ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IsStoppingTime.measurable_of_le
protected theorem measurable_of_le [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι]
    [OrderTopology ι] [SecondCountableTopology ι] (hτ : IsStoppingTime f τ) {i : ι}
    (hτ_le : ∀ ω, τ ω ≤ i) : Measurable[f i] τ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [SecondCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) {i : ι} (hτ_le : ∀ (ω : Ω), τ ω ≤ i)
Goal: Measurable τ
-- MeasureTheory.IsStoppingTime.measurableSpace_min
theorem measurableSpace_min (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    (hτ.min hπ).measurableSpace = hτ.measurableSpace ⊓ hπ.measurableSpace 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: ⋯.measurableSpace = hτ.measurableSpace ⊓ hπ.measurableSpace
-- MeasureTheory.IsStoppingTime.measurableSet_min_iff
theorem measurableSet_min_iff (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) (s : Set Ω) :
    MeasurableSet[(hτ.min hπ).measurableSpace] s ↔
      MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[hπ.measurableSpace] s 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π) (s : Set Ω)
Goal: MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s
-- MeasureTheory.IsStoppingTime.measurableSpace_min_const
theorem measurableSpace_min_const (hτ : IsStoppingTime f τ) {i : ι} :
    (hτ.min_const i).measurableSpace = hτ.measurableSpace ⊓ f i 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) {i : ι}
Goal: ⋯.measurableSpace = hτ.measurableSpace ⊓ ↑f i
-- MeasureTheory.IsStoppingTime.measurableSet_min_const_iff
theorem measurableSet_min_const_iff (hτ : IsStoppingTime f τ) (s : Set Ω) {i : ι} :
    MeasurableSet[(hτ.min_const i).measurableSpace] s ↔
      MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[f i] s 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (s : Set Ω) {i : ι}
Goal: MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s
-- MeasureTheory.IsStoppingTime.measurableSet_inter_le
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π) (s : Set Ω) (hs : MeasurableSet s)
Goal: MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
-- MeasureTheory.IsStoppingTime.measurableSet_inter_le_iff
theorem measurableSet_inter_le_iff [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) (s : Set Ω) :
    MeasurableSet[hτ.measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) ↔
      MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π) (s : Set Ω)
Goal: MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
-- MeasureTheory.IsStoppingTime.measurableSet_inter_le_const_iff
theorem measurableSet_inter_le_const_iff (hτ : IsStoppingTime f τ) (s : Set Ω) (i : ι) :
    MeasurableSet[hτ.measurableSpace] (s ∩ {ω | τ ω ≤ i}) ↔
      MeasurableSet[(hτ.min_const i).measurableSpace] (s ∩ {ω | τ ω ≤ i}) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (s : Set Ω) (i : ι)
Goal: MeasurableSet (s ∩ {ω | τ ω ≤ i}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ i})
-- MeasureTheory.IsStoppingTime.measurableSet_le_stopping_time
theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: MeasurableSet {ω | τ ω ≤ π ω}
-- MeasureTheory.IsStoppingTime.measurableSet_stopping_time_le
theorem measurableSet_stopping_time_le [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hπ.measurableSpace] {ω | τ ω ≤ π ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: MeasurableSet {ω | τ ω ≤ π ω}
-- MeasureTheory.IsStoppingTime.measurableSet_eq_stopping_time
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι] [MeasurableSub₂ ι] (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: MeasurableSet {ω | τ ω = π ω}
