-- Lean 4 proof step from Mathlib.Probability.ProductMeasure (goal)
Goal:
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
A : ℕ → Set ((i : ι) → X i)
A_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X
A_anti : Antitone A
A_inter : Eq (α := Set ((i : ι) → X i)) (⋂ (n : ℕ), A n) ∅
this : ∀ (i : ι), Nonempty.{u_2 + 1} (X i)
s : (n : ℕ) → Finset ι
S : (n : ℕ) → Set ((i : ↥(s n)) → X (↑i : ι))
mS : ∀ (n : ℕ), MeasurableSet (S n)
A_eq : ∀ (n : ℕ), A n = cylinder (α := X) (s n) (S n)
u : Set ι := ⋃ (n : ℕ), (↑(s n) : Set ι)
t : (n : ℕ) → Finset (↑u : Type u_1) := fun (n : ℕ) ↦ (s n).preimage (Subtype.val (p := fun (x : ι) ↦ x ∈ u)) ⋯
The The₁ The₂ The₃ The₄ :
  Tendsto (β := ℝ≥0∞) (fun (n : ℕ) ↦ (piContent (X := X) μ : (a : Set ((i : ι) → X i)) → ℝ≥0∞) (A n)) atTop (𝓝 0)
The₅ The₆ The₇ :
  let f : ((i : (↑u : Type u_1)) → X (↑i : ι)) → (i : ι) → X i := fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ι) ↦
    if hi : i ∈ u then x (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) i hi) else Classical.ofNonempty;
  let aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) := fun (n : ℕ) ↦
    { toFun := fun (i : ↥(t n)) ↦ Subtype.mk (p := fun (x : ι) ↦ x ∈ s n) (↑(↑i : (↑u : Type u_1)) : ι) ⋯,
      invFun := fun (i : ↥(s n)) ↦
        Subtype.mk (p := fun (x : (↑u : Type u_1)) ↦ x ∈ t n) (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) (↑i : ι) ⋯) ⋯,
      left_inv := ⋯, right_inv := ⋯ };
  let g : (n : ℕ) → ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) ≃ ((b : ↥(s n)) → X (↑b : ι)) :=
    fun (n : ℕ) ↦ Equiv.piCongrLeft.{u_2 + 1, u_1 + 1, u_1 + 1} (fun (i : ↥(s n)) ↦ X (↑i : ι)) (aux n);
  ∀
    (r_comp_f :
      ∀ (n : ℕ),
        (s n).restrict (π := X) ∘ f =
          Function.comp (β := (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι))
            (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι))
            fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ↥(t n)) ↦ x (↑i : (↑u : Type u_1))),
    Tendsto (β := ℝ≥0∞) (fun (n : ℕ) ↦ (piContent (X := X) μ : (a : Set ((i : ι) → X i)) → ℝ≥0∞) (A n)) atTop (𝓝 0)
The₈ :
  let f : ((i : (↑u : Type u_1)) → X (↑i : ι)) → (i : ι) → X i := fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ι) ↦
    if hi : i ∈ u then x (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) i hi) else Classical.ofNonempty;
  let aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) := fun (n : ℕ) ↦
    { toFun := fun (i : ↥(t n)) ↦ Subtype.mk (p := fun (x : ι) ↦ x ∈ s n) (↑(↑i : (↑u : Type u_1)) : ι) ⋯,
      invFun := fun (i : ↥(s n)) ↦
        Subtype.mk (p := fun (x : (↑u : Type u_1)) ↦ x ∈ t n) (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) (↑i : ι) ⋯) ⋯,
      left_inv := ⋯, right_inv := ⋯ };
  let g : (n : ℕ) → ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) ≃ ((b : ↥(s n)) → X (↑b : ι)) :=
    fun (n : ℕ) ↦ Equiv.piCongrLeft.{u_2 + 1, u_1 + 1, u_1 + 1} (fun (i : ↥(s n)) ↦ X (↑i : ι)) (aux n);
  ∀
    (r_comp_f :
      ∀ (n : ℕ),
        (s n).restrict (π := X) ∘ f =
          Function.comp (β := (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι))
            (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι))
            fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ↥(t n)) ↦ x (↑i : (↑u : Type u_1))),
    let B : (n : ℕ) → Set ((i : (↑u : Type u_1)) → X (↑i : ι)) := fun (n : ℕ) ↦ f ⁻¹' A n;
    let T : (n : ℕ) → Set ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) := fun (n : ℕ) ↦
      (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι)) ⁻¹' S n;
    ∀ (B_eq : ∀ (n : ℕ), B n = cylinder (α := fun (i : (↑u : Type u_1)) ↦ X (↑i : ι)) (t n) (T n)),
      Tendsto (β := ℝ≥0∞) (fun (n : ℕ) ↦ (piContent (X := X) μ : (a : Set ((i : ι) → X i)) → ℝ≥0∞) (A n)) atTop (𝓝 0)
The₉ :
  let f : ((i : (↑u : Type u_1)) → X (↑i : ι)) → (i : ι) → X i := fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ι) ↦
    if hi : i ∈ u then x (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) i hi) else Classical.ofNonempty;
  let aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) := fun (n : ℕ) ↦
    { toFun := fun (i : ↥(t n)) ↦ Subtype.mk (p := fun (x : ι) ↦ x ∈ s n) (↑(↑i : (↑u : Type u_1)) : ι) ⋯,
      invFun := fun (i : ↥(s n)) ↦
        Subtype.mk (p := fun (x : (↑u : Type u_1)) ↦ x ∈ t n) (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) (↑i : ι) ⋯) ⋯,
      left_inv := ⋯, right_inv := ⋯ };
  let g : (n : ℕ) → ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) ≃ ((b : ↥(s n)) → X (↑b : ι)) :=
    fun (n : ℕ) ↦ Equiv.piCongrLeft.{u_2 + 1, u_1 + 1, u_1 + 1} (fun (i : ↥(s n)) ↦ X (↑i : ι)) (aux n);
  ∀
    (r_comp_f :
      ∀ (n : ℕ),
        (s n).restrict (π := X) ∘ f =
          Function.comp (β := (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι))
            (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι))
            fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ↥(t n)) ↦ x (↑i : (↑u : Type u_1))),
    let B : (n : ℕ) → Set ((i : (↑u : Type u_1)) → X (↑i : ι)) := fun (n : ℕ) ↦ f ⁻¹' A n;
    let T : (n : ℕ) → Set ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) := fun (n : ℕ) ↦
      (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι)) ⁻¹' S n;
    ∀ (B_eq : ∀ (n : ℕ), B n = cylinder (α := fun (i : (↑u : Type u_1)) ↦ X (↑i : ι)) (t n) (T n))
      (mT : ∀ (n : ℕ), MeasurableSet (T n)),
      Tendsto (β := ℝ≥0∞) (fun (n : ℕ) ↦ (piContent (X := X) μ : (a : Set ((i : ι) → X i)) → ℝ≥0∞) (A n)) atTop (𝓝 0)
The₁₀ :
  let f : ((i : (↑u : Type u_1)) → X (↑i : ι)) → (i : ι) → X i := fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ι) ↦
    if hi : i ∈ u then x (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) i hi) else Classical.ofNonempty;
  let aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) := fun (n : ℕ) ↦
    { toFun := fun (i : ↥(t n)) ↦ Subtype.mk (p := fun (x : ι) ↦ x ∈ s n) (↑(↑i : (↑u : Type u_1)) : ι) ⋯,
      invFun := fun (i : ↥(s n)) ↦
        Subtype.mk (p := fun (x : (↑u : Type u_1)) ↦ x ∈ t n) (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) (↑i : ι) ⋯) ⋯,
      left_inv := ⋯, right_inv := ⋯ };
  let g : (n : ℕ) → ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) ≃ ((b : ↥(s n)) → X (↑b : ι)) :=
    fun (n : ℕ) ↦ Equiv.piCongrLeft.{u_2 + 1, u_1 + 1, u_1 + 1} (fun (i : ↥(s n)) ↦ X (↑i : ι)) (aux n);
  ∀
    (r_comp_f :
      ∀ (n : ℕ),
