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

Goal state before:
pp: case neg
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
⊢ AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯ =ᵐ[μ] ↑↑(condExpL1 hm μ f)

Tactic:
]

Goal state after:
case neg
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
⊢ AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯ =ᵐ[μ] ↑↑(condExpL1 hm μ f)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_3
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)
h_coe : ↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f)) =ᵐ[μ] fun a ↦ T (↑↑↑((condExpL2 E' 𝕜 hm) f) a)
⊢ AEStronglyMeasurable (↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f))) μ

Tactic:
]

Goal state after:
case refine_3
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)
h_coe : ↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f)) =ᵐ[μ] fun a ↦ T (↑↑↑((condExpL2 E' 𝕜 hm) f) a)
⊢ AEStronglyMeasurable (↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f))) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
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
⊢ 0 =ᵐ[μ] ↑↑0

Tactic:
]

Goal state after:
case neg
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
⊢ 0 =ᵐ[μ] ↑↑0
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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
hm : m ≤ m₀
inst✝ : SigmaFinite (μ.trim hm)
hf : Integrable f μ
⊢ μ[f | m] =ᵐ[μ] ↑↑((condExpL1CLM E hm μ) (Integrable.toL1 f hf))

Tactic:
refine (condExp_ae_eq_condExpL1 hm f).trans (Eventually.of_forall fun x => ?_)

Goal state after:
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
hm : m ≤ m₀
inst✝ : SigmaFinite (μ.trim hm)
hf : Integrable f μ
x : α
⊢ ↑↑(condExpL1 hm μ f) x = ↑↑((condExpL1CLM E hm μ) (Integrable.toL1 f hf)) x
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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
hm : m ≤ m₀
inst✝ : SigmaFinite (μ.trim hm)
hf : Integrable f μ
x : α
⊢ ↑↑((condExpL1CLM E hm μ) (Integrable.toL1 f hf)) x = ↑↑((condExpL1CLM E hm μ) (Integrable.toL1 f hf)) x

Tactic:
]

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

Goal state before:
pp: case refine_1
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 : (α → F) → Prop
h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add :
  ∀ ⦃f g : α → F⦄,
    Disjoint (Function.support f) (Function.support g) →
      MemLp f p μ → MemLp g p μ → StronglyMeasurable f → StronglyMeasurable g → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑↑f}
h_ae : ∀ ⦃f g : α → F⦄, f =ᵐ[μ] g → MemLp f p μ → P f → P g
f : α → F
hf : MemLp f p μ
hfm : AEStronglyMeasurable f μ
f_Lp : ↥(Lp F p μ) := toLp f hf
hfm_Lp : AEStronglyMeasurable (↑↑f_Lp) μ
c : F
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ P ↑↑(indicatorConstLp p ⋯ ⋯ c)

Tactic:
]

Goal state after:
case refine_1
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 : (α → F) → Prop
h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add :
  ∀ ⦃f g : α → F⦄,
    Disjoint (Function.support f) (Function.support g) →
      MemLp f p μ → MemLp g p μ → StronglyMeasurable f → StronglyMeasurable g → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑↑f}
h_ae : ∀ ⦃f g : α → F⦄, f =ᵐ[μ] g → MemLp f p μ → P f → P g
f : α → F
hf : MemLp f p μ
hfm : AEStronglyMeasurable f μ
f_Lp : ↥(Lp F p μ) := toLp f hf
hfm_Lp : AEStronglyMeasurable (↑↑f_Lp) μ
c : F
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ P ↑↑(indicatorConstLp p ⋯ ⋯ c)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
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
hf : ¬Integrable f μ
hm : ¬m ≤ m₀
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg
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
hf : ¬Integrable f μ
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 = 0

Tactic:
]

Goal state after:
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
hS_finite : ∀ (i : ℕ), μ (S i ∩ s) < ∞
h_mono : Monotone fun i ↦ S i ∩ s
h_eq_forall : (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) = fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ
h : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))
⊢ Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))

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
hS_finite : ∀ (i : ℕ), μ (S i ∩ s) < ∞
h_mono : Monotone fun i ↦ S i ∩ s
h_eq_forall : (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) = fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ
h : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))
⊢ Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))
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
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hf : ¬Integrable f μ
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg
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
⊢ 0 = 0

Tactic:
]

Goal state after:
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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
⊢ ↑↑↑((condExpL2 E' 𝕜 hm) ((toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1))) =ᵐ[μ] fun a ↦
    ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x

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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
⊢ ↑↑↑((condExpL2 E' 𝕜 hm) ((toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1))) =ᵐ[μ] fun a ↦
    ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
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)
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg
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
⊢ StronglyMeasurable 0

Tactic:
]

