mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : OrderBot ι
𝓕 : Filtration ι m
m' : MeasurableSpace (ι × Ω)
hm'bot : ∀ (A : Set Ω), MeasurableSet A → MeasurableSet ({⊥} ×ˢ A)
hm' : ∀ (i : ι) (A : Set Ω), MeasurableSet A → MeasurableSet (Set.Ioi i ×ˢ A)
i : ι
A : Set Ω
hA : MeasurableSet A
⊢ MeasurableSet (Set.Ioi i ×ˢ A)

Tactic:
·

Goal state after:
case inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : OrderBot ι
𝓕 : Filtration ι m
m' : MeasurableSpace (ι × Ω)
hm'bot : ∀ (A : Set Ω), MeasurableSet A → MeasurableSet ({⊥} ×ˢ A)
hm' : ∀ (i : ι) (A : Set Ω), MeasurableSet A → MeasurableSet (Set.Ioi i ×ˢ A)
i : ι
A : Set Ω
hA : MeasurableSet A
⊢ MeasurableSet (Set.Ioi i ×ˢ A)
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_1
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
⊢ ∀ (A : Set Ω), MeasurableSet A → MeasurableSet ({⊥} ×ˢ A)

pp: case refine_2
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
⊢ ∀ (i_1 : ι) (A : Set Ω), MeasurableSet A → MeasurableSet (Set.Ioi i_1 ×ˢ A)

Tactic:
·

Goal state after:
case refine_1
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
⊢ ∀ (A : Set Ω), MeasurableSet A → MeasurableSet ({⊥} ×ˢ A)
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_2
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
⊢ ∀ (i_1 : ι) (A : Set Ω), MeasurableSet A → MeasurableSet (Set.Ioi i_1 ×ˢ A)

Tactic:
·

Goal state after:
case refine_2
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
⊢ ∀ (i_1 : ι) (A : Set Ω), MeasurableSet A → MeasurableSet (Set.Ioi i_1 ×ˢ A)
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_2.inl
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i j : ι
A : Set Ω
hA : MeasurableSet A
hji : j ≤ i
⊢ MeasurableSet ((fun x ↦ (↑x.1, x.2)) ⁻¹' Set.Ioi j ×ˢ A)

pp: case refine_2.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i j : ι
A : Set Ω
hA : MeasurableSet A
hij : i ≤ j
⊢ MeasurableSet ((fun x ↦ (↑x.1, x.2)) ⁻¹' Set.Ioi j ×ˢ A)

Tactic:
·

Goal state after:
case refine_2.inl
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i j : ι
A : Set Ω
hA : MeasurableSet A
hji : j ≤ i
⊢ MeasurableSet ((fun x ↦ (↑x.1, x.2)) ⁻¹' Set.Ioi j ×ˢ A)
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_2.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i j : ι
A : Set Ω
hA : MeasurableSet A
hij : i ≤ j
⊢ MeasurableSet ((fun x ↦ (↑x.1, x.2)) ⁻¹' Set.Ioi j ×ˢ A)

Tactic:
·

Goal state after:
case refine_2.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i j : ι
A : Set Ω
hA : MeasurableSet A
hij : i ≤ j
⊢ MeasurableSet ((fun x ↦ (↑x.1, x.2)) ⁻¹' Set.Ioi j ×ˢ A)
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_2
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
⊢ ∀ (i_1 : ι) (A : Set Ω), MeasurableSet A → MeasurableSet (Set.Ioi i_1 ×ˢ A)

Tactic:
intros j A hA
    simp only [MeasurableSpace.map_def]
    obtain hji | hij := le_total j i
    · rw [(by grind : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' Set.Ioi j ×ˢ A
        = (Subtype.val ⁻¹' (Set.Ioc j i)) ×ˢ A)]
      exact (measurable_subtype_coe measurableSet_Ioc).prod (𝓕.mono hji _ hA)
    · simp [(by grind : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' Set.Ioi j ×ˢ A = ∅)]

