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:
· exact fun e ↦ measurable_inclusion_predictable <| (h𝓕.approx m).measurableSet_fiber e

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

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))