-- Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean
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

Proof state:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup 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 μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    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 μ)
hf✝ : AEStronglyMeasurable (↑↑f✝) μ
f g : α → F
hf : MemLp f p μ
hg : MemLp g p μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
h_disj : Disjoint (Function.support f) (Function.support g)
hPf : P (MemLp.toLp f hf)
hPg : P (MemLp.toLp g hg)
s_f : Set α := Function.support (AEStronglyMeasurable.mk f hfm)
hs_f : MeasurableSet s_f
hs_f_eq : s_f =ᶠ[ae μ] Function.support f
s_g : Set α := Function.support (AEStronglyMeasurable.mk g hgm)
hs_g : MeasurableSet s_g
hs_g_eq : s_g =ᶠ[ae μ] Function.support g
h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅
f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᶠ[ae μ] f'
⊢ P (MemLp.toLp f hf + MemLp.toLp g hg)

Tactic:
have hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g)

Result:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup 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 μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    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 μ)
hf✝ : AEStronglyMeasurable (↑↑f✝) μ
f g : α → F
hf : MemLp f p μ
hg : MemLp g p μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
h_disj : Disjoint (Function.support f) (Function.support g)
hPf : P (MemLp.toLp f hf)
hPg : P (MemLp.toLp g hg)
s_f : Set α := Function.support (AEStronglyMeasurable.mk f hfm)
hs_f : MeasurableSet s_f
hs_f_eq : s_f =ᶠ[ae μ] Function.support f
s_g : Set α := Function.support (AEStronglyMeasurable.mk g hgm)
hs_g : MeasurableSet s_g
hs_g_eq : s_g =ᶠ[ae μ] Function.support g
h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅
f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᶠ[ae μ] f'
hf'_meas : StronglyMeasurable f'
⊢ P (MemLp.toLp f hf + MemLp.toLp g hg)

Proof state:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup 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 μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    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 μ)
hf : AEStronglyMeasurable (↑↑f) μ
⊢ P f

Tactic:
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) from
  Lp.induction_stronglyMeasurable_aux hm hp_ne_top _ h_ind h_add_ae h_closed f hf

Result:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup 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 μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    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 μ)
hf : AEStronglyMeasurable (↑↑f) μ
⊢ ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable 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)