File: Mathlib/Probability/Independence/ZeroOne.lean

full_name: ProbabilityTheory.Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self
code: theorem Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
    (h_indep : Kernel.IndepSet t t κ μα) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 ∨ κ a t = ∞ 
goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤

full_name: ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self
code: theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
    (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ 
goal: μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤

full_name: ProbabilityTheory.Kernel.measure_eq_zero_or_one_of_indepSet_self
code: theorem Kernel.measure_eq_zero_or_one_of_indepSet_self [∀ a, IsFiniteMeasure (κ a)] {t : Set Ω}
    (h_indep : IndepSet t t κ μα) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1

full_name: ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self
code: theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω}
    (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 
goal: μ t = 0 ∨ μ t = 1

full_name: ProbabilityTheory.condexp_eq_zero_or_one_of_condIndepSet_self
code: theorem condexp_eq_zero_or_one_of_condIndepSet_self
    [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [hμ : IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t)
    (h_indep : CondIndepSet m hm t t μ) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1

full_name: ProbabilityTheory.Kernel.indep_biSup_compl
code: theorem Kernel.indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) (t : Set ι) :
    Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) κ μα 
goal: ProbabilityTheory.Kernel.Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) κ μα

full_name: ProbabilityTheory.indep_biSup_compl
code: theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t : Set ι) :
    Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ 
goal: ProbabilityTheory.Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ

full_name: ProbabilityTheory.condIndep_biSup_compl
code: theorem condIndep_biSup_compl [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (t : Set ι) :
    CondIndep m (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) hm μ 
goal: ProbabilityTheory.CondIndep m (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) hm μ

full_name: ProbabilityTheory.Kernel.indep_biSup_limsup
code: theorem Kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f) {t : Set ι} (ht : p t) :
    Indep (⨆ n ∈ t, s n) (limsup s f) κ μα 
goal: ProbabilityTheory.Kernel.Indep (⨆ n ∈ t, s n) (Filter.limsup s f) κ μα

full_name: ProbabilityTheory.indep_biSup_limsup
code: theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    {t : Set ι} (ht : p t) :
    Indep (⨆ n ∈ t, s n) (limsup s f) μ 
goal: ProbabilityTheory.Indep (⨆ n ∈ t, s n) (Filter.limsup s f) μ

full_name: ProbabilityTheory.condIndep_biSup_limsup
code: theorem condIndep_biSup_limsup [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    {t : Set ι} (ht : p t) :
    CondIndep m (⨆ n ∈ t, s n) (limsup s f) hm μ 
goal: ProbabilityTheory.CondIndep m (⨆ n ∈ t, s n) (Filter.limsup s f) hm μ

full_name: ProbabilityTheory.Kernel.indep_iSup_directed_limsup
code: theorem Kernel.indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
    Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) κ μα 
goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) → ProbabilityTheory.Kernel.Indep (⨆ a, ⨆ n ∈ ns a, s n) (Filter.limsup s f) κ μα

full_name: ProbabilityTheory.indep_iSup_directed_limsup
code: theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
    Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ 
goal: ProbabilityTheory.Indep (⨆ a, ⨆ n ∈ ns a, s n) (Filter.limsup s f) μ

full_name: ProbabilityTheory.condIndep_iSup_directed_limsup
code: theorem condIndep_iSup_directed_limsup [StandardBorelSpace Ω]
    [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
    CondIndep m (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) hm μ 
goal: ProbabilityTheory.CondIndep m (⨆ a, ⨆ n ∈ ns a, s n) (Filter.limsup s f) hm μ

full_name: ProbabilityTheory.Kernel.indep_iSup_limsup
code: theorem Kernel.indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (⨆ n, s n) (limsup s f) κ μα 
goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) →
            (∀ (n : ι), ∃ a, n ∈ ns a) → ProbabilityTheory.Kernel.Indep (⨆ n, s n) (Filter.limsup s f) κ μα

full_name: ProbabilityTheory.indep_iSup_limsup
code: theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (⨆ n, s n) (limsup s f) μ 
goal: ProbabilityTheory.Indep (⨆ n, s n) (Filter.limsup s f) μ

full_name: ProbabilityTheory.condIndep_iSup_limsup
code: theorem condIndep_iSup_limsup [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    CondIndep m (⨆ n, s n) (limsup s f) hm μ 
goal: ProbabilityTheory.CondIndep m (⨆ n, s n) (Filter.limsup s f) hm μ

full_name: ProbabilityTheory.Kernel.indep_limsup_self
code: theorem Kernel.indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (limsup s f) (limsup s f) κ μα 
goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) →
            (∀ (n : ι), ∃ a, n ∈ ns a) → ProbabilityTheory.Kernel.Indep (Filter.limsup s f) (Filter.limsup s f) κ μα

