-- Mathlib/MeasureTheory/Decomposition/Exhaustion.lean
lemma sigmaFinite_restrict_sigmaFiniteSetWRT' (μ ν : Measure α) [IsFiniteMeasure ν] :
    SigmaFinite (μ.restrict (μ.sigmaFiniteSetWRT' ν))

Proof state:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p =>
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm
n : ℕ
⊢ (μ.restrict (μ.sigmaFiniteSetWRT' ν)) (μ.sigmaFiniteSetWRT' ν)ᶜ < ⊤ ∧
    (μ.restrict (μ.sigmaFiniteSetWRT' ν))
        (spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair n).1)) (Nat.unpair n).2 ∩
          μ.sigmaFiniteSetGE ν (Nat.unpair n).1) <
      ⊤

Tactic:
rw [Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT', Set.compl_inter_self,
  Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT']

Result:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p =>
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm
n : ℕ
⊢ μ ∅ < ⊤ ∧
    μ
        (spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair n).1)) (Nat.unpair n).2 ∩
            μ.sigmaFiniteSetGE ν (Nat.unpair n).1 ∩
          μ.sigmaFiniteSetWRT' ν) <
      ⊤

Proof state:
case refine_2
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p =>
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm
⊢ (μ.sigmaFiniteSetWRT' ν)ᶜ ∪
      ⋃ i,
        spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair i).1)) (Nat.unpair i).2 ∩
          μ.sigmaFiniteSetGE ν (Nat.unpair i).1 =
    Set.univ

Tactic:
suffices ⋃ n, (spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair n).1)) n.unpair.2
    ∩ μ.sigmaFiniteSetGE ν n.unpair.1) = μ.sigmaFiniteSetWRT' ν by
  rw [this, Set.compl_union_self]

Result:
case refine_2
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p =>
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm
⊢ ⋃ n,
      spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair n).1)) (Nat.unpair n).2 ∩
        μ.sigmaFiniteSetGE ν (Nat.unpair n).1 =
    μ.sigmaFiniteSetWRT' ν

Proof state:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p =>
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm
n : ℕ
⊢ μ
      (spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair n).1)) (Nat.unpair n).2 ∩
        μ.sigmaFiniteSetGE ν (Nat.unpair n).1) <
    ⊤

Tactic:
rw [← Measure.restrict_apply' (measurableSet_sigmaFiniteSetGE _)]

Result:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p =>
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm
n : ℕ
⊢ (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair n).1))
      (spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν (Nat.unpair n).1)) (Nat.unpair n).2) <
    ⊤

Proof state:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
⊢ SigmaFinite (μ.restrict (μ.sigmaFiniteSetWRT' ν))

Tactic:
let f : ℕ × ℕ → Set α := fun p : ℕ × ℕ ↦ (μ.sigmaFiniteSetWRT' ν)ᶜ
  ∪ (spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ (μ.sigmaFiniteSetGE ν p.1))

Result:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p =>
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
⊢ SigmaFinite (μ.restrict (μ.sigmaFiniteSetWRT' ν))