-- MeasureTheory.IsStoppingTime.measurableSet_eq_stopping_time_of_countable
theorem measurableSet_eq_stopping_time_of_countable [Countable ι] [TopologicalSpace ι]
    [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι]
    [SecondCountableTopology ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} [Countable ι] [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: MeasurableSet {ω | τ ω = π ω}
-- MeasureTheory.stoppedValue_const
theorem stoppedValue_const (u : ι → Ω → β) (i : ι) : (stoppedValue u fun _ => i) = u i 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} (u : ι → Ω → β) (i : ι)
Goal: (MeasureTheory.stoppedValue u fun x => i) = u i
-- MeasureTheory.stoppedProcess_eq_stoppedValue
theorem stoppedProcess_eq_stoppedValue {u : ι → Ω → β} {τ : Ω → ι} :
    stoppedProcess u τ = fun i => stoppedValue u fun ω => min i (τ ω) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} [LinearOrder ι] {u : ι → Ω → β} {τ : Ω → ι}
Goal: MeasureTheory.stoppedProcess u τ = fun i => MeasureTheory.stoppedValue u fun ω => min i (τ ω)
-- MeasureTheory.stoppedValue_stoppedProcess
theorem stoppedValue_stoppedProcess {u : ι → Ω → β} {τ σ : Ω → ι} :
    stoppedValue (stoppedProcess u τ) σ = stoppedValue u fun ω => min (σ ω) (τ ω) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} [LinearOrder ι] {u : ι → Ω → β} {τ σ : Ω → ι}
Goal: MeasureTheory.stoppedValue (MeasureTheory.stoppedProcess u τ) σ = MeasureTheory.stoppedValue u fun ω => min (σ ω) (τ ω)
-- MeasureTheory.stoppedProcess_eq_of_le
theorem stoppedProcess_eq_of_le {u : ι → Ω → β} {τ : Ω → ι} {i : ι} {ω : Ω} (h : i ≤ τ ω) :
    stoppedProcess u τ i ω = u i ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} [LinearOrder ι] {u : ι → Ω → β} {τ : Ω → ι} {i : ι} {ω : Ω} (h : i ≤ τ ω)
Goal: MeasureTheory.stoppedProcess u τ i ω = u i ω
-- MeasureTheory.stoppedProcess_eq_of_ge
theorem stoppedProcess_eq_of_ge {u : ι → Ω → β} {τ : Ω → ι} {i : ι} {ω : Ω} (h : τ ω ≤ i) :
    stoppedProcess u τ i ω = u (τ ω) ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} [LinearOrder ι] {u : ι → Ω → β} {τ : Ω → ι} {i : ι} {ω : Ω} (h : τ ω ≤ i)
Goal: MeasureTheory.stoppedProcess u τ i ω = u (τ ω) ω
-- MeasureTheory.progMeasurable_min_stopping_time
theorem progMeasurable_min_stopping_time [MetrizableSpace ι] (hτ : IsStoppingTime f τ) :
    ProgMeasurable f fun i ω => min i (τ ω) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] [MeasurableSpace ι] [TopologicalSpace ι] [OrderTopology ι] [SecondCountableTopology ι] [BorelSpace ι] {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [TopologicalSpace.MetrizableSpace ι] (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: MeasureTheory.ProgMeasurable f fun i ω => min i (τ ω)
-- MeasureTheory.ProgMeasurable.stoppedProcess
theorem ProgMeasurable.stoppedProcess [MetrizableSpace ι] (h : ProgMeasurable f u)
    (hτ : IsStoppingTime f τ) : ProgMeasurable f (stoppedProcess u τ) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] [MeasurableSpace ι] [TopologicalSpace ι] [OrderTopology ι] [SecondCountableTopology ι] [BorelSpace ι] [TopologicalSpace β] {u : ι → Ω → β} {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [TopologicalSpace.MetrizableSpace ι] (h : MeasureTheory.ProgMeasurable f u) (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: MeasureTheory.ProgMeasurable f (MeasureTheory.stoppedProcess u τ)
-- MeasureTheory.ProgMeasurable.adapted_stoppedProcess
theorem ProgMeasurable.adapted_stoppedProcess [MetrizableSpace ι] (h : ProgMeasurable f u)
