ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<GOAL>MeasureTheory.IsStoppingTime f fun ω => min (τ ω) i
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>AddGroup ι<VAR>Preorder ι<VAR>CovariantClass ι ι (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1<VAR>CovariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<VAR>hi : 0 ≤ i<GOAL>MeasureTheory.IsStoppingTime f fun ω => τ ω + i
<VAR>Ω : Type u_1<VAR>m : MeasurableSpace Ω<VAR>f : MeasureTheory.Filtration ℕ m<VAR>τ : Ω → ℕ<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ℕ<GOAL>MeasureTheory.IsStoppingTime f fun ω => τ ω + i
<VAR>Ω : Type u_1<VAR>m : MeasurableSpace Ω<VAR>f : MeasureTheory.Filtration ℕ m<VAR>τ π : Ω → ℕ<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<GOAL>MeasureTheory.IsStoppingTime f (τ + π)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>s : Set Ω<GOAL>MeasurableSet s ↔ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<VAR>hle : τ ≤ π<GOAL>hτ.measurableSpace ≤ hπ.measurableSpace
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>Countable ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>hτ.measurableSpace ≤ m
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>Filter.atTop.IsCountablyGenerated<VAR>Filter.atTop.NeBot<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>hτ.measurableSpace ≤ m
<VAR>Ω : Type u_1<VAR>m : MeasurableSpace Ω<VAR>ι : Type u_4<VAR>SemilatticeSup ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>Filter.atTop.IsCountablyGenerated<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>hτ.measurableSpace ≤ m
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>i : ι<GOAL>⋯.measurableSpace = ↑f i
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>s : Set Ω<VAR>i : ι<GOAL>MeasurableSet (s ∩ {ω | τ ω = i}) ↔ MeasurableSet (s ∩ {ω | τ ω = i})
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<VAR>hτ_le : ∀ (ω : Ω), τ ω ≤ i<GOAL>hτ.measurableSpace ≤ ↑f i
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>n : ι<VAR>hτ_le : ∀ (ω : Ω), τ ω ≤ n<GOAL>hτ.measurableSpace ≤ m
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<VAR>hτ_le : ∀ (ω : Ω), i ≤ τ ω<GOAL>↑f i ≤ hτ.measurableSpace
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<GOAL>MeasurableSet {ω | τ ω ≤ i}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>h_countable : (Set.range τ).Countable<VAR>i : ι<GOAL>MeasurableSet {ω | τ ω = i}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>Countable ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<GOAL>MeasurableSet {ω | τ ω = i}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>h_countable : (Set.range τ).Countable<VAR>i : ι<GOAL>MeasurableSet {ω | i ≤ τ ω}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>Countable ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<GOAL>MeasurableSet {ω | i ≤ τ ω}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>h_countable : (Set.range τ).Countable<VAR>i : ι<GOAL>MeasurableSet {ω | τ ω < i}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>Countable ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<GOAL>MeasurableSet {ω | τ ω < i}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>h_countable : (Set.range τ).Countable<GOAL>hτ.measurableSpace ≤ m
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>TopologicalSpace ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>Measurable τ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>TopologicalSpace ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<VAR>hτ_le : ∀ (ω : Ω), τ ω ≤ i<GOAL>Measurable τ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<GOAL>⋯.measurableSpace = hτ.measurableSpace ⊓ hπ.measurableSpace
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<VAR>s : Set Ω<GOAL>MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<GOAL>⋯.measurableSpace = hτ.measurableSpace ⊓ ↑f i
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>s : Set Ω<VAR>i : ι<GOAL>MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>TopologicalSpace ι<VAR>SecondCountableTopology ι<VAR>OrderTopology ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<VAR>s : Set Ω<VAR>hs : MeasurableSet s<GOAL>MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>TopologicalSpace ι<VAR>SecondCountableTopology ι<VAR>OrderTopology ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<VAR>s : Set Ω<GOAL>MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>s : Set Ω<VAR>i : ι<GOAL>MeasurableSet (s ∩ {ω | τ ω ≤ i}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ i})