full_name: ProbabilityTheory.indep_limsup_self
code: theorem indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (limsup s f) (limsup s f) μ 
goal: ProbabilityTheory.Indep (Filter.limsup s f) (Filter.limsup s f) μ

full_name: ProbabilityTheory.condIndep_limsup_self
code: theorem condIndep_limsup_self [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    CondIndep m (limsup s f) (limsup s f) hm μ 
goal: ProbabilityTheory.CondIndep m (Filter.limsup s f) (Filter.limsup s f) hm μ

full_name: ProbabilityTheory.Kernel.measure_zero_or_one_of_measurableSet_limsup
code: theorem Kernel.measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) →
            (∀ (n : ι), ∃ a, n ∈ ns a) → ∀ {t : Set Ω}, MeasurableSet t → ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1

full_name: ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup
code: theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
    μ t = 0 ∨ μ t = 1 
goal: μ t = 0 ∨ μ t = 1

full_name: ProbabilityTheory.condexp_zero_or_one_of_measurableSet_limsup
code: theorem condexp_zero_or_one_of_measurableSet_limsup [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1

full_name: ProbabilityTheory.Kernel.indep_limsup_atTop_self
code: theorem Kernel.indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
    Indep (limsup s atTop) (limsup s atTop) κ μα 
goal: ProbabilityTheory.Kernel.Indep (Filter.limsup s Filter.atTop) (Filter.limsup s Filter.atTop) κ μα

full_name: ProbabilityTheory.indep_limsup_atTop_self
code: theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
    Indep (limsup s atTop) (limsup s atTop) μ 
goal: ProbabilityTheory.Indep (Filter.limsup s Filter.atTop) (Filter.limsup s Filter.atTop) μ

full_name: ProbabilityTheory.condIndep_limsup_atTop_self
code: theorem condIndep_limsup_atTop_self [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
    CondIndep m (limsup s atTop) (limsup s atTop) hm μ 
goal: ProbabilityTheory.CondIndep m (Filter.limsup s Filter.atTop) (Filter.limsup s Filter.atTop) hm μ

full_name: ProbabilityTheory.Kernel.measure_zero_or_one_of_measurableSet_limsup_atTop
code: theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1

full_name: ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop
code: /-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
sub-σ-algebras has probability 0 or 1.
The tail σ-algebra `limsup s atTop` is the same as `⋂ n, ⋃ i ≥ n, s i`. -/
theorem measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
    μ t = 0 ∨ μ t = 1 
goal: μ t = 0 ∨ μ t = 1

full_name: ProbabilityTheory.condexp_zero_or_one_of_measurableSet_limsup_atTop
code: theorem condexp_zero_or_one_of_measurableSet_limsup_atTop [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ] (h_le : ∀ n, s n ≤ m0)
    (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1

full_name: ProbabilityTheory.Kernel.indep_limsup_atBot_self
code: theorem Kernel.indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
    Indep (limsup s atBot) (limsup s atBot) κ μα 
goal: ProbabilityTheory.Kernel.Indep (Filter.limsup s Filter.atBot) (Filter.limsup s Filter.atBot) κ μα

full_name: ProbabilityTheory.indep_limsup_atBot_self
code: theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
    Indep (limsup s atBot) (limsup s atBot) μ 
goal: ProbabilityTheory.Indep (Filter.limsup s Filter.atBot) (Filter.limsup s Filter.atBot) μ

full_name: ProbabilityTheory.condIndep_limsup_atBot_self
code: theorem condIndep_limsup_atBot_self [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
    CondIndep m (limsup s atBot) (limsup s atBot) hm μ 
goal: ProbabilityTheory.CondIndep m (Filter.limsup s Filter.atBot) (Filter.limsup s Filter.atBot) hm μ

full_name: ProbabilityTheory.Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot
code: /-- **Kolmogorov's 0-1 law**, kernel version: any event in the tail σ-algebra of an independent
sequence of sub-σ-algebras has probability 0 or 1 almost surely. -/
theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1

full_name: ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atBot
code: /-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
sub-σ-algebras has probability 0 or 1. -/
theorem measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
    μ t = 0 ∨ μ t = 1 
goal: μ t = 0 ∨ μ t = 1

full_name: ProbabilityTheory.condexp_zero_or_one_of_measurableSet_limsup_atBot
code: /-- **Kolmogorov's 0-1 law**, conditional version: any event in the tail σ-algebra of a
conditinoally independent sequence of sub-σ-algebras has conditional probability 0 or 1. -/
theorem condexp_zero_or_one_of_measurableSet_limsup_atBot [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ] (h_le : ∀ n, s n ≤ m0)
    (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1