-- MeasureTheory.pdf_of_not_aemeasurable
theorem pdf_of_not_aemeasurable {_ : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {X : Ω → E} (hX : ¬AEMeasurable X ℙ) : pdf X ℙ μ =ᵐ[μ] 0 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {X : Ω → E},
  ¬AEMeasurable X ℙ → MeasureTheory.pdf X ℙ μ =ᵐ[μ] 0
-- MeasureTheory.pdf_of_not_haveLebesgueDecomposition
theorem pdf_of_not_haveLebesgueDecomposition {_ : MeasurableSpace Ω} {ℙ : Measure Ω}
    {μ : Measure E} {X : Ω → E} (h : ¬(map X ℙ).HaveLebesgueDecomposition μ) : pdf X ℙ μ = 0 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {X : Ω → E},
  ¬(MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ → MeasureTheory.pdf X ℙ μ = 0
-- MeasureTheory.aemeasurable_of_pdf_ne_zero
theorem aemeasurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    (X : Ω → E) (h : ¬pdf X ℙ μ =ᵐ[μ] 0) : AEMeasurable X ℙ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} (X : Ω → E) (h : ¬MeasureTheory.pdf X ℙ μ =ᵐ[μ] 0)
Goal: AEMeasurable X ℙ
-- MeasureTheory.hasPDF_of_pdf_ne_zero
theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
    (hac : map X ℙ ≪ μ) (hpdf : ¬pdf X ℙ μ =ᵐ[μ] 0) : HasPDF X ℙ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {X : Ω → E} (hac : MeasureTheory.Measure.map X ℙ ≪ μ) (hpdf : ¬MeasureTheory.pdf X ℙ μ =ᵐ[μ] 0)
Goal: MeasureTheory.HasPDF X ℙ μ
-- MeasureTheory.measurable_pdf
@[measurability]
theorem measurable_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E := by volume_tac) : Measurable (pdf X ℙ μ) 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : MeasureTheory.Measure Ω) (μ : MeasureTheory.Measure E := by volume_tac)
Goal: Measurable (MeasureTheory.pdf X ℙ μ)
-- MeasureTheory.withDensity_pdf_le_map
theorem withDensity_pdf_le_map {_ : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E := by volume_tac) : μ.withDensity (pdf X ℙ μ) ≤ map X ℙ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E]
Goal: ∀ {x : MeasurableSpace Ω} (X : Ω → E) (ℙ : MeasureTheory.Measure Ω) (μ : autoParam (MeasureTheory.Measure E) _auto✝),
  MeasureTheory.Measure.withDensity μ (MeasureTheory.pdf X ℙ μ) ≤ MeasureTheory.Measure.map X ℙ
-- MeasureTheory.setLIntegral_pdf_le_map
theorem setLIntegral_pdf_le_map {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E := by volume_tac) (s : Set E) :
    ∫⁻ x in s, pdf X ℙ μ x ∂μ ≤ map X ℙ s 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : MeasureTheory.Measure Ω) (μ : MeasureTheory.Measure E := by volume_tac) (s : Set E)
Goal: ∫⁻ (x : E) in s, MeasureTheory.pdf X ℙ μ x ∂μ ≤ (MeasureTheory.Measure.map X ℙ) s
-- MeasureTheory.map_eq_withDensity_pdf
theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E := by volume_tac) [hX : HasPDF X ℙ μ] :
    map X ℙ = μ.withDensity (pdf X ℙ μ) 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : MeasureTheory.Measure Ω) (μ : MeasureTheory.Measure E := by volume_tac) [hX : MeasureTheory.HasPDF X ℙ μ]
Goal: MeasureTheory.Measure.map X ℙ = MeasureTheory.Measure.withDensity μ (MeasureTheory.pdf X ℙ μ)
-- MeasureTheory.map_eq_setLIntegral_pdf
theorem map_eq_setLIntegral_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E := by volume_tac) [hX : HasPDF X ℙ μ] {s : Set E}
    (hs : MeasurableSet s) : map X ℙ s = ∫⁻ x in s, pdf X ℙ μ x ∂μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : MeasureTheory.Measure Ω) (μ : MeasureTheory.Measure E := by volume_tac) [hX : MeasureTheory.HasPDF X ℙ μ] {s : Set E} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.map X ℙ) s = ∫⁻ (x : E) in s, MeasureTheory.pdf X ℙ μ x ∂μ
-- MeasureTheory.pdf.congr
protected theorem congr {X Y : Ω → E} (hXY : X =ᵐ[ℙ] Y) : pdf X ℙ μ = pdf Y ℙ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {X Y : Ω → E} (hXY : X =ᵐ[ℙ] Y)
Goal: MeasureTheory.pdf X ℙ μ = MeasureTheory.pdf Y ℙ μ
-- MeasureTheory.pdf.lintegral_eq_measure_univ
theorem lintegral_eq_measure_univ {X : Ω → E} [HasPDF X ℙ μ] :
    ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ]
Goal: ∫⁻ (x : E), MeasureTheory.pdf X ℙ μ x ∂μ = ℙ Set.univ
