/-- Any pair of finite measures `μ` and `ν`, `HaveLebesgueDecomposition`. That is to say,
there exist a measure `ξ` and a measurable function `f`, such that `ξ` is mutually singular
with respect to `ν` and `μ = ξ + ν.withDensity f`.

This is not an instance since this is also shown for the more general σ-finite measures with
`MeasureTheory.Measure.haveLebesgueDecomposition_of_sigmaFinite`. -/
theorem haveLebesgueDecomposition_of_finiteMeasure [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    HaveLebesgueDecomposition μ ν :=
  ⟨by
    have h :=
      @exists_seq_tendsto_sSup _ _ _ _ _ (measurableLEEval ν μ)
        ⟨0, 0, zero_mem_measurableLE, by simp⟩ (OrderTop.bddAbove _)
    choose g _ hg₂ f hf₁ hf₂ using h
    -- we set `ξ` to be the supremum of an increasing sequence of functions obtained from above
    set ξ := ⨆ (n) (k) (_ : k ≤ n), f k with hξ
    -- we see that `ξ` has the largest integral among all functions in `measurableLE`
    have hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ a, ξ a ∂ν := by
      have :=
        @lintegral_tendsto_of_tendsto_of_monotone _ _ ν (fun n => ⨆ (k) (_ : k ≤ n), f k)
          (⨆ (n) (k) (_ : k ≤ n), f k) ?_ ?_ ?_
      · refine' tendsto_nhds_unique _ this
        refine' tendsto_of_tendsto_of_tendsto_of_le_of_le hg₂ tendsto_const_nhds _ _
        · intro n; rw [← hf₂ n]
          apply lintegral_mono
          simp only [iSup_apply, iSup_le_le f n n le_rfl]
        · intro n
          exact le_sSup ⟨⨆ (k : ℕ) (_ : k ≤ n), f k, iSup_mem_measurableLE' _ hf₁ _, rfl⟩
      · intro n
        refine' Measurable.aemeasurable _
        convert (iSup_mem_measurableLE _ hf₁ n).1
        refine Option.ext fun x => ?_; simp
      · refine' Filter.eventually_of_forall fun a => _
        simp [iSup_monotone' f _]
      · refine' Filter.eventually_of_forall fun a => _
        simp [tendsto_atTop_iSup (iSup_monotone' f a)]
    have hξm : Measurable ξ := by
      convert measurable_iSup fun n => (iSup_mem_measurableLE _ hf₁ n).1
      refine Option.ext fun x => ?_; simp [hξ]
    -- `ξ` is the `f` in the theorem statement and we set `μ₁` to be `μ - ν.withDensity ξ`
    -- since we need `μ₁ + ν.withDensity ξ = μ`
    set μ₁ := μ - ν.withDensity ξ with hμ₁
    have hle : ν.withDensity ξ ≤ μ := by
      intro B hB
      rw [hξ, withDensity_apply _ hB]
      simp_rw [iSup_apply]
      rw [lintegral_iSup (fun i => (iSup_mem_measurableLE _ hf₁ i).1) (iSup_monotone _)]
      exact iSup_le fun i => (iSup_mem_measurableLE _ hf₁ i).2 B hB
    have : IsFiniteMeasure (ν.withDensity ξ) := by
      refine' isFiniteMeasure_withDensity _
      have hle' := hle univ MeasurableSet.univ
      rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at hle'
      exact ne_top_of_le_ne_top (measure_ne_top _ _) hle'
    refine' ⟨⟨μ₁, ξ⟩, hξm, _, _⟩
    · by_contra h
      -- if they are not mutually singular, then from `exists_positive_of_not_mutuallySingular`,
      -- there exists some `ε > 0` and a measurable set `E`, such that `μ(E) > 0` and `E` is
      -- positive with respect to `ν - εμ`
      obtain ⟨ε, hε₁, E, hE₁, hE₂, hE₃⟩ := exists_positive_of_not_mutuallySingular μ₁ ν h
      simp_rw [hμ₁] at hE₃
