File: Mathlib/Probability/Kernel/Disintegration/Integral.lean

full_name: ProbabilityTheory.lintegral_condKernel_mem
code: lemma lintegral_condKernel_mem (a : α) {s : Set (β × Ω)} (hs : MeasurableSet s) :
    ∫⁻ x, Kernel.condKernel κ (a, x) {y | (x, y) ∈ s} ∂(Kernel.fst κ a) = κ a s 
goal: ∫⁻ (x : β), (κ.condKernel (a, x)) {y | (x, y) ∈ s} ∂κ.fst a = (κ a) s

full_name: ProbabilityTheory.setLIntegral_condKernel_eq_measure_prod
code: lemma setLIntegral_condKernel_eq_measure_prod (a : α) {s : Set β} (hs : MeasurableSet s)
    {t : Set Ω} (ht : MeasurableSet t) :
    ∫⁻ b in s, Kernel.condKernel κ (a, b) t ∂(Kernel.fst κ a) = κ a (s ×ˢ t) 
goal: ∫⁻ (b : β) in s, (κ.condKernel (a, b)) t ∂κ.fst a = (κ a) (s ×ˢ t)

full_name: ProbabilityTheory.lintegral_condKernel
code: lemma lintegral_condKernel (hf : Measurable f) (a : α) :
    ∫⁻ b, ∫⁻ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a) = ∫⁻ x, f x ∂(κ a) 
goal: ∫⁻ (b : β), ∫⁻ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω), f x ∂κ a

full_name: ProbabilityTheory.setLIntegral_condKernel
code: lemma setLIntegral_condKernel (hf : Measurable f) (a : α) {s : Set β}
    (hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t) :
    ∫⁻ b in s, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫⁻ x in s ×ˢ t, f x ∂(κ a) 
goal: ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in s ×ˢ t, f x ∂κ a

full_name: ProbabilityTheory.setLIntegral_condKernel_univ_right
code: lemma setLIntegral_condKernel_univ_right (hf : Measurable f) (a : α) {s : Set β}
    (hs : MeasurableSet s) :
    ∫⁻ b in s, ∫⁻ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫⁻ x in s ×ˢ Set.univ, f x ∂(κ a) 
goal: ∫⁻ (b : β) in s, ∫⁻ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in s ×ˢ Set.univ, f x ∂κ a

full_name: ProbabilityTheory.setLIntegral_condKernel_univ_left
code: lemma setLIntegral_condKernel_univ_left (hf : Measurable f) (a : α) {t : Set Ω}
    (ht : MeasurableSet t) :
    ∫⁻ b, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫⁻ x in Set.univ ×ˢ t, f x ∂(κ a) 
goal: ∫⁻ (b : β), ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in Set.univ ×ˢ t, f x ∂κ a

full_name: MeasureTheory.AEStronglyMeasurable.integral_kernel_condKernel
code: lemma _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_condKernel (a : α)
    (hf : AEStronglyMeasurable f (κ a)) :
    AEStronglyMeasurable (fun x ↦ ∫ y, f (x, y) ∂(Kernel.condKernel κ (a, x)))
      (Kernel.fst κ a) 
goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : Ω), f (x, y) ∂κ.condKernel (a, x)) (κ.fst a)

full_name: ProbabilityTheory.integral_condKernel
code: lemma integral_condKernel (a : α) (hf : Integrable f (κ a)) :
    ∫ b, ∫ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a) = ∫ x, f x ∂(κ a) 
goal: ∫ (b : β), ∫ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫ (x : β × Ω), f x ∂κ a

full_name: ProbabilityTheory.setIntegral_condKernel
code: lemma setIntegral_condKernel (a : α) {s : Set β} (hs : MeasurableSet s)
    {t : Set Ω} (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) (κ a)) :
    ∫ b in s, ∫ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫ x in s ×ˢ t, f x ∂(κ a) 
goal: ∫ (b : β) in s, ∫ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫ (x : β × Ω) in s ×ˢ t, f x ∂κ a