    (hτ : IsStoppingTime f τ) : Adapted f (MeasureTheory.stoppedProcess u τ) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] [MeasurableSpace ι] [TopologicalSpace ι] [OrderTopology ι] [SecondCountableTopology ι] [BorelSpace ι] [TopologicalSpace β] {u : ι → Ω → β} {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [TopologicalSpace.MetrizableSpace ι] (h : MeasureTheory.ProgMeasurable f u) (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: MeasureTheory.Adapted f (MeasureTheory.stoppedProcess u τ)
-- MeasureTheory.ProgMeasurable.stronglyMeasurable_stoppedProcess
theorem ProgMeasurable.stronglyMeasurable_stoppedProcess [MetrizableSpace ι]
    (hu : ProgMeasurable f u) (hτ : IsStoppingTime f τ) (i : ι) :
    StronglyMeasurable (MeasureTheory.stoppedProcess u τ i) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] [MeasurableSpace ι] [TopologicalSpace ι] [OrderTopology ι] [SecondCountableTopology ι] [BorelSpace ι] [TopologicalSpace β] {u : ι → Ω → β} {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [TopologicalSpace.MetrizableSpace ι] (hu : MeasureTheory.ProgMeasurable f u) (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedProcess u τ i)
-- MeasureTheory.stronglyMeasurable_stoppedValue_of_le
theorem stronglyMeasurable_stoppedValue_of_le (h : ProgMeasurable f u) (hτ : IsStoppingTime f τ)
    {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) : StronglyMeasurable[f n] (stoppedValue u τ) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] [MeasurableSpace ι] [TopologicalSpace ι] [OrderTopology ι] [SecondCountableTopology ι] [BorelSpace ι] [TopologicalSpace β] {u : ι → Ω → β} {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} (h : MeasureTheory.ProgMeasurable f u) (hτ : MeasureTheory.IsStoppingTime f τ) {n : ι} (hτ_le : ∀ (ω : Ω), τ ω ≤ n)
Goal: MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedValue u τ)
-- MeasureTheory.measurable_stoppedValue
theorem measurable_stoppedValue [MetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
    (hf_prog : ProgMeasurable f u) (hτ : IsStoppingTime f τ) :
    Measurable[hτ.measurableSpace] (stoppedValue u τ) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] [MeasurableSpace ι] [TopologicalSpace ι] [OrderTopology ι] [SecondCountableTopology ι] [BorelSpace ι] [TopologicalSpace β] {u : ι → Ω → β} {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [TopologicalSpace.MetrizableSpace β] [MeasurableSpace β] [BorelSpace β] (hf_prog : MeasureTheory.ProgMeasurable f u) (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: Measurable (MeasureTheory.stoppedValue u τ)
-- MeasureTheory.stoppedValue_eq_of_mem_finset
theorem stoppedValue_eq_of_mem_finset [AddCommMonoid E] {s : Finset ι} (hbdd : ∀ ω, τ ω ∈ s) :
    stoppedValue u τ = ∑ i ∈ s, Set.indicator {ω | τ ω = i} (u i) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [AddCommMonoid E] {s : Finset ι} (hbdd : ∀ (ω : Ω), τ ω ∈ s)
Goal: MeasureTheory.stoppedValue u τ = ∑ i ∈ s, {ω | τ ω = i}.indicator (u i)
-- MeasureTheory.stoppedValue_eq'
theorem stoppedValue_eq' [Preorder ι] [LocallyFiniteOrderBot ι] [AddCommMonoid E] {N : ι}
    (hbdd : ∀ ω, τ ω ≤ N) :
    stoppedValue u τ = ∑ i ∈ Finset.Iic N, Set.indicator {ω | τ ω = i} (u i) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [Preorder ι] [LocallyFiniteOrderBot ι] [AddCommMonoid E] {N : ι} (hbdd : ∀ (ω : Ω), τ ω ≤ N)
Goal: MeasureTheory.stoppedValue u τ = ∑ i ∈ Finset.Iic N, {ω | τ ω = i}.indicator (u i)
-- MeasureTheory.stoppedProcess_eq_of_mem_finset
theorem stoppedProcess_eq_of_mem_finset [LinearOrder ι] [AddCommMonoid E] {s : Finset ι} (n : ι)
    (hbdd : ∀ ω, τ ω < n → τ ω ∈ s) : stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) +
