theorem extracted_formal_statement_1.{u_2, u_1} {ι : Type u_1} {Ω : ι → Type u_2} [inst : Fintype ι]
  [inst_1 : DecidableEq ι] [inst_2 : (i : ι) → MeasurableSpace (Ω i)] {μ : (i : ι) → Measure (Ω i)}
  [inst_3 : ∀ (i : ι), IsProbabilityMeasure (μ i)] (k : ι) (s : Set (Ω k)) (hs : MeasurableSet s)
  (h_original_goal : (Measure.map (Function.eval k) (Measure.pi μ)) s = (μ k) s) :
  ∏ i, (μ i) (Function.update (fun x => Set.univ) k s i) = (μ k) s := sorry

State before:
ι : Type u_1
Ω : ι → Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
k : ι
s : Set (Ω k)
hs : MeasurableSet s
⊢ (Measure.map (Function.eval k) (Measure.pi μ)) s = (μ k) s

Tactic: rw [Measure.map_apply (measurable_pi_apply k) hs, Set.eval_preimage, Measure.pi_pi]

State after:
[ι : Type u_1
 Ω : ι → Type u_2
 inst✝³ : Fintype ι
 inst✝² : DecidableEq ι
 inst✝¹ : (i : ι) → MeasurableSpace (Ω i)
 μ : (i : ι) → Measure (Ω i)
 inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
 k : ι
 s : Set (Ω k)
 hs : MeasurableSet s
 ⊢ ∏ i, (μ i) (Function.update (fun x => Set.univ) k s i) = (μ k) s]

Relevant lemmas:
MeasureTheory.Measure.pi_pi : ∀ {ι : Type u_1} {α : ι → Type u_3} [inst : Fintype ι]
  [inst_4 : (i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → Measure (α i)) [inst_5 : ∀ (i : ι), SigmaFinite (μ i)]
  (s : (i : ι) → Set (α i)), (Measure.pi μ) (Set.univ.pi s) = ∏ i, (μ i) (s i)
Set.eval_preimage : ∀ {ι : Type u_1} {α : ι → Type u_2} {i : ι} [inst : DecidableEq ι] {s : Set (α i)},
  Function.eval i ⁻¹' s = Set.univ.pi (Function.update (fun x => Set.univ) i s)
MeasureTheory.Measure.map_apply : ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {μ : Measure α} {f : α → β}, Measurable f → ∀ {s : Set β}, MeasurableSet s → (Measure.map f μ) s = μ (f ⁻¹' s)
measurable_pi_apply : ∀ {δ : Type u_4} {X : δ → Type u_6} [inst : (a : δ) → MeasurableSpace (X a)] (a : δ),
  Measurable fun f => f a