full_name: ProbabilityTheory.setIntegral_condKernel_univ_right
code: lemma setIntegral_condKernel_univ_right (a : α) {s : Set β} (hs : MeasurableSet s)
    (hf : IntegrableOn f (s ×ˢ Set.univ) (κ a)) :
    ∫ b in s, ∫ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫ x in s ×ˢ Set.univ, f x ∂(κ a) 
goal: ∫ (b : β) in s, ∫ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫ (x : β × Ω) in s ×ˢ Set.univ, f x ∂κ a

full_name: ProbabilityTheory.setIntegral_condKernel_univ_left
code: lemma setIntegral_condKernel_univ_left (a : α) {t : Set Ω} (ht : MeasurableSet t)
    (hf : IntegrableOn f (Set.univ ×ˢ t) (κ a)) :
    ∫ b, ∫ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫ x in Set.univ ×ˢ t, f x ∂(κ a) 
goal: ∫ (b : β), ∫ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫ (x : β × Ω) in Set.univ ×ˢ t, f x ∂κ a

full_name: MeasureTheory.Measure.lintegral_condKernel_mem
code: lemma lintegral_condKernel_mem {s : Set (β × Ω)} (hs : MeasurableSet s) :
    ∫⁻ x, ρ.condKernel x {y | (x, y) ∈ s} ∂ρ.fst = ρ s 
goal: ∫⁻ (x : β), (ρ.condKernel x) {y | (x, y) ∈ s} ∂ρ.fst = ρ s

full_name: MeasureTheory.Measure.setLIntegral_condKernel_eq_measure_prod
code: lemma setLIntegral_condKernel_eq_measure_prod {s : Set β} (hs : MeasurableSet s) {t : Set Ω}
    (ht : MeasurableSet t) :
    ∫⁻ b in s, ρ.condKernel b t ∂ρ.fst = ρ (s ×ˢ t) 
goal: ∫⁻ (b : β) in s, (ρ.condKernel b) t ∂ρ.fst = ρ (s ×ˢ t)

full_name: MeasureTheory.Measure.lintegral_condKernel
code: lemma lintegral_condKernel (hf : Measurable f) :
    ∫⁻ b, ∫⁻ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫⁻ x, f x ∂ρ 
goal: ∫⁻ (b : β), ∫⁻ (ω : Ω), f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫⁻ (x : β × Ω), f x ∂ρ

full_name: MeasureTheory.Measure.setLIntegral_condKernel
code: lemma setLIntegral_condKernel (hf : Measurable f) {s : Set β}
    (hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t) :
    ∫⁻ b in s, ∫⁻ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst
      = ∫⁻ x in s ×ˢ t, f x ∂ρ 
goal: ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in t, f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫⁻ (x : β × Ω) in s ×ˢ t, f x ∂ρ

full_name: MeasureTheory.Measure.setLIntegral_condKernel_univ_right
code: lemma setLIntegral_condKernel_univ_right (hf : Measurable f) {s : Set β}
    (hs : MeasurableSet s) :
    ∫⁻ b in s, ∫⁻ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst
      = ∫⁻ x in s ×ˢ Set.univ, f x ∂ρ 
goal: ∫⁻ (b : β) in s, ∫⁻ (ω : Ω), f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫⁻ (x : β × Ω) in s ×ˢ Set.univ, f x ∂ρ

full_name: MeasureTheory.Measure.setLIntegral_condKernel_univ_left
code: lemma setLIntegral_condKernel_univ_left (hf : Measurable f) {t : Set Ω}
    (ht : MeasurableSet t) :
    ∫⁻ b, ∫⁻ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst
      = ∫⁻ x in Set.univ ×ˢ t, f x ∂ρ 
goal: ∫⁻ (b : β), ∫⁻ (ω : Ω) in t, f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫⁻ (x : β × Ω) in Set.univ ×ˢ t, f x ∂ρ

full_name: MeasureTheory.AEStronglyMeasurable.integral_condKernel
code: lemma _root_.MeasureTheory.AEStronglyMeasurable.integral_condKernel
    (hf : AEStronglyMeasurable f ρ) :
    AEStronglyMeasurable (fun x ↦ ∫ y, f (x, y) ∂ρ.condKernel x) ρ.fst 
goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : Ω), f (x, y) ∂ρ.condKernel x) ρ.fst