        (s n).restrict (π := X) ∘ f =
          Function.comp (β := (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι))
            (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι))
            fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ↥(t n)) ↦ x (↑i : (↑u : Type u_1))),
    let B : (n : ℕ) → Set ((i : (↑u : Type u_1)) → X (↑i : ι)) := fun (n : ℕ) ↦ f ⁻¹' A n;
    let T : (n : ℕ) → Set ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) := fun (n : ℕ) ↦
      (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι)) ⁻¹' S n;
    ∀ (B_eq : ∀ (n : ℕ), B n = cylinder (α := fun (i : (↑u : Type u_1)) ↦ X (↑i : ι)) (t n) (T n))
      (mT : ∀ (n : ℕ), MeasurableSet (T n))
      (B_mem :
        ∀ (n : ℕ),
          Membership.mem (γ := Set (Set ((i : (↑u : Type u_1)) → X (↑i : ι))))
            (measurableCylinders fun (i : (↑u : Type u_1)) ↦ X (↑i : ι)) (B n)),
      Tendsto (β := ℝ≥0∞) (fun (n : ℕ) ↦ (piContent (X := X) μ : (a : Set ((i : ι) → X i)) → ℝ≥0∞) (A n)) atTop (𝓝 0)
⊢ Tendsto (β := ℝ≥0∞) (fun (n : ℕ) ↦ (piContent (X := X) μ : (a : Set ((i : ι) → X i)) → ℝ≥0∞) (A n)) atTop (𝓝 0)
have The₁₁ : let f : ((i : (↑u : Type u_1)) → X (↑i : ι)) → (i : ι) → X i := fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ι) ↦
  if hi : i ∈ u then x (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) i hi) else Classical.ofNonempty;
let aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) := fun (n : ℕ) ↦
  { toFun := fun (i : ↥(t n)) ↦ Subtype.mk (p := fun (x : ι) ↦ x ∈ s n) (↑(↑i : (↑u : Type u_1)) : ι) sorry,
    invFun := fun (i : ↥(s n)) ↦
      Subtype.mk (p := fun (x : (↑u : Type u_1)) ↦ x ∈ t n) (Subtype.mk (p := fun (x : ι) ↦ x ∈ u) (↑i : ι) sorry) sorry,
    left_inv := sorry, right_inv := sorry };
let g : (n : ℕ) → ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) ≃ ((b : ↥(s n)) → X (↑b : ι)) :=
  fun (n : ℕ) ↦ Equiv.piCongrLeft.{u_2 + 1, u_1 + 1, u_1 + 1} (fun (i : ↥(s n)) ↦ X (↑i : ι)) (aux n);
∀
  (r_comp_f :
    ∀ (n : ℕ),
      (s n).restrict (π := X) ∘ f =
        Function.comp (β := (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι))
          (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι))
          fun (x : (i : (↑u : Type u_1)) → X (↑i : ι)) (i : ↥(t n)) ↦ x (↑i : (↑u : Type u_1))),
  let B : (n : ℕ) → Set ((i : (↑u : Type u_1)) → X (↑i : ι)) := fun (n : ℕ) ↦ f ⁻¹' A n;
  let T : (n : ℕ) → Set ((a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) := fun (n : ℕ) ↦
    (⇑(g n) : (a : (a : ↥(t n)) → X (↑((aux n : (a : ↥(t n)) → ↥(s n)) a) : ι)) → (b : ↥(s n)) → X (↑b : ι)) ⁻¹' S n;
  ∀ (B_eq : ∀ (n : ℕ), B n = cylinder (α := fun (i : (↑u : Type u_1)) ↦ X (↑i : ι)) (t n) (T n))
    (mT : ∀ (n : ℕ), MeasurableSet (T n))
    (B_mem :
      ∀ (n : ℕ),
        Membership.mem (γ := Set (Set ((i : (↑u : Type u_1)) → X (↑i : ι))))
          (measurableCylinders fun (i : (↑u : Type u_1)) ↦ X (↑i : ι)) (B n))
    (mB : ∀ (n : ℕ), MeasurableSet (B n)),
    Tendsto (β := ℝ≥0∞) (fun (n : ℕ) ↦ (piContent (X := X) μ : (a : Set ((i : ι) → X i)) → ℝ≥0∞) (A n)) atTop (𝓝 0)
Tactic: classical
  -- The map `f` allows to pull back `Aₙ`
  let f : (Π i : u, X i) → Π i, X i :=