      have hξle : ∀ A, MeasurableSet A → (∫⁻ a in A, ξ a ∂ν) ≤ μ A := by
        intro A hA; rw [hξ]
        simp_rw [iSup_apply]
        rw [lintegral_iSup (fun n => (iSup_mem_measurableLE _ hf₁ n).1) (iSup_monotone _)]
        exact iSup_le fun n => (iSup_mem_measurableLE _ hf₁ n).2 A hA
      -- since `E` is positive, we have `∫⁻ a in A ∩ E, ε + ξ a ∂ν ≤ μ (A ∩ E)` for all `A`
      have hε₂ : ∀ A : Set α, MeasurableSet A → (∫⁻ a in A ∩ E, ε + ξ a ∂ν) ≤ μ (A ∩ E) := by
        intro A hA
        have := subset_le_of_restrict_le_restrict _ _ hE₁ hE₃ (inter_subset_right A E)
        rwa [zero_apply, toSignedMeasure_sub_apply (hA.inter hE₁),
          Measure.sub_apply (hA.inter hE₁) hle,
          ENNReal.toReal_sub_of_le _ (ne_of_lt (measure_lt_top _ _)), sub_nonneg, le_sub_iff_add_le,
          ← ENNReal.toReal_add, ENNReal.toReal_le_toReal, Measure.coe_smul, Pi.smul_apply,
          withDensity_apply _ (hA.inter hE₁), show ε • ν (A ∩ E) = (ε : ℝ≥0∞) * ν (A ∩ E) by rfl,
          ← set_lintegral_const, ← lintegral_add_left measurable_const] at this
        · rw [Ne.def, ENNReal.add_eq_top, not_or]
          exact ⟨ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)⟩
        · exact ne_of_lt (measure_lt_top _ _)
        · exact ne_of_lt (measure_lt_top _ _)
        · exact ne_of_lt (measure_lt_top _ _)
        · rw [withDensity_apply _ (hA.inter hE₁)]
          exact hξle (A ∩ E) (hA.inter hE₁)
      -- from this, we can show `ξ + ε * E.indicator` is a function in `measurableLE` with
      -- integral greater than `ξ`
      have hξε : (ξ + E.indicator fun _ => (ε : ℝ≥0∞)) ∈ measurableLE ν μ := by
        refine' ⟨Measurable.add hξm (Measurable.indicator measurable_const hE₁), fun A hA => _⟩
        have :
          (∫⁻ a in A, (ξ + E.indicator fun _ => (ε : ℝ≥0∞)) a ∂ν) =
            (∫⁻ a in A ∩ E, ε + ξ a ∂ν) + ∫⁻ a in A \ E, ξ a ∂ν := by
          simp only [lintegral_add_left measurable_const, lintegral_add_left hξm,
            set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE₁, Pi.add_apply,
            lintegral_indicator _ hE₁, restrict_apply hE₁]
          rw [inter_comm, add_comm]
        rw [this, ← measure_inter_add_diff A hE₁]
        exact add_le_add (hε₂ A hA) (hξle (A \ E) (hA.diff hE₁))
      have : (∫⁻ a, ξ a + E.indicator (fun _ => (ε : ℝ≥0∞)) a ∂ν) ≤ sSup (measurableLEEval ν μ) :=
        le_sSup ⟨ξ + E.indicator fun _ => (ε : ℝ≥0∞), hξε, rfl⟩
      -- but this contradicts the maximality of `∫⁻ x, ξ x ∂ν`
      refine' not_lt.2 this _
      rw [hξ₁, lintegral_add_left hξm, lintegral_indicator _ hE₁, set_lintegral_const]
      refine' ENNReal.lt_add_right _ (ENNReal.mul_pos_iff.2 ⟨ENNReal.coe_pos.2 hε₁, hE₂⟩).ne'
      have := measure_ne_top (ν.withDensity ξ) univ
      rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at this
    -- since `ν.withDensity ξ ≤ μ`, it is clear that `μ = μ₁ + ν.withDensity ξ`
    · rw [hμ₁]; 

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
μ₁ : Measure α := μ - withDensity ν ξ
hμ₁ : μ₁ = μ - withDensity ν ξ
hle : withDensity ν ξ ≤ μ
this : IsFiniteMeasure (withDensity ν ξ)
⊢ μ = (μ - withDensity ν ξ, ξ).1 + withDensity ν (μ - withDensity ν ξ, ξ).2

Next tactic:
ext1 A hA