Theorem: MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le
File: Mathlib/Probability/Process/Stopping.lean

Proof steps:
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : LinearOrder ι
μ : Measure Ω
ℱ : Filtration ι m
τ σ : Ω → ι
E : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
f : Ω → E
inst✝⁶ : atTop.IsCountablyGenerated
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : MeasurableSpace ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
this : SigmaFinite (μ.trim ⋯)
⊢ μ[f|⋯.measurableSpace] =ᶠ[ae (μ.restrict {x | τ x ≤ σ x})] μ[f|hτ.measurableSpace]
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : LinearOrder ι
μ : Measure Ω
ℱ : Filtration ι m
τ σ : Ω → ι
E : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
f : Ω → E
inst✝⁶ : atTop.IsCountablyGenerated
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : MeasurableSpace ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
this : SigmaFinite (μ.trim ⋯)
t : Set Ω
⊢ MeasurableSet ({ω | τ ω ≤ σ ω} ∩ t) ↔ MeasurableSet ({ω | τ ω ≤ σ ω} ∩ t)
tactic: refine (condexp_ae_eq_restrict_of_measurableSpace_eq_on hτ.measurableSpace_le
    (hτ.min hσ).measurableSpace_le (hτ.measurableSet_le_stopping_time hσ) fun t => ?_).symm
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : LinearOrder ι
μ : Measure Ω
ℱ : Filtration ι m
τ σ : Ω → ι
E : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
f : Ω → E
inst✝⁶ : atTop.IsCountablyGenerated
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : MeasurableSpace ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
this : SigmaFinite (μ.trim ⋯)
t : Set Ω
⊢ MeasurableSet ({ω | τ ω ≤ σ ω} ∩ t) ↔ MeasurableSet ({ω | τ ω ≤ σ ω} ∩ t)
state_after: case hπ
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : LinearOrder ι
μ : Measure Ω
ℱ : Filtration ι m
τ σ : Ω → ι
E : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
f : Ω → E
inst✝⁶ : atTop.IsCountablyGenerated
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : MeasurableSpace ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
this : SigmaFinite (μ.trim ⋯)
t : Set Ω
⊢ IsStoppingTime ℱ σ
tactic: rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_le_iff]
state_before: case hπ
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : LinearOrder ι
μ : Measure Ω
ℱ : Filtration ι m
τ σ : Ω → ι
E : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
f : Ω → E
inst✝⁶ : atTop.IsCountablyGenerated
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : MeasurableSpace ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
this : SigmaFinite (μ.trim ⋯)
t : Set Ω
⊢ IsStoppingTime ℱ σ
state_after: no goals
tactic: simp_all only