Theorem: MeasureTheory.Lp.induction_stronglyMeasurable
File: Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean

Proof steps:
state_before: 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 : RCLike 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 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)), AEStronglyMeasurable' m (↑↑f) μ → P f
state_after: 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 : RCLike 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 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' m (↑↑f) μ
⊢ P f
tactic: intro f hf
state_before: 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 : RCLike 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 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' m (↑↑f) μ
⊢ P f
state_after: 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 : RCLike 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 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' m (↑↑f) μ
⊢ ∀ ⦃f g : α → F⦄ (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 (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)
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
  -- Porting note: `P` should be an explicit argument to `Lp.induction_stronglyMeasurable_aux`, but
  -- it isn't?
    Lp.induction_stronglyMeasurable_aux hm hp_ne_top h_ind h_add_ae h_closed f hf
state_before: 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 : RCLike 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 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' m (↑↑f) μ
⊢ ∀ ⦃f g : α → F⦄ (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 (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m g μ
h_disj : Disjoint (Function.support f) (Function.support g)
hPf : P (Memℒp.toLp f hf)
hPg : P (Memℒp.toLp g hg)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: intro f g hf hg hfm hgm h_disj hPf hPg
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m g μ
h_disj : Disjoint (Function.support f) (Function.support g)
hPf : P (Memℒp.toLp f hf)
hPg : P (Memℒp.toLp g hg)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: let s_f : Set α := Function.support (hfm.mk f)
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: have hs_f : MeasurableSet[m] s_f := hfm.stronglyMeasurable_mk.measurableSet_support
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: have hs_f_eq : s_f =ᵐ[μ] Function.support f := hfm.ae_eq_mk.symm.support
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: let s_g : Set α := Function.support (hgm.mk g)
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: have hs_g : MeasurableSet[m] s_g := hgm.stronglyMeasurable_mk.measurableSet_support
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: have hs_g_eq : s_g =ᵐ[μ] Function.support g := hgm.ae_eq_mk.symm.support
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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] ∅
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: 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
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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] ∅
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: let f' := (s_f \ s_g).indicator (hfm.mk f)
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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 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
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
tactic: have hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g)
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: have hf'_Lp : Memℒp f' p μ := hf.ae_eq hff'
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: let g' := (s_g \ s_f).indicator (hgm.mk g)
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: 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
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: have hg'_meas : StronglyMeasurable[m] g' := hgm.stronglyMeasurable_mk.indicator (hs_g.diff hs_f)
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: have hg'_Lp : Memℒp g' p μ := hg.ae_eq hgg'
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
h_disj : Disjoint (Function.support f') (Function.support g')
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
tactic: 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
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m 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'
hf'_Lp : Memℒp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
h_disj : Disjoint (Function.support f') (Function.support g')
⊢ P (Memℒp.toLp f hf + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m g μ
h_disj✝ : Disjoint (Function.support f) (Function.support g)
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'
hf'_Lp : Memℒp f' p μ
hPf : P (Memℒp.toLp f' hf'_Lp)
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
h_disj : Disjoint (Function.support f') (Function.support g')
⊢ P (Memℒp.toLp f' hf'_Lp + Memℒp.toLp g hg)
tactic: rw [← Memℒp.toLp_congr hf'_Lp hf hff'.symm] at hPf ⊢
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m g μ
h_disj✝ : Disjoint (Function.support f) (Function.support g)
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'
hf'_Lp : Memℒp f' p μ
hPf : P (Memℒp.toLp f' hf'_Lp)
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
h_disj : Disjoint (Function.support f') (Function.support g')
⊢ P (Memℒp.toLp f' hf'_Lp + Memℒp.toLp g hg)
state_after: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
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 =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'
hf'_Lp : Memℒp f' p μ
hPf : P (Memℒp.toLp f' hf'_Lp)
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
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)
tactic: rw [← Memℒp.toLp_congr hg'_Lp hg hgg'.symm] at hPg ⊢
state_before: 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 : RCLike 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 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' m (↑↑f✝) μ
f g : α → F
hf : Memℒp f p μ
hg : Memℒp g p μ
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 =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'
hf'_Lp : Memℒp f' p μ
hPf : P (Memℒp.toLp f' hf'_Lp)
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)
hgg' : g =ᶠ[ae μ] g'
hg'_meas : StronglyMeasurable g'
hg'_Lp : Memℒp g' p μ
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)
state_after: no goals
tactic: exact h_add hf'_Lp hg'_Lp hf'_meas hg'_meas h_disj hPf hPg