theorem extracted_formal_statement_12.{u_2, u_1} {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_1 : IsProbabilityMeasure μ] [inst_2 : Fintype ι] [inst_3 : DecidableEq ι] (s : Finset ι) {f : ι → Set (ι → Ω)}
  (hf : ∀ i ∈ s, ∃ s', inst.MeasurableSet' s' ∧ Function.eval i ⁻¹' s' = f i) :
  let f' := fun i hi => Classical.choose (hf i hi);
  let hf' := fun i hi => Classical.choose_spec (hf i hi);
  let f'' := fun i => if h : i ∈ s then f' i h else Set.univ;
  ⋂ i ∈ s, f i = Set.univ.pi f'' →
    ∀ (i : ι) (h : i ∈ s),
      (μ (if h : i ∈ s then f' i h else Set.univ) = if i ∈ s then (Measure.pi fun x => μ) (f i) else 1) →
        μ (f' i h) = (Measure.pi fun x => μ) (Function.eval i ⁻¹' Classical.choose (hf i h)) := sorry

State before:
Ω : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace Ω
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
s : Finset ι
f : ι → Set (ι → Ω)
hf : ∀ i ∈ s, ∃ s', inst✝³.MeasurableSet' s' ∧ Function.eval i ⁻¹' s' = f i
f' : (i : ι) → i ∈ s → Set Ω := fun i hi => Classical.choose ⋯
hf' : ∀ (i : ι) (hi : i ∈ s),
  inst✝³.MeasurableSet' (Classical.choose ⋯) ∧ Function.eval i ⁻¹' Classical.choose ⋯ = f i :=
  fun i hi => Classical.choose_spec (hf i hi)
f'' : ι → Set Ω := fun i => if h : i ∈ s then f' i h else Set.univ
this : ⋂ i ∈ s, f i = Set.univ.pi f''
i : ι
h : i ∈ s
⊢ μ (if h : i ∈ s then f' i h else Set.univ) = if i ∈ s then (Measure.pi fun x => μ) (f i) else 1

Tactic: rw [dif_pos h, if_pos h, ← (hf' i h).2]

State after:
[Ω : Type u_1
 ι : Type u_2
 inst✝³ : MeasurableSpace Ω
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Fintype ι
 inst✝ : DecidableEq ι
 s : Finset ι
 f : ι → Set (ι → Ω)
 hf : ∀ i ∈ s, ∃ s', inst✝³.MeasurableSet' s' ∧ Function.eval i ⁻¹' s' = f i
 f' : (i : ι) → i ∈ s → Set Ω := fun i hi => Classical.choose ⋯
 hf' : ∀ (i : ι) (hi : i ∈ s),
   inst✝³.MeasurableSet' (Classical.choose ⋯) ∧ Function.eval i ⁻¹' Classical.choose ⋯ = f i :=
   fun i hi => Classical.choose_spec (hf i hi)
 f'' : ι → Set Ω := fun i => if h : i ∈ s then f' i h else Set.univ
 this : ⋂ i ∈ s, f i = Set.univ.pi f''
 i : ι
 h : i ∈ s
 ⊢ μ (f' i h) = (Measure.pi fun x => μ) (Function.eval i ⁻¹' Classical.choose ⋯)]

Relevant lemmas:
dif_pos : ∀ {c : Prop} {h : Decidable c} (hc : c) {α : Sort u} {t : c → α} {e : ¬c → α}, dite c t e = t hc
if_pos : ∀ {c : Prop} {h : Decidable c}, c → ∀ {α : Sort u} {t e : α}, (if c then t else e) = t