ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.condDistrib_apply_of_ne_zero
lemma condDistrib_apply_of_ne_zero [MeasurableSingletonClass β]
    (hY : Measurable Y) (x : β) (hX : μ.map X {x} ≠ 0) (s : Set Ω) :
    condDistrib Y X μ x s = (μ.map X {x})⁻¹ * μ.map (fun a => (X a, Y a)) ({x} ×ˢ s) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} [MeasurableSingletonClass β] (hY : Measurable Y) (x : β) (hX : (MeasureTheory.Measure.map X μ) {x} ≠ 0) (s : Set Ω)
Goal: ((ProbabilityTheory.condDistrib Y X μ) x) s =
  ((MeasureTheory.Measure.map X μ) {x})⁻¹ * (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)
-- ProbabilityTheory.measurable_condDistrib
theorem measurable_condDistrib (hs : MeasurableSet s) :
    Measurable[mβ.comap X] fun a => condDistrib Y X μ (X a) s 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {s : Set Ω} (hs : MeasurableSet s)
Goal: Measurable fun a => ((ProbabilityTheory.condDistrib Y X μ) (X a)) s
-- MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
theorem _root_.MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
    (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
    (∀ᵐ a ∂μ.map X, Integrable (fun ω => f (a, ω)) (condDistrib Y X μ a)) ∧
      Integrable (fun a => ∫ ω, ‖f (a, ω)‖ ∂condDistrib Y X μ a) (μ.map X) ↔
    Integrable f (μ.map fun a => (X a, Y a)) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} (hY : AEMeasurable Y μ) (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: (∀ᵐ (a : β) ∂MeasureTheory.Measure.map X μ,
      MeasureTheory.Integrable (fun ω => f (a, ω)) ((ProbabilityTheory.condDistrib Y X μ) a)) ∧
    MeasureTheory.Integrable (fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂(ProbabilityTheory.condDistrib Y X μ) a)
      (MeasureTheory.Measure.map X μ) ↔
  MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ)
-- MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
    (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
    AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂condDistrib Y X μ x) (μ.map X) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] (hY : AEMeasurable Y μ) (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : Ω), f (x, y) ∂(ProbabilityTheory.condDistrib Y X μ) x)
  (MeasureTheory.Measure.map X μ)
-- MeasureTheory.AEStronglyMeasurable.integral_condDistrib
theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condDistrib (hX : AEMeasurable X μ)
    (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
    AEStronglyMeasurable (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.AEStronglyMeasurable (fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ
-- ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib
theorem aestronglyMeasurable'_integral_condDistrib (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
    (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
    AEStronglyMeasurable' (mβ.comap X) (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.AEStronglyMeasurable' (MeasurableSpace.comap X mβ)
  (fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ
-- ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd
theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
    (κ : Kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
    ∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} (hX : Measurable X) (hY : Measurable Y) (κ : ProbabilityTheory.Kernel β Ω) [ProbabilityTheory.IsFiniteKernel κ] (hκ : MeasureTheory.Measure.map (fun x => (X x, Y x)) μ = MeasureTheory.Measure.map X μ ⊗ₘ κ)
Goal: ∀ᵐ (x : β) ∂MeasureTheory.Measure.map X μ, κ x = (ProbabilityTheory.condDistrib Y X μ) x
-- ProbabilityTheory.integrable_toReal_condDistrib
theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableSet s) :
    Integrable (fun a => (condDistrib Y X μ (X a) s).toReal) μ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {s : Set Ω} (hX : AEMeasurable X μ) (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (fun a => (((ProbabilityTheory.condDistrib Y X μ) (X a)) s).toReal) μ
