/-- **Hahn decomposition theorem** -/
theorem hahn_decomposition [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    ∃ s,
      MeasurableSet s ∧
        (∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t := by
  let d : Set α → ℝ := fun s => ((μ s).toNNReal : ℝ) - (ν s).toNNReal
  let c : Set ℝ := d '' { s | MeasurableSet s }
  let γ : ℝ := sSup c
  have hμ : ∀ s, μ s ≠ ∞ := measure_ne_top μ
  have hν : ∀ s, ν s ≠ ∞ := measure_ne_top ν
  have to_nnreal_μ : ∀ s, ((μ s).toNNReal : ℝ≥0∞) = μ s := fun s => ENNReal.coe_toNNReal <| hμ _
  have to_nnreal_ν : ∀ s, ((ν s).toNNReal : ℝ≥0∞) = ν s := fun s => ENNReal.coe_toNNReal <| hν _
  have d_split : ∀ s t, MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) := by
    intro s t _hs ht
    dsimp only
    rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht,
      ENNReal.toNNReal_add (hμ _) (hμ _), ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add,
      NNReal.coe_add]
    simp only [sub_eq_add_neg, neg_add]
    abel
  have d_Union :
    ∀ s : ℕ → Set α, Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) := by
    intro s hm
    refine' Tendsto.sub _ _ <;>
      refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm
    exact hμ _
    exact hν _
  have d_Inter :
    ∀ s : ℕ → Set α,
      (∀ n, MeasurableSet (s n)) →
        (∀ n m, n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) := by
    intro s hs hm
    refine' Tendsto.sub _ _ <;>
      refine'
        NNReal.tendsto_coe.2 <|
          (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _
    exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩]
  have bdd_c : BddAbove c := by
    use (μ univ).toNNReal
    rintro r ⟨s, _hs, rfl⟩
    refine' le_trans (sub_le_self _ <| NNReal.coe_nonneg _) _
    rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ]
    exact measure_mono (subset_univ _)
  have c_nonempty : c.Nonempty := Nonempty.image _ ⟨_, MeasurableSet.empty⟩
  have d_le_γ : ∀ s, MeasurableSet s → d s ≤ γ := fun s hs => le_csSup bdd_c ⟨s, hs, rfl⟩
  have : ∀ n : ℕ, ∃ s : Set α, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s := by
    intro n
    have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n)
    rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩
    exact ⟨s, hs, hlt⟩
  rcases Classical.axiom_of_choice this with ⟨e, he⟩
  change ℕ → Set α at e
  have he₁ : ∀ n, MeasurableSet (e n) := fun n => (he n).1
  have he₂ : ∀ n, γ - (1 / 2) ^ n < d (e n) := fun n => (he n).2
  let f : ℕ → ℕ → Set α := fun n m => (Finset.Ico n (m + 1)).inf e
  have hf : ∀ n m, MeasurableSet (f n m) := by
    intro n m
    simp only [Finset.inf_eq_iInf]
    exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _
  have f_subset_f : ∀ {a b c d}, a ≤ b → c ≤ d → f a d ⊆ f b c := by
    intro a b c d hab hcd
    simp_rw [Finset.inf_eq_iInf]
    exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)
  have f_succ : ∀ n m, n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) := by
    intro n m hnm
    have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm)
    simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm]
    rfl
  have le_d_f : ∀ n m, m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) := by
    intro n m h
    refine' Nat.le_induction _ _ n h
    · have := he₂ m
      simp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton]
      linarith
    · intro n (hmn : m ≤ n) ih
      have : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) := by
        calc
          γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤
              γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by
            refine' add_le_add_left (add_le_add_left _ _) γ
            

Proof state:
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ (1 / 2) ^ (n + 1) ≤ (1 / 2) ^ n - (1 / 2) ^ (n + 1)

Next tactic:
simp only [pow_add, pow_one, le_sub_iff_add_le]