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_1
Ω : Type u_1
ι : Type u_2
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m✝
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m x ↦ (StronglyMeasurable.approx h𝓕 m) (↑x.1, x.2)
m : ℕ
⊢ ∀ (x : E), MeasurableSet (X m ⁻¹' {x})

pp: case refine_2
Ω : Type u_1
ι : Type u_2
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m✝
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m x ↦ (StronglyMeasurable.approx h𝓕 m) (↑x.1, x.2)
m : ℕ
⊢ (Set.range (X m)).Finite

pp: case refine_3
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m_1 x ↦ (StronglyMeasurable.approx h𝓕 m_1) (↑x.1, x.2)
⊢ ∀ (x : { x // x ∈ Set.Iic i } × Ω),
    Tendsto (fun n ↦ ((fun m_1 ↦ { toFun := X m_1, measurableSet_fiber' := ⋯, finite_range' := ⋯ }) n) x) atTop
      (𝓝 ((fun p ↦ u (↑p.1) p.2) x))

Tactic:
·

Goal state after:
case refine_1
Ω : Type u_1
ι : Type u_2
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m✝
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m x ↦ (StronglyMeasurable.approx h𝓕 m) (↑x.1, x.2)
m : ℕ
⊢ ∀ (x : E), MeasurableSet (X m ⁻¹' {x})
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_2
Ω : Type u_1
ι : Type u_2
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m✝
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m x ↦ (StronglyMeasurable.approx h𝓕 m) (↑x.1, x.2)
m : ℕ
⊢ (Set.range (X m)).Finite

pp: case refine_3
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m_1 x ↦ (StronglyMeasurable.approx h𝓕 m_1) (↑x.1, x.2)
⊢ ∀ (x : { x // x ∈ Set.Iic i } × Ω),
    Tendsto (fun n ↦ ((fun m_1 ↦ { toFun := X m_1, measurableSet_fiber' := ⋯, finite_range' := ⋯ }) n) x) atTop
      (𝓝 ((fun p ↦ u (↑p.1) p.2) x))

Tactic:
·

Goal state after:
case refine_2
Ω : Type u_1
ι : Type u_2
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m✝
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m x ↦ (StronglyMeasurable.approx h𝓕 m) (↑x.1, x.2)
m : ℕ
⊢ (Set.range (X m)).Finite
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_3
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m_1 x ↦ (StronglyMeasurable.approx h𝓕 m_1) (↑x.1, x.2)
⊢ ∀ (x : { x // x ∈ Set.Iic i } × Ω),
    Tendsto (fun n ↦ ((fun m_1 ↦ { toFun := X m_1, measurableSet_fiber' := ⋯, finite_range' := ⋯ }) n) x) atTop
      (𝓝 ((fun p ↦ u (↑p.1) p.2) x))

Tactic:
·

Goal state after:
case refine_3
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m_1 x ↦ (StronglyMeasurable.approx h𝓕 m_1) (↑x.1, x.2)
⊢ ∀ (x : { x // x ∈ Set.Iic i } × Ω),
    Tendsto (fun n ↦ ((fun m_1 ↦ { toFun := X m_1, measurableSet_fiber' := ⋯, finite_range' := ⋯ }) n) x) atTop
      (𝓝 ((fun p ↦ u (↑p.1) p.2) x))
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_1
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
n : ℕ
this✝ : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
this : MeasurableSpace Ω := ↑𝓕 n
X : ℕ → Ω → E := fun m ↦ Function.curry (⇑(StronglyMeasurable.approx h𝓕 m)) (n + 1)
m : ℕ
⊢ ∀ (x : E), MeasurableSet (X m ⁻¹' {x})

pp: case refine_2
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
n : ℕ
this✝ : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
this : MeasurableSpace Ω := ↑𝓕 n
X : ℕ → Ω → E := fun m ↦ Function.curry (⇑(StronglyMeasurable.approx h𝓕 m)) (n + 1)
m : ℕ
⊢ (Set.range (X m)).Finite