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>TopologicalSpace ι<VAR>SecondCountableTopology ι<VAR>OrderTopology ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<GOAL>MeasurableSet {ω | τ ω ≤ π ω}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>AddGroup ι<VAR>TopologicalSpace ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>OrderTopology ι<VAR>MeasurableSingletonClass ι<VAR>SecondCountableTopology ι<VAR>MeasurableSub₂ ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<GOAL>MeasurableSet {ω | τ ω = π ω}
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>f : MeasureTheory.Filtration ι m<VAR>τ π : Ω → ι<VAR>Countable ι<VAR>TopologicalSpace ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>OrderTopology ι<VAR>MeasurableSingletonClass ι<VAR>SecondCountableTopology ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>hπ : MeasureTheory.IsStoppingTime f π<GOAL>MeasurableSet {ω | τ ω = π ω}
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>u : ι → Ω → β<VAR>i : ι<GOAL>(MeasureTheory.stoppedValue u fun x => i) = u i
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>LinearOrder ι<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<GOAL>MeasureTheory.stoppedProcess u τ = fun i => MeasureTheory.stoppedValue u fun ω => min i (τ ω)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>LinearOrder ι<VAR>u : ι → Ω → β<VAR>τ σ : Ω → ι<GOAL>MeasureTheory.stoppedValue (MeasureTheory.stoppedProcess u τ) σ = MeasureTheory.stoppedValue u fun ω => min (σ ω) (τ ω)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>LinearOrder ι<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>i : ι<VAR>ω : Ω<VAR>h : i ≤ τ ω<GOAL>MeasureTheory.stoppedProcess u τ i ω = u i ω
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>LinearOrder ι<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>i : ι<VAR>ω : Ω<VAR>h : τ ω ≤ i<GOAL>MeasureTheory.stoppedProcess u τ i ω = u (τ ω) ω
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>MeasurableSpace ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>BorelSpace ι<VAR>τ : Ω → ι<VAR>f : MeasureTheory.Filtration ι m<VAR>TopologicalSpace.MetrizableSpace ι<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>MeasureTheory.ProgMeasurable f fun i ω => min i (τ ω)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>MeasurableSpace ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>BorelSpace ι<VAR>TopologicalSpace β<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>f : MeasureTheory.Filtration ι m<VAR>TopologicalSpace.MetrizableSpace ι<VAR>h : MeasureTheory.ProgMeasurable f u<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>MeasureTheory.ProgMeasurable f (MeasureTheory.stoppedProcess u τ)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>MeasurableSpace ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>BorelSpace ι<VAR>TopologicalSpace β<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>f : MeasureTheory.Filtration ι m<VAR>TopologicalSpace.MetrizableSpace ι<VAR>h : MeasureTheory.ProgMeasurable f u<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>MeasureTheory.Adapted f (MeasureTheory.stoppedProcess u τ)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>MeasurableSpace ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>BorelSpace ι<VAR>TopologicalSpace β<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>f : MeasureTheory.Filtration ι m<VAR>TopologicalSpace.MetrizableSpace ι<VAR>hu : MeasureTheory.ProgMeasurable f u<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>i : ι<GOAL>MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedProcess u τ i)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>MeasurableSpace ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>BorelSpace ι<VAR>TopologicalSpace β<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>f : MeasureTheory.Filtration ι m<VAR>h : MeasureTheory.ProgMeasurable f u<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>n : ι<VAR>hτ_le : ∀ (ω : Ω), τ ω ≤ n<GOAL>MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedValue u τ)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>MeasurableSpace ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>SecondCountableTopology ι<VAR>BorelSpace ι<VAR>TopologicalSpace β<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>f : MeasureTheory.Filtration ι m<VAR>TopologicalSpace.MetrizableSpace β<VAR>MeasurableSpace β<VAR>BorelSpace β<VAR>hf_prog : MeasureTheory.ProgMeasurable f u<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>Measurable (MeasureTheory.stoppedValue u τ)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>AddCommMonoid E<VAR>s : Finset ι<VAR>hbdd : ∀ (ω : Ω), τ ω ∈ s<GOAL>MeasureTheory.stoppedValue u τ = ∑ i ∈ s, {ω | τ ω = i}.indicator (u i)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>Preorder ι<VAR>LocallyFiniteOrderBot ι<VAR>AddCommMonoid