awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
e_a :
  ∫⁻ (b : 𝓨), μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a =
    ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a
⊢ f (∫⁻ (b : 𝓨), μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a).toReal =
    f (∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a).toReal

Tactic:
congr 2
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
⊢ ∫⁻ (b : 𝓨), μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a =
    ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a

Tactic:
lintegral_congr_ae ?_

Premises used: MeasureTheory.lintegral_congr_ae
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
e_a :
  (ae (F := Measure 𝓨) ((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨)).EventuallyEq (β := ℝ≥0∞)
    (fun (b : 𝓨) ↦ μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b)) fun (b : 𝓨) ↦ (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
⊢ ∫⁻ (b : 𝓨), μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a =
    ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a

Tactic:
refine lintegral_congr_ae ?_

Premises used: MeasureTheory.lintegral_congr_ae
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
b : 𝓨
hb : μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
⊢ μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)

Tactic:
hb
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h :
  ∀ (a_1 : 𝓨),
    Membership.mem (γ := Set 𝓨)
      {x : 𝓨 |
        Membership.mem (γ := Set 𝓨)
            {x : 𝓨 |
              (fun (b : 𝓨) ↦ μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)) x}
            x →
          Membership.mem (γ := Set 𝓨)
            {x : 𝓨 |
              (fun (x : 𝓨) ↦
                  Eq (α := ℝ≥0∞) ((fun (b : 𝓨) ↦ μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b)) x)
                    ((fun (b : 𝓨) ↦ (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)) x))
                x}
            x}
      a_1
h₁ h₂ :
  ∀ (b : 𝓨) (hb : μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
⊢ (ae (F := Measure 𝓨) ((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨)).EventuallyEq (β := ℝ≥0∞)
    (fun (b : 𝓨) ↦ μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b)) fun (b : 𝓨) ↦ (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)

Tactic:
filter_upwards [h_mul_eq] with b hb using hb
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
⊢ AEMeasurable (β := ℝ≥0∞) (_m := m𝓨) (fun (x : 𝓨) ↦ (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, x)) ((η : (a : 𝓧) → Measure 𝓨) a)

Tactic:
fun_prop
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
rw rw₁ :
  f (∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a).toReal =
    f (∫⁻ (a_1 : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, a_1) ∂(η : (a : 𝓧) → Measure 𝓨) a).toReal
⊢ f (∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a).toReal =
    f (@integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) a) fun (b : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)).toReal)

Tactic:
rw [integral_toReal (by fun_prop) h_lt_top]

Premises used: MeasureTheory.integral_toReal
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
rw :
  f (⨍ (x : 𝓨), ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, x)).toReal ∂(η : (a : 𝓧) → Measure 𝓨) a) ≤
    ⨍ (x : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, x)).toReal ∂(η : (a : 𝓧) → Measure 𝓨) a
⊢ f (@integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) a) fun (b : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)).toReal) ≤
    @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) a) fun (b : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)).toReal

Tactic:
rw [← average_eq_integral, ← average_eq_integral]

Premises used: MeasureTheory.average_eq_integral
-- Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
inst_2 : IsMarkovKernel κ
inst_3 : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int_1 :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal)
        ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top_1 : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν,
      @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) x)) ∧
    Integrable (ε := ℝ)
      (fun (x : 𝓧) ↦
        @integral _ _ _ _ m𝓨 ((η : (a : 𝓧) → Measure 𝓨) x) fun (y : 𝓨) ↦ ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖)
      ν
h_ae1 :
  ∀ᵐ (a : 𝓧) ∂ν,
    μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν,
    ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
      μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂(η : (a : 𝓧) → Measure 𝓨) a = μ.rnDeriv ν a
h_mul_eq :
  ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
h_int : @Integrable ℝ _ _ _ m𝓨 (fun (y : 𝓨) ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) ((η : (a : 𝓧) → Measure 𝓨) a)
⊢ ∀ᵐ (x : 𝓨) ∂((η : (a : 𝓧) → Measure 𝓨) a : Measure 𝓨),
    Membership.mem (γ := Set ℝ) (Ici 0) ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, x)).toReal

Tactic:
simp