Theorem: ProbabilityTheory.kernel.singularPart_eq_zero_iff_absolutelyContinuous
File: Mathlib/Probability/Kernel/RadonNikodym.lean

tactic: conv_rhs => rw [← rnDeriv_add_singularPart κ η, coeFn_add, Pi.add_apply]
state_before: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ (singularPart κ η) a = 0 ↔ κ a ≪ η a
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ (singularPart κ η) a = 0 ↔ (withDensity η (rnDeriv κ η)) a + (singularPart κ η) a ≪ η a

tactic: refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
state_before: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ (singularPart κ η) a = 0 ↔ (withDensity η (rnDeriv κ η)) a + (singularPart κ η) a ≪ η a
state_after: case refine_1
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (singularPart κ η) a = 0
⊢ (withDensity η (rnDeriv κ η)) a + (singularPart κ η) a ≪ η a

case refine_2
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (withDensity η (rnDeriv κ η)) a + (singularPart κ η) a ≪ η a
⊢ (singularPart κ η) a = 0

tactic: rw [Measure.AbsolutelyContinuous.add_left_iff] at h
state_before: case refine_2
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (withDensity η (rnDeriv κ η)) a + (singularPart κ η) a ≪ η a
⊢ (singularPart κ η) a = 0
state_after: case refine_2
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (withDensity η (rnDeriv κ η)) a ≪ η a ∧ (singularPart κ η) a ≪ η a
⊢ (singularPart κ η) a = 0

tactic: exact Measure.eq_zero_of_absolutelyContinuous_of_mutuallySingular h.2
  (mutuallySingular_singularPart _ _ _)
state_before: case refine_2
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (withDensity η (rnDeriv κ η)) a ≪ η a ∧ (singularPart κ η) a ≪ η a
⊢ (singularPart κ η) a = 0
state_after: no goals

tactic: rw [h, add_zero]
state_before: case refine_1
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (singularPart κ η) a = 0
⊢ (withDensity η (rnDeriv κ η)) a + (singularPart κ η) a ≪ η a
state_after: case refine_1
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (singularPart κ η) a = 0
⊢ (withDensity η (rnDeriv κ η)) a ≪ η a

tactic: exact withDensity_absolutelyContinuous _ _
state_before: case refine_1
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (singularPart κ η) a = 0
⊢ (withDensity η (rnDeriv κ η)) a ≪ η a
state_after: no goals