mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: 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 =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 := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : MemLp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable.mk g hgm)
⊢ s_g \ (s_f ∩ s_g) =ᵐ[μ] s_g

Tactic:
]

Goal state after:
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 =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 := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : MemLp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable.mk g hgm)
⊢ s_g \ (s_f ∩ s_g) =ᵐ[μ] s_g
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
f g : α → F'
hf_Lp : MemLp f 1 μ
hg_Lp : MemLp g 1 μ
a✝ : Disjoint (Function.support f) (Function.support g)
hf : ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp) x ∂μ
hg : ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp g hg_Lp) x ∂μ
⊢ ∫ (x : α) in s,
      (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp)) + ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg_Lp))) x ∂μ =
    ∫ (x : α) in s, (↑↑(MemLp.toLp f hf_Lp) + ↑↑(MemLp.toLp g hg_Lp)) x ∂μ

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
f g : α → F'
hf_Lp : MemLp f 1 μ
hg_Lp : MemLp g 1 μ
a✝ : Disjoint (Function.support f) (Function.support g)
hf : ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp) x ∂μ
hg : ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp g hg_Lp) x ∂μ
⊢ ∫ (x : α) in s,
      (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp)) + ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg_Lp))) x ∂μ =
    ∫ (x : α) in s, (↑↑(MemLp.toLp f hf_Lp) + ↑↑(MemLp.toLp g hg_Lp)) x ∂μ
Module: Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum

Goal state before:
pp: X : Type u_1
E : Type u_2
x✝ : MeasurableSpace X
μ : Measure X
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
hp : Fact (1 ≤ p)
ι : Type u_3
inst✝¹ : Countable ι
inst✝ : CompleteSpace E
f : ι → ↥(Lp E p μ)
hf : ∑' (n : ι), ‖f n‖ₑ ≠ ∞
hι : Finite ι
this : Fintype ι := Fintype.ofFinite ι
x : X
hx : ↑↑(∑ i, f i) x = ∑ i, ↑↑(f i) x
⊢ HasSum (fun n ↦ ↑↑(f n) x) (∑ i, ↑↑(f i) x)

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_2
x✝ : MeasurableSpace X
μ : Measure X
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
hp : Fact (1 ≤ p)
ι : Type u_3
inst✝¹ : Countable ι
inst✝ : CompleteSpace E
f : ι → ↥(Lp E p μ)
hf : ∑' (n : ι), ‖f n‖ₑ ≠ ∞
hι : Finite ι
this : Fintype ι := Fintype.ofFinite ι
x : X
hx : ↑↑(∑ i, f i) x = ∑ i, ↑↑(f i) x
⊢ HasSum (fun n ↦ ↑↑(f n) x) (∑ i, ↑↑(f i) x)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: 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 =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 := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : MemLp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable.mk g hgm)
⊢ s_g =ᵐ[μ] s_g

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
f g : α → F'
hf_Lp : MemLp f 1 μ
hg_Lp : MemLp g 1 μ
a✝ : Disjoint (Function.support f) (Function.support g)
hf : ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp) x ∂μ
hg : ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp g hg_Lp) x ∂μ
⊢ ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp) x ∂μ + ∫ (x : α) in s, ↑↑(MemLp.toLp g hg_Lp) x ∂μ =
    ∫ (a : α) in s, ↑↑(MemLp.toLp f hf_Lp) a ∂μ + ∫ (a : α) in s, ↑↑(MemLp.toLp g hg_Lp) a ∂μ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: 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 =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 := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : MemLp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable.mk g hgm)
this : s_g \ s_f =ᵐ[μ] s_g
⊢ AEStronglyMeasurable.mk g hgm =ᵐ[μ] g

Tactic:
]

Goal state after:
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 =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 := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : MemLp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable.mk g hgm)
this : s_g \ s_f =ᵐ[μ] s_g
⊢ AEStronglyMeasurable.mk g hgm =ᵐ[μ] g
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
⊢ ∫ (a : α) in s, (↑↑↑((condExpL2 E' 𝕜 hm) f) - ↑↑f) a ∂μ = 0

Tactic:
]

Goal state after:
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
⊢ ∫ (a : α) in s, (↑↑↑((condExpL2 E' 𝕜 hm) f) - ↑↑f) a ∂μ = 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_1
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
⊢ Integrable (↑↑(↑((condExpL2 E' 𝕜 hm) f) - f)) (μ.restrict s)

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
⊢ Integrable (↑↑(↑((condExpL2 E' 𝕜 hm) f) - f)) (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_2
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E'
⊢ ∫ (a : α) in s, ⟪c, ↑↑↑((condExpL2 E' 𝕜 hm) f) a⟫ ∂μ - ∫ (a : α) in s, ⟪c, ↑↑f a⟫ ∂μ = 0

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E'
⊢ ∫ (a : α) in s, ⟪c, ↑↑↑((condExpL2 E' 𝕜 hm) f) a⟫ ∂μ - ∫ (a : α) in s, ⟪c, ↑↑f a⟫ ∂μ = 0
Module: Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum

Goal state before:
pp: X : Type u_1
E : Type u_2
x✝ : MeasurableSpace X
μ : Measure X
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
hp : Fact (1 ≤ p)
ι : Type u_3
inst✝¹ : Countable ι
inst✝ : CompleteSpace E
f : ι → ↥(Lp E p μ)
hf : ∑' (n : ι), ‖f n‖ₑ ≠ ∞
hι : Infinite ι
e : ℕ ≃ ι
this : ∀ᵐ (a : X) ∂μ, HasSum (fun n ↦ ↑↑(f (e n)) a) (↑↑(∑' (n : ℕ), f (e n)) a)
x : X
hx : HasSum (fun n ↦ ↑↑(f (e n)) x) (↑↑(∑' (b : ι), f b) x)
⊢ HasSum (fun n ↦ ↑↑(f n) x) (↑↑(∑' (n : ι), f n) x)

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_2
x✝ : MeasurableSpace X
μ : Measure X
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
hp : Fact (1 ≤ p)
ι : Type u_3
inst✝¹ : Countable ι
inst✝ : CompleteSpace E
f : ι → ↥(Lp E p μ)
hf : ∑' (n : ι), ‖f n‖ₑ ≠ ∞
hι : Infinite ι
e : ℕ ≃ ι
this : ∀ᵐ (a : X) ∂μ, HasSum (fun n ↦ ↑↑(f (e n)) a) (↑↑(∑' (n : ℕ), f (e n)) a)
x : X
hx : HasSum (fun n ↦ ↑↑(f (e n)) x) (↑↑(∑' (b : ι), f b) x)
⊢ HasSum (fun n ↦ ↑↑(f n) x) (↑↑(∑' (n : ι), f n) x)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_2
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E'
h_ae_eq_f : ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯) =ᵐ[μ] fun a ↦ ⟪c, ↑↑f a⟫
⊢ ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) (MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯)) x ∂μ =
    ∫ (x : α) in s, ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯) x ∂μ

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E'
h_ae_eq_f : ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯) =ᵐ[μ] fun a ↦ ⟪c, ↑↑f a⟫
⊢ ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) (MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯)) x ∂μ =
    ∫ (x : α) in s, ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯) x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm_not : ¬m ≤ m₀
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm_not : ¬SigmaFinite (μ.trim hm)
⊢ 0 = 0

pp: case hnc
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm_not : ¬SigmaFinite (μ.trim hm)
⊢ ¬(SigmaFinite (μ.trim hm) ∧ Integrable f μ)

Tactic:
]

Goal state after:
case hnc
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm_not : ¬SigmaFinite (μ.trim hm)
⊢ ¬(SigmaFinite (μ.trim hm) ∧ Integrable f μ)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
hs : MeasurableSet s
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_meas : ∀ (i : ℕ), MeasurableSet (S i)
hS_meas0 : ∀ (i : ℕ), MeasurableSet (S i)
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ (if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
      if StronglyMeasurable f then f
      else
        have this := ⋯;
        AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =
    if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯ else 0

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ (if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
      if StronglyMeasurable f then f
      else
        have this := ⋯;
        AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =
    if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯ else 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
hs : MeasurableSet s
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_meas : ∀ (i : ℕ), MeasurableSet (S i)
hS_meas0 : ∀ (i : ℕ), MeasurableSet (S i)
hs_eq : s = ⋃ i, S i ∩ s
i : ℕ
hS_finite_trim : μ (S i) < ∞
⊢ μ (S i) < ∞

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
hs : MeasurableSet s
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_meas : ∀ (i : ℕ), MeasurableSet (S i)
hS_meas0 : ∀ (i : ℕ), MeasurableSet (S i)
hs_eq : s = ⋃ i, S i ∩ s
i : ℕ
hS_finite_trim : μ (S i) < ∞
⊢ μ (S i) < ∞
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ f = f

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
⊢ (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =ᵐ[μ]
    ↑↑(condExpL1 hm μ f)

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
⊢ (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =ᵐ[μ]
    ↑↑(condExpL1 hm μ f)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_1
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : InnerProductSpace 𝕜 E'
inst✝⁶ : CompleteSpace E'
inst✝⁵ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : RCLike 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2 μ)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ T (∫ (x : α) in s, ↑↑f x ∂μ) = T (∫ (x : α) in s, ↑↑f x ∂μ)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : Integrable f μ
⊢ (if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯) =ᵐ[μ] ↑↑(condExpL1 hm μ f)

Tactic:
]

Goal state after:
case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : Integrable f μ
⊢ (if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯) =ᵐ[μ] ↑↑(condExpL1 hm μ f)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : Integrable f μ
⊢ (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =ᵐ[μ]
    ↑↑(condExpL1 hm μ f)

Tactic:
rw [if_pos hfi]

Goal state after:
case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : Integrable f μ
⊢ (if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯) =ᵐ[μ] ↑↑(condExpL1 hm μ f)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : Integrable f μ
hfm : StronglyMeasurable f
⊢ f =ᵐ[μ] ↑↑(condExpL1 hm μ f)

Tactic:
]

Goal state after:
case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : Integrable f μ
hfm : StronglyMeasurable f
⊢ f =ᵐ[μ] ↑↑(condExpL1 hm μ f)