-- MeasureTheory.Integrable.condDistrib_ae_map
theorem _root_.MeasureTheory.Integrable.condDistrib_ae_map
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    ∀ᵐ b ∂μ.map X, Integrable (fun ω => f (b, ω)) (condDistrib Y X μ b) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: ∀ᵐ (b : β) ∂MeasureTheory.Measure.map X μ,
  MeasureTheory.Integrable (fun ω => f (b, ω)) ((ProbabilityTheory.condDistrib Y X μ) b)
-- MeasureTheory.Integrable.condDistrib_ae
theorem _root_.MeasureTheory.Integrable.condDistrib_ae (hX : AEMeasurable X μ)
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    ∀ᵐ a ∂μ, Integrable (fun ω => f (X a, ω)) (condDistrib Y X μ (X a)) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: ∀ᵐ (a : α) ∂μ, MeasureTheory.Integrable (fun ω => f (X a, ω)) ((ProbabilityTheory.condDistrib Y X μ) (X a))
-- MeasureTheory.Integrable.integral_norm_condDistrib_map
theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib_map
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂condDistrib Y X μ x) (μ.map X) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.Integrable (fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂(ProbabilityTheory.condDistrib Y X μ) x)
  (MeasureTheory.Measure.map X μ)
-- MeasureTheory.Integrable.integral_norm_condDistrib
theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib (hX : AEMeasurable X μ)
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable (fun a => ∫ y, ‖f (X a, y)‖ ∂condDistrib Y X μ (X a)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.Integrable (fun a => ∫ (y : Ω), ‖f (X a, y)‖ ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ
-- MeasureTheory.Integrable.norm_integral_condDistrib_map
theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib_map
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable (fun x => ‖∫ y, f (x, y) ∂condDistrib Y X μ x‖) (μ.map X) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.Integrable (fun x => ‖∫ (y : Ω), f (x, y) ∂(ProbabilityTheory.condDistrib Y X μ) x‖)
  (MeasureTheory.Measure.map X μ)
-- MeasureTheory.Integrable.norm_integral_condDistrib
theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib (hX : AEMeasurable X μ)
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable (fun a => ‖∫ y, f (X a, y) ∂condDistrib Y X μ (X a)‖) μ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.Integrable (fun a => ‖∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)‖) μ
-- MeasureTheory.Integrable.integral_condDistrib_map
theorem _root_.MeasureTheory.Integrable.integral_condDistrib_map
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable (fun x => ∫ y, f (x, y) ∂condDistrib Y X μ x) (μ.map X) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.Integrable (fun x => ∫ (y : Ω), f (x, y) ∂(ProbabilityTheory.condDistrib Y X μ) x)
  (MeasureTheory.Measure.map X μ)
-- MeasureTheory.Integrable.integral_condDistrib
theorem _root_.MeasureTheory.Integrable.integral_condDistrib (hX : AEMeasurable X μ)
    (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: MeasureTheory.Integrable (fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ
-- ProbabilityTheory.setLIntegral_preimage_condDistrib
theorem setLIntegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurable Y μ)
    (hs : MeasurableSet s) (ht : MeasurableSet t) :
    ∫⁻ a in X ⁻¹' t, condDistrib Y X μ (X a) s ∂μ = μ (X ⁻¹' t ∩ Y ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {s : Set Ω} {t : Set β} (hX : Measurable X) (hY : AEMeasurable Y μ) (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: ∫⁻ (a : α) in X ⁻¹' t, ((ProbabilityTheory.condDistrib Y X μ) (X a)) s ∂μ = μ (X ⁻¹' t ∩ Y ⁻¹' s)
-- ProbabilityTheory.setLIntegral_condDistrib_of_measurableSet
theorem setLIntegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEMeasurable Y μ)
    (hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet[mβ.comap X] t) :
    ∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ = μ (t ∩ Y ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {s : Set Ω} (hX : Measurable X) (hY : AEMeasurable Y μ) (hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet t)
Goal: ∫⁻ (a : α) in t, ((ProbabilityTheory.condDistrib Y X μ) (X a)) s ∂μ = μ (t ∩ Y ⁻¹' s)
-- ProbabilityTheory.condDistrib_ae_eq_condexp
theorem condDistrib_ae_eq_condexp (hX : Measurable X) (hY : Measurable Y) (hs : MeasurableSet s) :
    (fun a => (condDistrib Y X μ (X a) s).toReal) =ᵐ[μ] μ⟦Y ⁻¹' s|mβ.comap X⟧ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {s : Set Ω} (hX : Measurable X) (hY : Measurable Y) (hs : MeasurableSet s)
Goal: (fun a => (((ProbabilityTheory.condDistrib Y X μ) (X a)) s).toReal) =ᵐ[μ]
  μ[(Y ⁻¹' s).indicator fun ω => 1|MeasurableSpace.comap X mβ]
-- ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib'
theorem condexp_prod_ae_eq_integral_condDistrib' [NormedSpace ℝ F] [CompleteSpace F]
    (hX : Measurable X) (hY : AEMeasurable Y μ)
    (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
    μ[fun a => f (X a, Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f (X a,y) ∂condDistrib Y X μ (X a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] [CompleteSpace F] (hX : Measurable X) (hY : AEMeasurable Y μ) (hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))
Goal: μ[fun a => f (X a, Y a)|MeasurableSpace.comap X mβ] =ᵐ[μ] fun a =>
  ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)
-- ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib₀
theorem condexp_prod_ae_eq_integral_condDistrib₀ [NormedSpace ℝ F] [CompleteSpace F]
    (hX : Measurable X) (hY : AEMeasurable Y μ)
    (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a)))
    (hf_int : Integrable (fun a => f (X a, Y a)) μ) :
    μ[fun a => f (X a, Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] [CompleteSpace F] (hX : Measurable X) (hY : AEMeasurable Y μ) (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ)) (hf_int : MeasureTheory.Integrable (fun a => f (X a, Y a)) μ)
Goal: μ[fun a => f (X a, Y a)|MeasurableSpace.comap X mβ] =ᵐ[μ] fun a =>
  ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)
-- ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib
theorem condexp_prod_ae_eq_integral_condDistrib [NormedSpace ℝ F] [CompleteSpace F]
    (hX : Measurable X) (hY : AEMeasurable Y μ) (hf : StronglyMeasurable f)
    (hf_int : Integrable (fun a => f (X a, Y a)) μ) :
    μ[fun a => f (X a, Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [NormedSpace ℝ F] [CompleteSpace F] (hX : Measurable X) (hY : AEMeasurable Y μ) (hf : MeasureTheory.StronglyMeasurable f) (hf_int : MeasureTheory.Integrable (fun a => f (X a, Y a)) μ)
Goal: μ[fun a => f (X a, Y a)|MeasurableSpace.comap X mβ] =ᵐ[μ] fun a =>
  ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)
-- ProbabilityTheory.condexp_ae_eq_integral_condDistrib
theorem condexp_ae_eq_integral_condDistrib [NormedSpace ℝ F] [CompleteSpace F] (hX : Measurable X)
    (hY : AEMeasurable Y μ) {f : Ω → F} (hf : StronglyMeasurable f)
    (hf_int : Integrable (fun a => f (Y a)) μ) :
    μ[fun a => f (Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f y ∂condDistrib Y X μ (X a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} [NormedSpace ℝ F] [CompleteSpace F] (hX : Measurable X) (hY : AEMeasurable Y μ) {f : Ω → F} (hf : MeasureTheory.StronglyMeasurable f) (hf_int : MeasureTheory.Integrable (fun a => f (Y a)) μ)
Goal: μ[fun a => f (Y a)|MeasurableSpace.comap X mβ] =ᵐ[μ] fun a =>
  ∫ (y : Ω), f y ∂(ProbabilityTheory.condDistrib Y X μ) (X a)
-- ProbabilityTheory.condexp_ae_eq_integral_condDistrib'
theorem condexp_ae_eq_integral_condDistrib' {Ω} [NormedAddCommGroup Ω] [NormedSpace ℝ Ω]
    [CompleteSpace Ω] [MeasurableSpace Ω] [BorelSpace Ω] [SecondCountableTopology Ω] {Y : α → Ω}
    (hX : Measurable X) (hY_int : Integrable Y μ) :