full_name: MeasureTheory.Measure.integral_condKernel
code: lemma integral_condKernel (hf : Integrable f ρ) :
    ∫ b, ∫ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x, f x ∂ρ 
goal: ∫ (b : β), ∫ (ω : Ω), f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω), f x ∂ρ

full_name: MeasureTheory.Measure.setIntegral_condKernel
code: lemma setIntegral_condKernel {s : Set β} (hs : MeasurableSet s)
    {t : Set Ω} (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) ρ) :
    ∫ b in s, ∫ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in s ×ˢ t, f x ∂ρ 
goal: ∫ (b : β) in s, ∫ (ω : Ω) in t, f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω) in s ×ˢ t, f x ∂ρ

full_name: MeasureTheory.Measure.setIntegral_condKernel_univ_right
code: lemma setIntegral_condKernel_univ_right {s : Set β} (hs : MeasurableSet s)
    (hf : IntegrableOn f (s ×ˢ Set.univ) ρ) :
    ∫ b in s, ∫ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in s ×ˢ Set.univ, f x ∂ρ 
goal: ∫ (b : β) in s, ∫ (ω : Ω), f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω) in s ×ˢ Set.univ, f x ∂ρ

full_name: MeasureTheory.Measure.setIntegral_condKernel_univ_left
code: lemma setIntegral_condKernel_univ_left {t : Set Ω} (ht : MeasurableSet t)
    (hf : IntegrableOn f (Set.univ ×ˢ t) ρ) :
    ∫ b, ∫ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in Set.univ ×ˢ t, f x ∂ρ 
goal: ∫ (b : β), ∫ (ω : Ω) in t, f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω) in Set.univ ×ˢ t, f x ∂ρ

full_name: MeasureTheory.AEStronglyMeasurable.ae_integrable_condKernel_iff
code: theorem AEStronglyMeasurable.ae_integrable_condKernel_iff {f : α × Ω → F}
    (hf : AEStronglyMeasurable f ρ) :
    (∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a)) ∧
      Integrable (fun a ↦ ∫ ω, ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔ Integrable f ρ 
goal: (∀ᵐ (a : α) ∂ρ.fst, MeasureTheory.Integrable (fun ω => f (a, ω)) (ρ.condKernel a)) ∧
    MeasureTheory.Integrable (fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔
  MeasureTheory.Integrable f ρ

full_name: MeasureTheory.Integrable.condKernel_ae
code: theorem Integrable.condKernel_ae {f : α × Ω → F} (hf_int : Integrable f ρ) :
    ∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a) 
goal: ∀ᵐ (a : α) ∂ρ.fst, MeasureTheory.Integrable (fun ω => f (a, ω)) (ρ.condKernel a)

full_name: MeasureTheory.Integrable.integral_norm_condKernel
code: theorem Integrable.integral_norm_condKernel {f : α × Ω → F} (hf_int : Integrable f ρ) :
    Integrable (fun x ↦ ∫ y, ‖f (x, y)‖ ∂ρ.condKernel x) ρ.fst 
goal: MeasureTheory.Integrable (fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂ρ.condKernel x) ρ.fst

full_name: MeasureTheory.Integrable.norm_integral_condKernel
code: theorem Integrable.norm_integral_condKernel {f : α × Ω → E} (hf_int : Integrable f ρ) :
    Integrable (fun x ↦ ‖∫ y, f (x, y) ∂ρ.condKernel x‖) ρ.fst 
goal: MeasureTheory.Integrable (fun x => ‖∫ (y : Ω), f (x, y) ∂ρ.condKernel x‖) ρ.fst

full_name: MeasureTheory.Integrable.integral_condKernel
code: theorem Integrable.integral_condKernel {f : α × Ω → E} (hf_int : Integrable f ρ) :
    Integrable (fun x ↦ ∫ y, f (x, y) ∂ρ.condKernel x) ρ.fst 
goal: MeasureTheory.Integrable (fun x => ∫ (y : Ω), f (x, y) ∂ρ.condKernel x) ρ.fst