/-- In the space `ι → ℝ`, the Hausdorff measure coincides exactly with the Lebesgue measure. -/
@[simp]
theorem hausdorffMeasure_pi_real {ι : Type*} [Fintype ι] :
    (μH[Fintype.card ι] : Measure (ι → ℝ)) = volume := by
  classical
  -- it suffices to check that the two measures coincide on products of rational intervals
  refine'
    (pi_eq_generateFrom (fun _ => Real.borel_eq_generateFrom_Ioo_rat.symm)
        (fun _ => Real.isPiSystem_Ioo_rat) (fun _ => Real.finiteSpanningSetsInIooRat _) _).symm
  simp only [mem_iUnion, mem_singleton_iff]
  -- fix such a product `s` of rational intervals, of the form `Π (a i, b i)`.
  intro s hs
  choose a b H using hs
  obtain rfl : s = fun i => Ioo (α := ℝ) (a i) (b i)
  exact funext fun i => (H i).2
  replace H := fun i => (H i).1
  apply le_antisymm _
  -- first check that `volume s ≤ μH s`
  · have Hle : volume ≤ (μH[Fintype.card ι] : Measure (ι → ℝ)) := by
      refine' le_hausdorffMeasure _ _ ∞ ENNReal.coe_lt_top fun s _ => _
      rw [ENNReal.rpow_nat_cast]
      exact Real.volume_pi_le_diam_pow s
    rw [← volume_pi_pi fun i => Ioo (a i : ℝ) (b i)]
    exact Measure.le_iff'.1 Hle _
  /- For the other inequality `μH s ≤ volume s`, we use a covering of `s` by sets of small diameter
    `1/n`, namely cubes with left-most point of the form `a i + f i / n` with `f i` ranging between
    `0` and `⌈(b i - a i) * n⌉`. Their number is asymptotic to `n^d * Π (b i - a i)`. -/
  have I : ∀ i, 0 ≤ (b i : ℝ) - a i := fun i => by
    simpa only [sub_nonneg, Rat.cast_le] using (H i).le
  let γ := fun n : ℕ => ∀ i : ι, Fin ⌈((b i : ℝ) - a i) * n⌉₊
  let t : ∀ n : ℕ, γ n → Set (ι → ℝ) := fun n f =>
    Set.pi univ fun i => Icc (a i + f i / n) (a i + (f i + 1) / n)
  have A : Tendsto (fun n : ℕ => 1 / (n : ℝ≥0∞)) atTop (𝓝 0) := by
    simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero]
  have B : ∀ᶠ n in atTop, ∀ i : γ n, diam (t n i) ≤ 1 / n := by
    refine' eventually_atTop.2 ⟨1, fun n hn => _⟩
    intro f
    refine' diam_pi_le_of_le fun b => _
    simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl,
      add_sub_add_left_eq_sub, add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat]
  have C : ∀ᶠ n in atTop, (Set.pi univ fun i : ι => Ioo (a i : ℝ) (b i)) ⊆ ⋃ i : γ n, t n i := by
    refine' eventually_atTop.2 ⟨1, fun n hn => _⟩
    have npos : (0 : ℝ) < n := Nat.cast_pos.2 hn
    intro x hx
    simp only [mem_Ioo, mem_univ_pi] at hx
    simp only [mem_iUnion, mem_Ioo, mem_univ_pi]
    let f : γ n := fun i =>
      ⟨⌊(x i - a i) * n⌋₊, by
        apply Nat.floor_lt_ceil_of_lt_of_pos
        · refine' (mul_lt_mul_right npos).2 _
          simp only [(hx i).right, sub_lt_sub_iff_right]
        · refine' mul_pos _ npos
          simpa only [Rat.cast_lt, sub_pos] using H i⟩
    refine' ⟨f, fun i => ⟨_, _⟩⟩
    · calc
        (a i : ℝ) + ⌊(x i - a i) * n⌋₊ / n ≤ (a i : ℝ) + (x i - a i) * n / n := by
          refine' add_le_add le_rfl ((div_le_div_right npos).2 _)
          exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)
        _ = x i := by field_simp [npos.ne']
    · calc
        x i = (a i : ℝ) + (x i - a i) * n / n := by field_simp [npos.ne']
        _ ≤ (a i : ℝ) + (⌊(x i - a i) * n⌋₊ + 1) / n :=
          add_le_add le_rfl ((div_le_div_right npos).2 (Nat.lt_floor_add_one _).le)
  calc
    μH[Fintype.card ι] (Set.pi univ fun i : ι => Ioo (a i : ℝ) (b i)) ≤
        liminf (fun n : ℕ => ∑ i : γ n, diam (t n i) ^ ((Fintype.card ι) : ℝ)) atTop :=
      hausdorffMeasure_le_liminf_sum _ (Set.pi univ fun i => Ioo (a i : ℝ) (b i))
        (fun n : ℕ => 1 / (n : ℝ≥0∞)) A t B C
    _ ≤ liminf (fun n : ℕ => ∑ i : γ n, (1 / (n : ℝ≥0∞)) ^ Fintype.card ι) atTop := by
      refine' liminf_le_liminf _ _
      · filter_upwards [B] with _ hn
        apply Finset.sum_le_sum fun i _ => _
        simp only [ENNReal.rpow_nat_cast]
        intros i _
        exact pow_le_pow_left' (hn i) _
      · isBoundedDefault
    _ = liminf (fun n : ℕ => ∏ i : ι, (⌈((b i : ℝ) - a i) * n⌉₊ : ℝ≥0∞) / n) atTop := by
      simp only [Finset.card_univ, Nat.cast_prod, one_mul, Fintype.card_fin, Finset.sum_const,
        nsmul_eq_mul, Fintype.card_pi, div_eq_mul_inv, Finset.prod_mul_distrib, Finset.prod_const]
    _ = ∏ i : ι, volume (Ioo (a i : ℝ) (b i)) := by
      

Proof state:
ι✝ : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁶ : EMetricSpace X
inst✝⁵ : EMetricSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) :=
  fun n f => Set.pi univ fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (Set.pi univ fun i => Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
⊢ liminf (fun n => ∏ i : ι, ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) atTop = ∏ i : ι, ↑↑volume (Ioo ↑(a i) ↑(b i))

Next tactic:
simp only [Real.volume_Ioo]