Goal state after:
case neg
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
⊢ StronglyMeasurable 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
hS_finite : ∀ (i : ℕ), μ (S i ∩ s) < ∞
h_mono : Monotone fun i ↦ S i ∩ s
h_eq_forall : (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) = fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ
h_right : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))
h :
  Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) atTop
    (𝓝 (∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ))
⊢ Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) atTop
    (𝓝 (∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ))

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
hS_finite : ∀ (i : ℕ), μ (S i ∩ s) < ∞
h_mono : Monotone fun i ↦ S i ∩ s
h_eq_forall : (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) = fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ
h_right : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))
h :
  Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) atTop
    (𝓝 (∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ))
⊢ Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) atTop
    (𝓝 (∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ))
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
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)
⊢ StronglyMeasurable 0

Tactic:
]

Goal state after:
case neg
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)
⊢ StronglyMeasurable 0
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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
h_comp :
  ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) =ᵐ[μ] fun a ↦
    (toSpanSingleton ℝ x) (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a)
⊢ (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) =ᵐ[μ]
    ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)))

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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
h_comp :
  ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) =ᵐ[μ] fun a ↦
    (toSpanSingleton ℝ x) (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a)
⊢ (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) =ᵐ[μ]
    ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)))
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
hS_finite : ∀ (i : ℕ), μ (S i ∩ s) < ∞
h_mono : Monotone fun i ↦ S i ∩ s
h_eq_forall : (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) = fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ
h_right : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))
h_left : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ))
⊢ ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ

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
hS_finite : ∀ (i : ℕ), μ (S i ∩ s) < ∞
h_mono : Monotone fun i ↦ S i ∩ s
h_eq_forall : (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ) = fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ
h_right : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑f x ∂μ))
h_left : Tendsto (fun i ↦ ∫ (x : α) in S i ∩ s, ↑↑f x ∂μ) atTop (𝓝 (∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ))
⊢ ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ
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
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ StronglyMeasurable
    (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯ else 0)

Tactic:
]

Goal state after:
case pos
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)
⊢ StronglyMeasurable
    (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯ else 0)
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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
a : α
ha :
  ↑↑↑((condExpL2 E' 𝕜 hm) (indicatorConstLp 2 hs hμs x)) a =
    (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) a
x✝ : a ∈ t
⊢ ↑‖(fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) a‖₊ =
    ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x‖₊

Tactic:
]

Goal state after:
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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ (∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ) * ↑‖x‖₊ =
    (∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ) * ↑‖x‖₊

pp: case hf
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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ Measurable fun a ↦ ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊

Tactic:
]

Goal state after:
case hf
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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ Measurable fun a ↦ ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case refine_1
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)
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
c : F'
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ AEStronglyMeasurable (↑↑((condExpInd F' hm μ s) c)) μ

Tactic:
]

Goal state after:
case refine_1
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)
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
c : F'
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ AEStronglyMeasurable (↑↑((condExpInd F' hm μ s) c)) μ
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : Integrable (fun a ↦ ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) a).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) a).toReal) (ν ⊗ₘ η)
hμν_ac : μ ≪ ν
hκη_ac : μ ⊗ₘ κ ≪ μ ⊗ₘ η
⊢ Integrable (llr μ ν) μ

Tactic:
]

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : Integrable (fun a ↦ ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) a).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) a).toReal) (ν ⊗ₘ η)
hμν_ac : μ ≪ ν
hκη_ac : μ ⊗ₘ κ ≪ μ ⊗ₘ η
⊢ Integrable (llr μ ν) μ
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
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ μ (s ∩ t) * ↑‖x‖₊ ≤ μ (s ∩ t) * ↑‖x‖₊

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 g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
h : f =ᵐ[μ] g
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ ↑↑(condExpL1 hm μ g) =ᵐ[μ] ↑↑(condExpL1 hm μ g)

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
hf : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ AEStronglyMeasurable.mk f hf =ᵐ[μ] AEStronglyMeasurable.mk f hf

Tactic:
]

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : ¬m ≤ m₀
⊢ Integrable 0 μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : ¬m ≤ m₀
⊢ Integrable 0 μ
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)
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
f g : α → F'
hf : MemLp f 1 μ
hg : MemLp g 1 μ
a✝ : Disjoint (Function.support f) (Function.support g)
hfm : AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf))) μ
hgm : AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg))) μ
⊢ AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf) + (condExpL1CLM F' hm μ) (MemLp.toLp g hg))) 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)
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
f g : α → F'
hf : MemLp f 1 μ
hg : MemLp g 1 μ
a✝ : Disjoint (Function.support f) (Function.support g)
hfm : AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf))) μ
hgm : AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg))) μ
⊢ AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf) + (condExpL1CLM F' hm μ) (MemLp.toLp g hg))) μ