/-- To prove something for an `Lp` function a.e. strongly measurable with respect to a
sub-σ-algebra `m` in a normed space, it suffices to show that
* the property holds for (multiples of) characteristic functions which are measurable w.r.t. `m`;
* is closed under addition;
* the set of functions in `Lp` strongly measurable w.r.t. `m` for which the property holds is
  closed.
-/
@[elab_as_elim]
theorem Lp.induction_stronglyMeasurable (hm : m ≤ m0) (hp_ne_top : p ≠ ∞) (P : Lp F p μ → Prop)
    (h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet[m] s) (hμs : μ s < ∞),
      P (Lp.simpleFunc.indicatorConst p (hm s hs) hμs.ne c))
    (h_add : ∀ ⦃f g⦄, ∀ hf : Memℒp f p μ, ∀ hg : Memℒp g p μ, ∀ _ : StronglyMeasurable[m] f,
      ∀ _ : StronglyMeasurable[m] g, Disjoint (Function.support f) (Function.support g) →
        P (hf.toLp f) → P (hg.toLp g) → P (hf.toLp f + hg.toLp g))
    (h_closed : IsClosed {f : lpMeas F ℝ m p μ | P f}) :
    ∀ f : Lp F p μ, AEStronglyMeasurable' m f μ → P f := by
  intro f hf
  suffices h_add_ae :
    ∀ ⦃f g⦄, ∀ hf : Memℒp f p μ, ∀ hg : Memℒp g p μ, ∀ _ : AEStronglyMeasurable' m f μ,
      ∀ _ : AEStronglyMeasurable' m g μ, Disjoint (Function.support f) (Function.support g) →
        P (hf.toLp f) → P (hg.toLp g) → P (hf.toLp f + hg.toLp g)
  -- Porting note: `P` should be an explicit argument to `Lp.induction_stronglyMeasurable_aux`, but
  -- it isn't?
  

Proof state:
α : Type u_1
E' : Type u_2
F : Type u_3
F' : Type u_4
𝕜 : Type u_5
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : InnerProductSpace 𝕜 E'
inst✝⁹ : CompleteSpace E'
inst✝⁸ : NormedSpace ℝ E'
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : NormedSpace 𝕜 F'
inst✝³ : NormedSpace ℝ F'
inst✝² : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ⊤
P : ↥(Lp F p) → Prop
h_ind :
  ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤),
    P ↑(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : ↑↑μ s ≠ ⊤) c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : Memℒp f p) (hg : Memℒp g p),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f : ↥(Lp F p)
hf : AEStronglyMeasurable' m (↑↑f) μ
h_add_ae :
  ∀ ⦃f g : α → F⦄ (hf : Memℒp f p) (hg : Memℒp g p),
    AEStronglyMeasurable' m f μ →
      AEStronglyMeasurable' m g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)
⊢ P f
case h_add_ae
α : Type u_1
E' : Type u_2
F : Type u_3
F' : Type u_4
𝕜 : Type u_5
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : InnerProductSpace 𝕜 E'
inst✝⁹ : CompleteSpace E'
inst✝⁸ : NormedSpace ℝ E'
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : NormedSpace 𝕜 F'
inst✝³ : NormedSpace ℝ F'
inst✝² : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ⊤
P : ↥(Lp F p) → Prop
h_ind :
  ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤),
    P ↑(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : ↑↑μ s ≠ ⊤) c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : Memℒp f p) (hg : Memℒp g p),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f : ↥(Lp F p)
hf : AEStronglyMeasurable' m (↑↑f) μ
⊢ ∀ ⦃f g : α → F⦄ (hf : Memℒp f p) (hg : Memℒp g p),
    AEStronglyMeasurable' m f μ →
      AEStronglyMeasurable' m g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)

Next tactic:
exact Lp.induction_stronglyMeasurable_aux hm hp_ne_top h_ind h_add_ae h_closed f hf