Tactic:
·

Goal state after:
case refine_1
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
n : ℕ
this✝ : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
this : MeasurableSpace Ω := ↑𝓕 n
X : ℕ → Ω → E := fun m ↦ Function.curry (⇑(StronglyMeasurable.approx h𝓕 m)) (n + 1)
m : ℕ
⊢ ∀ (x : E), MeasurableSet (X m ⁻¹' {x})
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_2
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
n : ℕ
this✝ : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
this : MeasurableSpace Ω := ↑𝓕 n
X : ℕ → Ω → E := fun m ↦ Function.curry (⇑(StronglyMeasurable.approx h𝓕 m)) (n + 1)
m : ℕ
⊢ (Set.range (X m)).Finite

Tactic:
·

Goal state after:
case refine_2
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
n : ℕ
this✝ : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
this : MeasurableSpace Ω := ↑𝓕 n
X : ℕ → Ω → E := fun m ↦ Function.curry (⇑(StronglyMeasurable.approx h𝓕 m)) (n + 1)
m : ℕ
⊢ (Set.range (X m)).Finite
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_1
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
⊢ ∀ (x : E), MeasurableSet (Y m ⁻¹' {x})

pp: case refine_2
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
⊢ (Set.range (Y m)).Finite

pp: case refine_3
Ω : Type u_1
m : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m_1 x ↦
    match x.1 with
    | 0 => (h₀.approx m_1) x.2
    | n.succ => (⋯.approx m_1) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
⊢ ∀ (x : ℕ × Ω),
    Tendsto (fun n ↦ ((fun m_1 ↦ { toFun := Y m_1, measurableSet_fiber' := ⋯, finite_range' := ⋯ }) n) x) atTop
      (𝓝 (Function.uncurry u x))

Tactic:
·

Goal state after:
case refine_1
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
⊢ ∀ (x : E), MeasurableSet (Y m ⁻¹' {x})
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_1.zero
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
⊢ MeasurableSet ({0} ×ˢ (Function.curry (Y m) 0 ⁻¹' {s}))

pp: case refine_1.succ
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
⊢ MeasurableSet ({n + 1} ×ˢ (Function.curry (Y m) (n + 1) ⁻¹' {s}))

Tactic:
·

Goal state after:
case refine_1.zero
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
⊢ MeasurableSet ({0} ×ˢ (Function.curry (Y m) 0 ⁻¹' {s}))
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_1.succ
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
⊢ MeasurableSet ({n + 1} ×ˢ (Function.curry (Y m) (n + 1) ⁻¹' {s}))

Tactic:
·

Goal state after:
case refine_1.succ
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
⊢ MeasurableSet ({n + 1} ×ˢ (Function.curry (Y m) (n + 1) ⁻¹' {s}))
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case pos
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : n + 1 ≤ m
⊢ MeasurableSet (Function.curry (Y m) (n + 1) ⁻¹' {s})

pp: case neg
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : m < n + 1
⊢ MeasurableSet (Function.curry (Y m) (n + 1) ⁻¹' {s})

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : n + 1 ≤ m
⊢ MeasurableSet (Function.curry (Y m) (n + 1) ⁻¹' {s})
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf_int : ∀ (n : ℕ), Integrable (f n) μ
n : ℕ
⊢ Integrable (martingalePart f ℱ μ n) μ

Tactic:
rw [martingalePart_eq_sum]

Goal state after:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf_int : ∀ (n : ℕ), Integrable (f n) μ
n : ℕ
⊢ Integrable ((fun n ↦ f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i | ↑ℱ i])) n) μ
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case neg
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : m < n + 1
⊢ MeasurableSet (Function.curry (Y m) (n + 1) ⁻¹' {s})

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : m < n + 1
⊢ MeasurableSet (Function.curry (Y m) (n + 1) ⁻¹' {s})
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝¹ : CompleteSpace E
hf : StronglyAdapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
⊢ μ[martingalePart f ℱ μ j | ↑ℱ i] =ᵐ[μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k | ↑ℱ i] - μ[μ[f (k + 1) - f k | ↑ℱ k] | ↑ℱ i])

Tactic:
rw [martingalePart_eq_sum]

Goal state after:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝¹ : CompleteSpace E
hf : StronglyAdapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
⊢ μ[(fun n ↦ f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i | ↑ℱ i])) j | ↑ℱ i] =ᵐ[μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k | ↑ℱ i] - μ[μ[f (k + 1) - f k | ↑ℱ k] | ↑ℱ i])