      ∑ i ∈ s.filter (· < n), Set.indicator {ω | τ ω = i} (u i) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [LinearOrder ι] [AddCommMonoid E] {s : Finset ι} (n : ι) (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)
-- MeasureTheory.stoppedProcess_eq''
theorem stoppedProcess_eq'' [LinearOrder ι] [LocallyFiniteOrderBot ι] [AddCommMonoid E] (n : ι) :
    stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) +
      ∑ i ∈ Finset.Iio n, Set.indicator {ω | τ ω = i} (u i) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [LinearOrder ι] [LocallyFiniteOrderBot ι] [AddCommMonoid E] (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)
-- MeasureTheory.memℒp_stoppedValue_of_mem_finset
theorem memℒp_stoppedValue_of_mem_finset (hτ : IsStoppingTime ℱ τ) (hu : ∀ n, Memℒp (u n) p μ)
    {s : Finset ι} (hbdd : ∀ ω, τ ω ∈ s) : Memℒp (stoppedValue u τ) p μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {p : ℝ≥0∞} {u : ι → Ω → E} [PartialOrder ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ) {s : Finset ι} (hbdd : ∀ (ω : Ω), τ ω ∈ s)
Goal: MeasureTheory.Memℒp (MeasureTheory.stoppedValue u τ) p μ
-- MeasureTheory.memℒp_stoppedValue
theorem memℒp_stoppedValue [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
    (hu : ∀ n, Memℒp (u n) p μ) {N : ι} (hbdd : ∀ ω, τ ω ≤ N) : Memℒp (stoppedValue u τ) p μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {p : ℝ≥0∞} {u : ι → Ω → E} [PartialOrder ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] [LocallyFiniteOrderBot ι] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ) {N : ι} (hbdd : ∀ (ω : Ω), τ ω ≤ N)
Goal: MeasureTheory.Memℒp (MeasureTheory.stoppedValue u τ) p μ
-- MeasureTheory.integrable_stoppedValue_of_mem_finset
theorem integrable_stoppedValue_of_mem_finset (hτ : IsStoppingTime ℱ τ)
    (hu : ∀ n, Integrable (u n) μ) {s : Finset ι} (hbdd : ∀ ω, τ ω ∈ s) :
    Integrable (stoppedValue u τ) μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [PartialOrder ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ) {s : Finset ι} (hbdd : ∀ (ω : Ω), τ ω ∈ s)
Goal: MeasureTheory.Integrable (MeasureTheory.stoppedValue u τ) μ
-- MeasureTheory.integrable_stoppedValue
theorem integrable_stoppedValue [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
    (hu : ∀ n, Integrable (u n) μ) {N : ι} (hbdd : ∀ ω, τ ω ≤ N) :
    Integrable (stoppedValue u τ) μ 
Arguments: {Ω : Type u_1} (ι : Type u_3) {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [PartialOrder ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] [LocallyFiniteOrderBot ι] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ) {N : ι} (hbdd : ∀ (ω : Ω), τ ω ≤ N)
Goal: MeasureTheory.Integrable (MeasureTheory.stoppedValue u τ) μ
-- MeasureTheory.memℒp_stoppedProcess_of_mem_finset
theorem memℒp_stoppedProcess_of_mem_finset (hτ : IsStoppingTime ℱ τ) (hu : ∀ n, Memℒp (u n) p μ)
    (n : ι) {s : Finset ι} (hbdd : ∀ ω, τ ω < n → τ ω ∈ s) : Memℒp (stoppedProcess u τ n) p μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {p : ℝ≥0∞} {u : ι → Ω → E} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ) (n : ι) {s : Finset ι} (hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s)
Goal: MeasureTheory.Memℒp (MeasureTheory.stoppedProcess u τ n) p μ
-- MeasureTheory.memℒp_stoppedProcess
theorem memℒp_stoppedProcess [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
    (hu : ∀ n, Memℒp (u n) p μ) (n : ι) : Memℒp (stoppedProcess u τ n) p μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {p : ℝ≥0∞} {u : ι → Ω → E} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] [LocallyFiniteOrderBot ι] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Memℒp (u n) p μ) (n : ι)
Goal: MeasureTheory.Memℒp (MeasureTheory.stoppedProcess u τ n) p μ
