-- Mathlib/Probability/Kernel/RadonNikodym.lean
/-- Lebesgue decomposition of a finite kernel `κ` with respect to another one `η`.
`κ` is the sum of an absolutely continuous part `withDensity η (rnDeriv κ η)` and a singular part
`singularPart κ η`. -/
lemma rnDeriv_add_singularPart (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
    withDensity η (rnDeriv κ η) + singularPart κ η = κ

Proof state:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
hm : MeasurableSet (κ.mutuallySingularSetSlice η a)
⊢ ((η.withDensity (κ.rnDeriv η)) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) +
      ((κ.singularPart η) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a)

Tactic:
simp only [measure_union (Disjoint.mono inter_subset_right le_rfl disjoint_sdiff_right)
  (hs.diff hm)]

Result:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
hm : MeasurableSet (κ.mutuallySingularSetSlice η a)
⊢ ((η.withDensity (κ.rnDeriv η)) a) (s ∩ κ.mutuallySingularSetSlice η a) +
        ((η.withDensity (κ.rnDeriv η)) a) (s \ κ.mutuallySingularSetSlice η a) +
      (((κ.singularPart η) a) (s ∩ κ.mutuallySingularSetSlice η a) +
        ((κ.singularPart η) a) (s \ κ.mutuallySingularSetSlice η a)) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a) + (κ a) (s \ κ.mutuallySingularSetSlice η a)

Proof state:
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
⊢ η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ

Tactic:
ext a s hs

Result:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) s = (κ a) s

Proof state:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
hm : MeasurableSet (κ.mutuallySingularSetSlice η a)
⊢ ((η.withDensity (κ.rnDeriv η)) a) (s ∩ κ.mutuallySingularSetSlice η a) +
        ((η.withDensity (κ.rnDeriv η)) a) (s \ κ.mutuallySingularSetSlice η a) +
      (((κ.singularPart η) a) (s ∩ κ.mutuallySingularSetSlice η a) +
        ((κ.singularPart η) a) (s \ κ.mutuallySingularSetSlice η a)) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a) + (κ a) (s \ κ.mutuallySingularSetSlice η a)

Tactic:
rw [singularPart_of_subset_mutuallySingularSetSlice (hs.inter hm) inter_subset_right,
  singularPart_of_subset_compl_mutuallySingularSetSlice (diff_subset_iff.mpr (by simp)),
  add_zero, withDensity_rnDeriv_of_subset_mutuallySingularSetSlice inter_subset_right,
  zero_add, withDensity_rnDeriv_of_subset_compl_mutuallySingularSetSlice (hs.diff hm)
    (diff_subset_iff.mpr (by simp)), add_comm]

Result:
no goals

Proof state:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) s = (κ a) s

Tactic:
rw [← inter_union_diff s (mutuallySingularSetSlice κ η a)]

Result:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a)
      (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a)

Proof state:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((η.withDensity (κ.rnDeriv η)) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) +
      ((κ.singularPart η) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a)

Tactic:
have hm := measurableSet_mutuallySingularSetSlice κ η a

Result:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
hm : MeasurableSet (κ.mutuallySingularSetSlice η a)
⊢ ((η.withDensity (κ.rnDeriv η)) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) +
      ((κ.singularPart η) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a)

Proof state:
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
hm : MeasurableSet (κ.mutuallySingularSetSlice η a)
⊢ s ⊆ κ.mutuallySingularSetSlice η a ∪ (κ.mutuallySingularSetSlice η a)ᶜ

Tactic:
simp

Result:
no goals

Proof state:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a)
      (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a)

Tactic:
simp only [coe_add, Pi.add_apply, Measure.coe_add]

Result:
case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((η.withDensity (κ.rnDeriv η)) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) +
      ((κ.singularPart η) a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a ∪ s \ κ.mutuallySingularSetSlice η a)