theorem filtrationOfSet_eq_natural [MulZeroOneClass β] [Nontrivial β] {s : ι → Set Ω}
    (hsm : ∀ i, MeasurableSet[m] (s i)) :
    filtrationOfSet hsm = natural (fun i => (s i).indicator (fun _ => 1 : Ω → β)) fun i =>
      stronglyMeasurable_one.indicator (hsm i) := by
  simp only [filtrationOfSet, natural, measurableSpace_iSup_eq, exists_prop, mk.injEq]
  ext1 i
  refine' le_antisymm (generateFrom_le _) (generateFrom_le _)
  · rintro _ ⟨j, hij, rfl⟩
    refine' measurableSet_generateFrom ⟨j, measurableSet_generateFrom ⟨hij, _⟩⟩
    rw [comap_eq_generateFrom]
    refine' measurableSet_generateFrom ⟨{1}, measurableSet_singleton 1, _⟩
    ext x
    simp [Set.indicator_const_preimage_eq_union]
  · rintro t ⟨n, ht⟩
    suffices MeasurableSpace.generateFrom {t | n ≤ i ∧
      MeasurableSet[MeasurableSpace.comap ((s n).indicator (fun _ => 1 : Ω → β)) mβ] t} ≤
        MeasurableSpace.generateFrom {t | ∃ (j : ι), j ≤ i ∧ s j = t} by
      exact this _ ht
    refine' generateFrom_le _
    rintro t ⟨hn, u, _, hu'⟩
    obtain heq | heq | heq | heq := Set.indicator_const_preimage (s n) u (1 : β)
    pick_goal 4; rw [Set.mem_singleton_iff] at heq
    

Proof state:
case h.refine'_2.intro.intro.intro.intro.inr.inr.inr
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : (Set.indicator (s n) fun x => 1) ⁻¹' u = t
heq : (Set.indicator (s n) fun x => 1) ⁻¹' u = ∅
⊢ MeasurableSet t
case h.refine'_2.intro.intro.intro.intro.inl
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : (Set.indicator (s n) fun x => 1) ⁻¹' u = t
heq : (Set.indicator (s n) fun x => 1) ⁻¹' u = Set.univ
⊢ MeasurableSet t
case h.refine'_2.intro.intro.intro.intro.inr.inl
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : (Set.indicator (s n) fun x => 1) ⁻¹' u = t
heq : (Set.indicator (s n) fun x => 1) ⁻¹' u = s n
⊢ MeasurableSet t
case h.refine'_2.intro.intro.intro.intro.inr.inr.inl
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : (Set.indicator (s n) fun x => 1) ⁻¹' u = t
heq : (Set.indicator (s n) fun x => 1) ⁻¹' u = (s n)ᶜ
⊢ MeasurableSet t

Next tactic:
all_goals rw [heq] at hu'; rw [← hu']