    fun x i ↦ if hi : i ∈ u then x ⟨i, hi⟩ else Classical.ofNonempty
  -- `aux` is the obvious equivalence between `sₙ` and `tₙ`
  let aux n : t n ≃ s n :=
    { toFun := fun i ↦ ⟨i.1.1, mem_preimage.1 i.2⟩
      invFun := fun i ↦ ⟨⟨i.1, Set.mem_iUnion.2 ⟨n, i.2⟩⟩, mem_preimage.2 i.2⟩
      left_inv := fun i ↦ by simp
      right_inv := fun i ↦ by simp }
  -- Finally `gₙ` is the equivalence between the product indexed by `tₙ` and the one indexed by `sₙ`
  let g n := (aux n).piCongrLeft (fun i : s n ↦ X i)
  -- Mapping from the product indexed by `u` by `f` and then restricting to `sₙ` is the same as
  -- first restricting to `tₙ` and then mapping by `gₙ`
  have r_comp_f n : (s n).restrict ∘ f = (g n) ∘ (fun (x : Π i : u, X i) i ↦ x i) := by
    ext x i
    simp only [Function.comp_apply, Finset.restrict,
      Equiv.piCongrLeft_apply, Equiv.coe_fn_symm_mk, f, aux, g, t]
    rw [dif_pos (Set.mem_iUnion.2 ⟨n, i.2⟩)]
  -- `Bₙ` is the same as `Aₙ` but in the product indexed by `u`
  let B n := f ⁻¹' (A n)
  -- `Tₙ` is the same as `Sₙ` but in the product indexed by `u`
  let T n := (g n) ⁻¹' (S n)
  -- We now transfer the properties of `Aₙ` and `Sₙ` to `Bₙ` and `Tₙ`
  have B_eq n : B n = cylinder (t n) (T n) := by
    simp_rw [B, A_eq, cylinder, ← Set.preimage_comp, r_comp_f]; rfl
  have mT n : MeasurableSet (T n) := (mS n).preimage (by fun_prop)
  have B_mem n : B n ∈ measurableCylinders (fun i : u ↦ X i) :=
    (mem_measurableCylinders (B n)).2 ⟨t n, T n, mT n, B_eq n⟩
  have mB n : MeasurableSet (B n) := .of_mem_measurableCylinders (B_mem n)
  have B_anti : Antitone B := fun m n hmn ↦ Set.preimage_mono <| A_anti hmn
  have B_inter : ⋂ n, B n = ∅ := by
    simp_rw [B, ← Set.preimage_iInter, A_inter, Set.preimage_empty]
  -- We now rewrite `piContent μ (A n)` as `piContent (fun i : u ↦ μ i) (B n)`. Then there are two
  -- cases: either `u` is finite and we rewrite it to the finite product measure, either
  -- it is countable and we rewrite it to the pushforward measure of `infinitePiNat`. In both cases
  -- we have an actual measure and we can conclude with `tendsto_measure_iInter_atTop`.
  conv =>
    enter [1]; ext n
    rw [A_eq, piContent_cylinder μ (mS n), ← pi_map_piCongrLeft (aux n),
      map_apply (by fun_prop) (mS n)]
    change (Measure.pi (fun i : t n ↦ μ i)) (T n)
    rw [← piContent_cylinder (fun i : u ↦ μ i) (mT n), ← B_eq n]
  obtain u_fin | u_inf := finite_or_infinite u
  · let _ := Fintype.ofFinite u
    simp_rw [fun n ↦ piContent_eq_measure_pi (fun i : u ↦ μ i) (mB n)]
    convert!
      tendsto_measure_iInter_atTop (fun n ↦ (mB n).nullMeasurableSet) B_anti ⟨0, measure_ne_top _ _⟩
    · rw [B_inter, measure_empty]
    · infer_instance
  · -- If `u` is infinite, then we have an equivalence with `ℕ` so we can apply `secondLemma`.
    have count_u : Countable u := Set.countable_iUnion (fun n ↦ (s n).countable_toSet)
    obtain ⟨φ, -⟩ := Classical.exists_true_of_nonempty (α := ℕ ≃ u) nonempty_equiv_of_countable
    conv => enter [1]; ext n; rw [← infinitePiNat_map_piCongrLeft _ φ (B_mem n)]
    convert!
      tendsto_measure_iInter_atTop (fun n ↦ (mB n).nullMeasurableSet) B_anti ⟨0, measure_ne_top _ _⟩
    · rw [B_inter, measure_empty]
    · infer_instance