    μ[Y|mβ.comap X] =ᵐ[μ] fun a => ∫ y, y ∂condDistrib Y X μ (X a) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {X : α → β} {mβ : MeasurableSpace β} {Ω : Type u_5} [NormedAddCommGroup Ω] [NormedSpace ℝ Ω] [CompleteSpace Ω] [MeasurableSpace Ω] [BorelSpace Ω] [SecondCountableTopology Ω] {Y : α → Ω} (hX : Measurable X) (hY_int : MeasureTheory.Integrable Y μ)
Goal: μ[Y|MeasurableSpace.comap X mβ] =ᵐ[μ] fun a => ∫ (y : Ω), y ∂(ProbabilityTheory.condDistrib Y X μ) (X a)
-- MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk
theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk
    {Ω F} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : Measure Ω} [TopologicalSpace F] {f : Ω → F}
    (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) 
Arguments: {β : Type u_2} {mβ : MeasurableSpace β} {Ω : Type u_5} {F : Type u_6} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : MeasureTheory.Measure Ω} [TopologicalSpace F] {f : Ω → F} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (X ω, ω)) μ)
-- MeasureTheory.Integrable.comp_snd_map_prod_mk
theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_mk
    {Ω} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : Measure Ω} {f : Ω → F} (hf_int : Integrable f μ) :
    Integrable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) 
Arguments: {β : Type u_2} {F : Type u_4} [NormedAddCommGroup F] {mβ : MeasurableSpace β} {Ω : Type u_5} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : MeasureTheory.Measure Ω} {f : Ω → F} (hf_int : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (X ω, ω)) μ)
-- ProbabilityTheory.aestronglyMeasurable_comp_snd_map_prod_mk_iff
theorem aestronglyMeasurable_comp_snd_map_prod_mk_iff {Ω F} {_ : MeasurableSpace Ω}
    [TopologicalSpace F] {X : Ω → β} {μ : Measure Ω} (hX : Measurable X) {f : Ω → F} :
    AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) ↔
    AEStronglyMeasurable f μ 
Arguments: {β : Type u_2} {mβ : MeasurableSpace β} {Ω : Type u_5} {F : Type u_6}
Goal: ∀ {x : MeasurableSpace Ω} [inst : TopologicalSpace F] {X : Ω → β} {μ : MeasureTheory.Measure Ω},
  Measurable X →
    ∀ {f : Ω → F},
      MeasureTheory.AEStronglyMeasurable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (X ω, ω)) μ) ↔
        MeasureTheory.AEStronglyMeasurable f μ
-- ProbabilityTheory.integrable_comp_snd_map_prod_mk_iff
theorem integrable_comp_snd_map_prod_mk_iff {Ω} {_ : MeasurableSpace Ω} {X : Ω → β} {μ : Measure Ω}
    (hX : Measurable X) {f : Ω → F} :
    Integrable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) ↔ Integrable f μ 
Arguments: {β : Type u_2} {F : Type u_4} [NormedAddCommGroup F] {mβ : MeasurableSpace β} {Ω : Type u_5}
Goal: ∀ {x : MeasurableSpace Ω} {X : Ω → β} {μ : MeasureTheory.Measure Ω},
  Measurable X →
    ∀ {f : Ω → F},
      MeasureTheory.Integrable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (X ω, ω)) μ) ↔
        MeasureTheory.Integrable f μ
-- ProbabilityTheory.condexp_ae_eq_integral_condDistrib_id
theorem condexp_ae_eq_integral_condDistrib_id [NormedSpace ℝ F] [CompleteSpace F] {X : Ω → β}
    {μ : Measure Ω} [IsFiniteMeasure μ] (hX : Measurable X) {f : Ω → F} (hf_int : Integrable f μ) :
    μ[f|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f y ∂condDistrib id X μ (X a) 
Arguments: {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [NormedAddCommGroup F] {mβ : MeasurableSpace β} [NormedSpace ℝ F] [CompleteSpace F] {X : Ω → β} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) {f : Ω → F} (hf_int : MeasureTheory.Integrable f μ)
Goal: μ[f|MeasurableSpace.comap X mβ] =ᵐ[μ] fun a => ∫ (y : Ω), f y ∂(ProbabilityTheory.condDistrib id X μ) (X a)
-- ProbabilityTheory.cdf_nonneg
lemma cdf_nonneg (x : ℝ) : 0 ≤ cdf μ x 
Arguments: (μ : MeasureTheory.Measure ℝ) (x : ℝ)
Goal: 0 ≤ ↑(ProbabilityTheory.cdf μ) x
-- ProbabilityTheory.cdf_le_one
lemma cdf_le_one (x : ℝ) : cdf μ x ≤ 1 
Arguments: (μ : MeasureTheory.Measure ℝ) (x : ℝ)
Goal: ↑(ProbabilityTheory.cdf μ) x ≤ 1