/-- 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?
  exact Lp.induction_stronglyMeasurable_aux hm hp_ne_top h_ind h_add_ae h_closed f hf
  intro f g hf hg hfm hgm h_disj hPf hPg
  let s_f : Set α := Function.support (hfm.mk f)
  have hs_f : MeasurableSet[m] s_f := hfm.stronglyMeasurable_mk.measurableSet_support
  have hs_f_eq : s_f =ᵐ[μ] Function.support f := hfm.ae_eq_mk.symm.support
  let s_g : Set α := Function.support (hgm.mk g)
  have hs_g : MeasurableSet[m] s_g := hgm.stronglyMeasurable_mk.measurableSet_support
  have hs_g_eq : s_g =ᵐ[μ] Function.support g := hgm.ae_eq_mk.symm.support
  have h_inter_empty : (s_f ∩ s_g : Set α) =ᵐ[μ] (∅ : Set α) := by
    refine' (hs_f_eq.inter hs_g_eq).trans _
    suffices Function.support f ∩ Function.support g = ∅ by rw [this]
    exact Set.disjoint_iff_inter_eq_empty.mp h_disj
  let f' := (s_f \ s_g).indicator (hfm.mk f)
  have hff' : f =ᵐ[μ] f' := by
    have : s_f \ s_g =ᵐ[μ] s_f := by
      rw [← Set.diff_inter_self_eq_diff, Set.inter_comm]
      refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _
      rw [Set.diff_empty]
    refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm
    rw [Set.indicator_support]
    exact hfm.ae_eq_mk.symm
  have hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g)
  have hf'_Lp : Memℒp f' p μ := hf.ae_eq hff'
  let g' := (s_g \ s_f).indicator (hgm.mk g)
  have hgg' : g =ᵐ[μ] g' := by
    have : s_g \ s_f =ᵐ[μ] s_g := by
      rw [← Set.diff_inter_self_eq_diff]
      refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _
      rw [Set.diff_empty]
    refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm
    rw [Set.indicator_support]
    exact hgm.ae_eq_mk.symm
  have hg'_meas : StronglyMeasurable[m] g' := hgm.stronglyMeasurable_mk.indicator (hs_g.diff hs_f)
  have hg'_Lp : Memℒp g' p μ := hg.ae_eq hgg'
  have h_disj : Disjoint (Function.support f') (Function.support g') :=
    haveI : Disjoint (s_f \ s_g) (s_g \ s_f) := disjoint_sdiff_sdiff
    this.mono Set.support_indicator_subset Set.support_indicator_subset
  rw [← Memℒp.toLp_congr hf'_Lp hf hff'.symm] at hPf ⊢
  rw [← Memℒp.toLp_congr hg'_Lp hg hgg'.symm] at hPg ⊢
  

Proof state:
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
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m g μ
h_disj✝ : Disjoint (Function.support f) (Function.support g)
s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)
hs_f : MeasurableSet s_f
hs_f_eq : s_f =ᵐ[μ] Function.support f
s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)
hs_g : MeasurableSet s_g
hs_g_eq : s_g =ᵐ[μ] Function.support g
h_inter_empty : s_f ∩ s_g =ᵐ[μ] ∅
f' : α → F := Set.indicator (s_f \ s_g) (AEStronglyMeasurable'.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : Memℒp f' p
hPf : P (Memℒp.toLp f' hf'_Lp)
g' : α → F := Set.indicator (s_g \ s_f) (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᵐ[μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p
hPg : P (Memℒp.toLp g' hg'_Lp)
h_disj : Disjoint (Function.support f') (Function.support g')
⊢ P (Memℒp.toLp f' hf'_Lp + Memℒp.toLp g' hg'_Lp)

Next tactic:
exact h_add hf'_Lp hg'_Lp hf'_meas hg'_meas h_disj hPf hPg