Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: 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✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun a ↦ f (μ.rnDeriv ν a).toReal) ν

Tactic:
refine integrable_of_le_of_le (f := fun a ↦ f (μ.rnDeriv ν a).toReal)
    (g₁ := fun x ↦ c * (μ.rnDeriv ν x).toReal + c')
    (g₂ := fun x ↦ ∫ b, f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, b)).toReal ∂(η x))
    ?_ ?_ ?_ (by fun_prop) ?_

Goal state after:
case refine_1
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✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ AEStronglyMeasurable (fun a ↦ f (μ.rnDeriv ν a).toReal) ν

case refine_2
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✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun x ↦ c * (μ.rnDeriv ν x).toReal + c') ≤ᵐ[ν] fun a ↦ f (μ.rnDeriv ν a).toReal

case refine_3
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✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun a ↦ f (μ.rnDeriv ν a).toReal) ≤ᵐ[ν] fun x ↦ ∫ (b : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, b)).toReal ∂η x

case refine_4
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✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ ∫ (b : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, b)).toReal ∂η x) ν