/-- The variance of a finite sum of pairwise independent random variables is the sum of the
variances. -/
theorem IndepFun.variance_sum [@IsProbabilityMeasure Ω _ ℙ] {ι : Type*} {X : ι → Ω → ℝ}
    {s : Finset ι} (hs : ∀ i ∈ s, @Memℒp _ _ _ (_) (X i) 2 ℙ)
    (h : Set.Pairwise ↑s fun i j => @IndepFun _ _ _ (_) _ _ (X i) (X j) ℙ) :
    Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by
  classical
  induction' s using Finset.induction_on with k s ks IH
  · simp only [Finset.sum_empty, variance_zero]
  rw [variance_def' (memℒp_finset_sum' _ hs), sum_insert ks, sum_insert ks]
  simp only [add_sq']
  calc
    𝔼[X k ^ 2 + (∑ i in s, X i) ^ 2 + 2 * X k * ∑ i in s, X i] - 𝔼[X k + ∑ i in s, X i] ^ 2 =
        𝔼[X k ^ 2] + 𝔼[(∑ i in s, X i) ^ 2] + 𝔼[2 * X k * ∑ i in s, X i] -
          (𝔼[X k] + 𝔼[∑ i in s, X i]) ^ 2 := by
      rw [integral_add', integral_add', integral_add']
      · exact Memℒp.integrable one_le_two (hs _ (mem_insert_self _ _))
      · apply integrable_finset_sum' _ fun i hi => ?_
        exact Memℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi))
      · exact Memℒp.integrable_sq (hs _ (mem_insert_self _ _))
      · apply Memℒp.integrable_sq
        exact memℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
      · apply Integrable.add
        · exact Memℒp.integrable_sq (hs _ (mem_insert_self _ _))
        · apply Memℒp.integrable_sq
          exact memℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
      · rw [mul_assoc]
        apply Integrable.const_mul _ (2 : ℝ)
        simp only [mul_sum, sum_apply, Pi.mul_apply]
        apply integrable_finset_sum _ fun i hi => ?_
        apply IndepFun.integrable_mul _ (Memℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
          (Memℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
        exact fun hki => ks (hki.symm ▸ hi)
    _ = Var[X k] + Var[∑ i in s, X i] +
        (𝔼[2 * X k * ∑ i in s, X i] - 2 * 𝔼[X k] * 𝔼[∑ i in s, X i]) := by
      rw [variance_def' (hs _ (mem_insert_self _ _)),
        variance_def' (memℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi))]
      ring
    _ = Var[X k] + Var[∑ i in s, X i] := by
      simp only [mul_assoc, integral_mul_left, Pi.mul_apply, Pi.one_apply, sum_apply,
        add_right_eq_self, mul_sum]
      rw [integral_finset_sum s fun i hi => ?_]; swap
      · apply Integrable.const_mul _ (2 : ℝ)
        apply IndepFun.integrable_mul _ (Memℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
          (Memℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
        exact fun hki => ks (hki.symm ▸ hi)
      rw [integral_finset_sum s fun i hi =>
          Memℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)),
        mul_sum, mul_sum, ← sum_sub_distrib]
      

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X✝ : Ω → ℝ
μ : Measure Ω
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
ι : Type u_2
X : ι → Ω → ℝ
k : ι
s : Finset ι
ks : k ∉ s
IH :
  (∀ i ∈ s, Memℒp (X i) 2) →
    (Set.Pairwise ↑s fun i j => IndepFun (X i) (X j)) → variance (∑ i in s, X i) ℙ = ∑ i in s, variance (X i) ℙ
hs : ∀ i ∈ insert k s, Memℒp (X i) 2
h : Set.Pairwise ↑(insert k s) fun i j => IndepFun (X i) (X j)
⊢ ∑ x in s, ((∫ (a : Ω), OfNat.ofNat 2 a * (X k a * X x a)) - 2 * ((∫ (a : Ω), X k a) * ∫ (a : Ω), X x a)) = 0

Next tactic:
apply Finset.sum_eq_zero fun i hi => ?_