E<VAR>N : ι<VAR>hbdd : ∀ (ω : Ω), τ ω ≤ N<GOAL>MeasureTheory.stoppedValue u τ = ∑ i ∈ Finset.Iic N, {ω | τ ω = i}.indicator (u i)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>LinearOrder ι<VAR>AddCommMonoid E<VAR>s : Finset ι<VAR>n : ι<VAR>hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s<GOAL>MeasureTheory.stoppedProcess u τ n = {a | n ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = i}.indicator (u i)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>LinearOrder ι<VAR>LocallyFiniteOrderBot ι<VAR>AddCommMonoid E<VAR>n : ι<GOAL>MeasureTheory.stoppedProcess u τ n = {a | n ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.Iio n, {ω | τ ω = i}.indicator (u i)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>p : ℝ≥0∞<VAR>u : ι → Ω → E<VAR>PartialOrder ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ<VAR>s : Finset ι<VAR>hbdd : ∀ (ω : Ω), τ ω ∈ s<GOAL>MeasureTheory.Memℒp (MeasureTheory.stoppedValue u τ) p μ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>p : ℝ≥0∞<VAR>u : ι → Ω → E<VAR>PartialOrder ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>LocallyFiniteOrderBot ι<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ<VAR>N : ι<VAR>hbdd : ∀ (ω : Ω), τ ω ≤ N<GOAL>MeasureTheory.Memℒp (MeasureTheory.stoppedValue u τ) p μ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>PartialOrder ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ<VAR>s : Finset ι<VAR>hbdd : ∀ (ω : Ω), τ ω ∈ s<GOAL>MeasureTheory.Integrable (MeasureTheory.stoppedValue u τ) μ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>PartialOrder ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>LocallyFiniteOrderBot ι<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ<VAR>N : ι<VAR>hbdd : ∀ (ω : Ω), τ ω ≤ N<GOAL>MeasureTheory.Integrable (MeasureTheory.stoppedValue u τ) μ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>p : ℝ≥0∞<VAR>u : ι → Ω → E<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>FirstCountableTopology ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ<VAR>n : ι<VAR>s : Finset ι<VAR>hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s<GOAL>MeasureTheory.Memℒp (MeasureTheory.stoppedProcess u τ n) p μ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>p : ℝ≥0∞<VAR>u : ι → Ω → E<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>FirstCountableTopology ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>LocallyFiniteOrderBot ι<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ<VAR>n : ι<GOAL>MeasureTheory.Memℒp (MeasureTheory.stoppedProcess u τ n) p μ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>FirstCountableTopology ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ<VAR>n : ι<VAR>s : Finset ι<VAR>hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s<GOAL>MeasureTheory.Integrable (MeasureTheory.stoppedProcess u τ n) μ
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>μ : MeasureTheory.Measure Ω<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>u : ι → Ω → E<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>OrderTopology ι<VAR>FirstCountableTopology ι<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>NormedAddCommGroup E<VAR>LocallyFiniteOrderBot ι<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ<VAR>n : ι<GOAL>MeasureTheory.Integrable (MeasureTheory.stoppedProcess u τ n) μ
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>TopologicalSpace β<VAR>TopologicalSpace.PseudoMetrizableSpace β<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>SecondCountableTopology ι<VAR>OrderTopology ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>f : MeasureTheory.Filtration ι m<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>TopologicalSpace.MetrizableSpace ι<VAR>hu : MeasureTheory.Adapted f u<VAR>hu_cont : ∀ (ω : Ω), Continuous fun i => u i ω<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>MeasureTheory.Adapted f (MeasureTheory.stoppedProcess u τ)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>TopologicalSpace β<VAR>TopologicalSpace.PseudoMetrizableSpace β<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>SecondCountableTopology ι<VAR>OrderTopology ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>f : MeasureTheory.Filtration ι m<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>DiscreteTopology ι<VAR>hu : MeasureTheory.Adapted f u<VAR>hτ : MeasureTheory.IsStoppingTime f τ<GOAL>MeasureTheory.Adapted f (MeasureTheory.stoppedProcess u τ)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>TopologicalSpace β<VAR>TopologicalSpace.PseudoMetrizableSpace β<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>SecondCountableTopology ι<VAR>OrderTopology ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>f : MeasureTheory.Filtration ι m<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>TopologicalSpace.MetrizableSpace ι<VAR>hu : MeasureTheory.Adapted f u<VAR>hu_cont : ∀ (ω : Ω), Continuous fun i => u i ω<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>n : ι<GOAL>MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedProcess u τ n)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>TopologicalSpace β<VAR>TopologicalSpace.PseudoMetrizableSpace β<VAR>LinearOrder ι<VAR>TopologicalSpace ι<VAR>SecondCountableTopology ι<VAR>OrderTopology ι<VAR>MeasurableSpace ι<VAR>BorelSpace ι<VAR>f : MeasureTheory.Filtration ι m<VAR>u : ι → Ω → β<VAR>τ : Ω → ι<VAR>DiscreteTopology ι<VAR>hu : MeasureTheory.Adapted f u<VAR>hτ : MeasureTheory.IsStoppingTime f τ<VAR>n : ι<GOAL>MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedProcess u τ n)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>u : ℕ → Ω → β<VAR>τ π : Ω → ℕ<VAR>AddCommGroup β<VAR>hle : τ ≤ π<GOAL>MeasureTheory.stoppedValue u π - MeasureTheory.stoppedValue u τ = fun ω => (∑ i ∈ Finset.Ico (τ ω) (π ω), (u (i + 1) - u i)) ω
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>u : ℕ → Ω → β<VAR>τ π : Ω → ℕ<VAR>AddCommGroup β<VAR>hle : τ ≤ π<VAR>N : ℕ<VAR>hbdd : ∀ (ω : Ω), π ω ≤ N<GOAL>MeasureTheory.stoppedValue u π - MeasureTheory.stoppedValue u τ = fun ω => (∑ i ∈ Finset.range (N + 1), {ω | τ ω ≤ i ∧ i < π ω}.indicator (u (i + 1) - u i)) ω
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>u : ℕ → Ω → β<VAR>τ : Ω → ℕ<VAR>AddCommMonoid β<VAR>N : ℕ<VAR>hbdd : ∀ (ω : Ω), τ ω ≤ N<GOAL>MeasureTheory.stoppedValue u τ = fun x => (∑ i ∈ Finset.range (N + 1), {ω | τ ω = i}.indicator (u i)) x
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>u : ℕ → Ω → β<VAR>τ : Ω → ℕ<VAR>AddCommMonoid β<VAR>n : ℕ<GOAL>MeasureTheory.stoppedProcess u τ n = {a | n ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.range n, {ω | τ ω = i}.indicator (u i)
<VAR>Ω : Type u_1<VAR>β : Type u_2<VAR>u : ℕ → Ω → β<VAR>τ : Ω → ℕ<VAR>AddCommMonoid β<VAR>n : ℕ<GOAL>MeasureTheory.stoppedProcess u τ n = {a | n + 1 ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.range (n + 1), {a | τ a = i}.indicator (u i)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>𝒢 : MeasureTheory.Filtration ι m<VAR>τ η : Ω → ι<VAR>i : ι<VAR>s : Set Ω<VAR>DecidablePred fun x => x ∈ s<VAR>hτ_st : MeasureTheory.IsStoppingTime 𝒢 τ<VAR>hη_st : MeasureTheory.IsStoppingTime 𝒢 η<VAR>hτ : ∀ (ω : Ω), i ≤ τ ω<VAR>hη : ∀ (ω : Ω), i ≤ η ω<VAR>hs : MeasurableSet s<GOAL>MeasureTheory.IsStoppingTime 𝒢 (s.piecewise τ η)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>Preorder ι<VAR>𝒢 : MeasureTheory.Filtration ι m<VAR>i j : ι<VAR>s : Set Ω<VAR>DecidablePred fun x => x ∈ s<VAR>hij : i ≤ j<VAR>hs : MeasurableSet s<GOAL>MeasureTheory.IsStoppingTime 𝒢 (s.piecewise (fun x => i) fun x => j)
<VAR>Ω : Type u_1<VAR>s : Set Ω<VAR>DecidablePred fun x => x ∈ s<VAR>ι' : Type u_4<VAR>i j : ι'<VAR>f : ι' → Ω → ℝ<GOAL>MeasureTheory.stoppedValue f (s.piecewise (fun x => i) fun x => j) = s.piecewise (f i) (f j)
<VAR>Ω : Type u_1<VAR>s : Set Ω<VAR>DecidablePred fun x => x ∈ s<VAR>ι' : Type u_4<VAR>i j : ι'<VAR>f : ι' → Ω → ℝ<GOAL>MeasureTheory.stoppedValue f (s.piecewise (fun x => i) fun x => j) = s.indicator (f i) + sᶜ.indicator (f j)
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>μ : MeasureTheory.Measure Ω<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>NormedAddCommGroup E<VAR>NormedSpace ℝ E<VAR>CompleteSpace E<VAR>f : Ω → E<VAR>MeasureTheory.SigmaFiniteFiltration μ ℱ<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>h_countable : (Set.range τ).Countable<VAR>MeasureTheory.SigmaFinite (μ.trim ⋯)<VAR>i : ι<GOAL>μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|↑ℱ i]
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>μ : MeasureTheory.Measure Ω<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>NormedAddCommGroup E<VAR>NormedSpace ℝ E<VAR>CompleteSpace E<VAR>f : Ω → E<VAR>Countable ι<VAR>MeasureTheory.SigmaFiniteFiltration μ ℱ<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>MeasureTheory.SigmaFinite (μ.trim ⋯)<VAR>i : ι<GOAL>μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|↑ℱ i]
<VAR>Ω : Type u_1<VAR>ι : Type u_3<VAR>m : MeasurableSpace Ω<VAR>LinearOrder ι<VAR>μ : MeasureTheory.Measure Ω<VAR>ℱ : MeasureTheory.Filtration ι m<VAR>τ : Ω → ι<VAR>E : Type u_4<VAR>NormedAddCommGroup E<VAR>NormedSpace ℝ E<VAR>CompleteSpace E<VAR>f : Ω → E<VAR>Filter.atTop.IsCountablyGenerated<VAR>hτ : MeasureTheory.IsStoppingTime ℱ τ<VAR>i : ι<VAR>MeasureTheory.SigmaFinite (μ.trim ⋯)<GOAL>μ[f|⋯.measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ i}] μ[f|hτ.measurableSpace]