-- MeasureTheory.integrable_stoppedProcess_of_mem_finset
theorem integrable_stoppedProcess_of_mem_finset (hτ : IsStoppingTime ℱ τ)
    (hu : ∀ n, Integrable (u n) μ) (n : ι) {s : Finset ι} (hbdd : ∀ ω, τ ω < n → τ ω ∈ s) :
    Integrable (stoppedProcess u τ n) μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ) (n : ι) {s : Finset ι} (hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s)
Goal: MeasureTheory.Integrable (MeasureTheory.stoppedProcess u τ n) μ
-- MeasureTheory.integrable_stoppedProcess
theorem integrable_stoppedProcess [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
    (hu : ∀ n, Integrable (u n) μ) (n : ι) : Integrable (stoppedProcess u τ n) μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {τ : Ω → ι} {E : Type u_4} {u : ι → Ω → E} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [NormedAddCommGroup E] [LocallyFiniteOrderBot ι] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hu : ∀ (n : ι), MeasureTheory.Integrable (u n) μ) (n : ι)
Goal: MeasureTheory.Integrable (MeasureTheory.stoppedProcess u τ n) μ
-- MeasureTheory.Adapted.stoppedProcess
theorem Adapted.stoppedProcess [MetrizableSpace ι] (hu : Adapted f u)
    (hu_cont : ∀ ω, Continuous fun i => u i ω) (hτ : IsStoppingTime f τ) :
    Adapted f (stoppedProcess u τ) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [LinearOrder ι] [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] {f : MeasureTheory.Filtration ι m} {u : ι → Ω → β} {τ : Ω → ι} [TopologicalSpace.MetrizableSpace ι] (hu : MeasureTheory.Adapted f u) (hu_cont : ∀ (ω : Ω), Continuous fun i => u i ω) (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: MeasureTheory.Adapted f (MeasureTheory.stoppedProcess u τ)
-- MeasureTheory.Adapted.stoppedProcess_of_discrete
theorem Adapted.stoppedProcess_of_discrete [DiscreteTopology ι] (hu : Adapted f u)
    (hτ : IsStoppingTime f τ) : Adapted f (MeasureTheory.stoppedProcess u τ) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [LinearOrder ι] [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] {f : MeasureTheory.Filtration ι m} {u : ι → Ω → β} {τ : Ω → ι} [DiscreteTopology ι] (hu : MeasureTheory.Adapted f u) (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: MeasureTheory.Adapted f (MeasureTheory.stoppedProcess u τ)
-- MeasureTheory.Adapted.stronglyMeasurable_stoppedProcess
theorem Adapted.stronglyMeasurable_stoppedProcess [MetrizableSpace ι] (hu : Adapted f u)
    (hu_cont : ∀ ω, Continuous fun i => u i ω) (hτ : IsStoppingTime f τ) (n : ι) :
    StronglyMeasurable (MeasureTheory.stoppedProcess u τ n) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [LinearOrder ι] [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] {f : MeasureTheory.Filtration ι m} {u : ι → Ω → β} {τ : Ω → ι} [TopologicalSpace.MetrizableSpace ι] (hu : MeasureTheory.Adapted f u) (hu_cont : ∀ (ω : Ω), Continuous fun i => u i ω) (hτ : MeasureTheory.IsStoppingTime f τ) (n : ι)
Goal: MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedProcess u τ n)
-- MeasureTheory.Adapted.stronglyMeasurable_stoppedProcess_of_discrete
theorem Adapted.stronglyMeasurable_stoppedProcess_of_discrete [DiscreteTopology ι]
    (hu : Adapted f u) (hτ : IsStoppingTime f τ) (n : ι) :
    StronglyMeasurable (MeasureTheory.stoppedProcess u τ n) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [LinearOrder ι] [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] {f : MeasureTheory.Filtration ι m} {u : ι → Ω → β} {τ : Ω → ι} [DiscreteTopology ι] (hu : MeasureTheory.Adapted f u) (hτ : MeasureTheory.IsStoppingTime f τ) (n : ι)
Goal: MeasureTheory.StronglyMeasurable (MeasureTheory.stoppedProcess u τ n)