-- MeasureTheory.pdf.eq_of_map_eq_withDensity
theorem eq_of_map_eq_withDensity [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞)
    (hmf : AEMeasurable f μ) : map X ℙ = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ] (f : E → ℝ≥0∞) (hmf : AEMeasurable f μ)
Goal: MeasureTheory.Measure.map X ℙ = μ.withDensity f ↔ MeasureTheory.pdf X ℙ μ =ᵐ[μ] f
-- MeasureTheory.pdf.eq_of_map_eq_withDensity'
theorem eq_of_map_eq_withDensity' [SigmaFinite μ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞)
    (hmf : AEMeasurable f μ) : map X ℙ = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} [MeasureTheory.SigmaFinite μ] {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ] (f : E → ℝ≥0∞) (hmf : AEMeasurable f μ)
Goal: MeasureTheory.Measure.map X ℙ = μ.withDensity f ↔ MeasureTheory.pdf X ℙ μ =ᵐ[μ] f
-- MeasureTheory.pdf.ae_lt_top
nonrec theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} :
    ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure ℙ] {μ : MeasureTheory.Measure E} {X : Ω → E}
Goal: ∀ᵐ (x : E) ∂μ, MeasureTheory.pdf X ℙ μ x < ⊤
-- MeasureTheory.pdf.ofReal_toReal_ae_eq
nonrec theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
    (fun x => ENNReal.ofReal (pdf X ℙ μ x).toReal) =ᵐ[μ] pdf X ℙ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → E}
Goal: (fun x => ENNReal.ofReal (MeasureTheory.pdf X ℙ μ x).toReal) =ᵐ[μ] MeasureTheory.pdf X ℙ μ
-- MeasureTheory.pdf.lintegral_pdf_mul
theorem lintegral_pdf_mul {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ≥0∞}
    (hf : AEMeasurable f μ) : ∫⁻ x, pdf X ℙ μ x * f x ∂μ = ∫⁻ x, f (X x) ∂ℙ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ] {f : E → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: ∫⁻ (x : E), MeasureTheory.pdf X ℙ μ x * f x ∂μ = ∫⁻ (x : Ω), f (X x) ∂ℙ
-- MeasureTheory.pdf.integrable_pdf_smul_iff
theorem integrable_pdf_smul_iff [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → F}
    (hf : AEStronglyMeasurable f μ) :
    Integrable (fun x => (pdf X ℙ μ x).toReal • f x) μ ↔ Integrable (fun x => f (X x)) ℙ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ] {f : E → F} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.Integrable (fun x => (MeasureTheory.pdf X ℙ μ x).toReal • f x) μ ↔
  MeasureTheory.Integrable (fun x => f (X x)) ℙ
-- MeasureTheory.pdf.integral_pdf_smul
theorem integral_pdf_smul [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → F}
    (hf : AEStronglyMeasurable f μ) : ∫ x, (pdf X ℙ μ x).toReal • f x ∂μ = ∫ x, f (X x) ∂ℙ 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ] {f : E → F} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: ∫ (x : E), (MeasureTheory.pdf X ℙ μ x).toReal • f x ∂μ = ∫ (x : Ω), f (X x) ∂ℙ
-- MeasureTheory.pdf.quasiMeasurePreserving_hasPDF
theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] (hX : AEMeasurable X ℙ) {g : E → F}
    (hg : QuasiMeasurePreserving g μ ν) (hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν) :
    HasPDF (g ∘ X) ℙ ν 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {F : Type u_3} [MeasurableSpace F] {ν : MeasureTheory.Measure F} {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ] (hX : AEMeasurable X ℙ) {g : E → F} (hg : MeasureTheory.Measure.QuasiMeasurePreserving g μ ν) (hmap : (MeasureTheory.Measure.map g (MeasureTheory.Measure.map X ℙ)).HaveLebesgueDecomposition ν)
Goal: MeasureTheory.HasPDF (g ∘ X) ℙ ν
-- MeasureTheory.pdf.quasiMeasurePreserving_hasPDF'
theorem quasiMeasurePreserving_hasPDF' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
    [HasPDF X ℙ μ] (hX : AEMeasurable X ℙ) {g : E → F} (hg : QuasiMeasurePreserving g μ ν) :
    HasPDF (g ∘ X) ℙ ν 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {F : Type u_3} [MeasurableSpace F] {ν : MeasureTheory.Measure F} [MeasureTheory.IsFiniteMeasure ℙ] [MeasureTheory.SigmaFinite ν] {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ] (hX : AEMeasurable X ℙ) {g : E → F} (hg : MeasureTheory.Measure.QuasiMeasurePreserving g μ ν)
Goal: MeasureTheory.HasPDF (g ∘ X) ℙ ν