File: Mathlib/Probability/Process/PartitionFiltration.lean

full_name: ProbabilityTheory.partitionFiltration
code: /-- A filtration built from the measurable spaces generated by the partitions `memPartition t n` for
all `n : ℕ`. -/
def partitionFiltration (ht : ∀ n, MeasurableSet (t n)) :
    Filtration ℕ m 
goal: MeasureTheory.Filtration ℕ m

full_name: ProbabilityTheory.measurableSet_partitionFiltration_of_mem
code: lemma measurableSet_partitionFiltration_of_mem (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) {s : Set α}
    (hs : s ∈ memPartition t n) :
    MeasurableSet[partitionFiltration ht n] s 
goal: MeasurableSet s

full_name: ProbabilityTheory.measurableSet_partitionFiltration_memPartitionSet
code: lemma measurableSet_partitionFiltration_memPartitionSet (ht : ∀ n, MeasurableSet (t n))
    (n : ℕ) (a : α) :
    MeasurableSet[partitionFiltration ht n] (memPartitionSet t n a) 
goal: MeasurableSet (memPartitionSet t n a)

full_name: ProbabilityTheory.measurable_memPartitionSet_subtype
code: lemma measurable_memPartitionSet_subtype (ht : ∀ n, MeasurableSet (t n)) (n : ℕ)
    (m : MeasurableSpace (memPartition t n)) :
    @Measurable α (memPartition t n) (partitionFiltration ht n) m
      (fun a ↦ ⟨memPartitionSet t n a, memPartitionSet_mem t n a⟩) 
goal: ∀ (m_1 : MeasurableSpace ↑(memPartition t n)), Measurable fun a => ⟨memPartitionSet t n a, ⋯⟩

full_name: ProbabilityTheory.measurable_partitionFiltration_memPartitionSet
code: lemma measurable_partitionFiltration_memPartitionSet (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) :
    Measurable[partitionFiltration ht n] (memPartitionSet t n) 
goal: Measurable (memPartitionSet t n)

full_name: ProbabilityTheory.measurable_memPartitionSet
code: lemma measurable_memPartitionSet (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) :
    Measurable (memPartitionSet t n) 
goal: Measurable (memPartitionSet t n)

full_name: ProbabilityTheory.iSup_partitionFiltration_eq_generateFrom_range
code: lemma iSup_partitionFiltration_eq_generateFrom_range (ht : ∀ n, MeasurableSet (t n)) :
    ⨆ n, partitionFiltration ht n = generateFrom (Set.range t) 
goal: ⨆ n, ↑(ProbabilityTheory.partitionFiltration ht) n = MeasurableSpace.generateFrom (Set.range t)

full_name: ProbabilityTheory.iSup_partitionFiltration
code: lemma iSup_partitionFiltration (ht : ∀ n, MeasurableSet (t n))
    (ht_range : generateFrom (Set.range t) = m) :
    ⨆ n, partitionFiltration ht n = m 
goal: ⨆ n, ↑(ProbabilityTheory.partitionFiltration ht) n = m

full_name: ProbabilityTheory.countableFiltration
code: /-- A filtration built from the measurable spaces generated by `countablePartition α n` for
all `n : ℕ`. -/
def countableFiltration (α : Type*) [m : MeasurableSpace α] [CountablyGenerated α] :
    Filtration ℕ m 
goal: MeasureTheory.Filtration ℕ m

full_name: ProbabilityTheory.measurableSet_countableFiltration_of_mem
code: lemma measurableSet_countableFiltration_of_mem (n : ℕ) {s : Set α}
    (hs : s ∈ countablePartition α n) :
    MeasurableSet[countableFiltration α n] s 
goal: MeasurableSet s

full_name: ProbabilityTheory.measurableSet_countableFiltration_countablePartitionSet
code: lemma measurableSet_countableFiltration_countablePartitionSet (n : ℕ) (t : α) :
    MeasurableSet[countableFiltration α n] (countablePartitionSet n t) 
goal: MeasurableSet (MeasurableSpace.countablePartitionSet n t)

full_name: ProbabilityTheory.measurable_countablePartitionSet_subtype
code: lemma measurable_countablePartitionSet_subtype (n : ℕ)
    (m : MeasurableSpace (countablePartition α n)) :
    @Measurable α (countablePartition α n) (countableFiltration α n) m
      (fun a ↦ ⟨countablePartitionSet n a, countablePartitionSet_mem n a⟩) 
goal: Measurable fun a => ⟨MeasurableSpace.countablePartitionSet n a, ⋯⟩

full_name: ProbabilityTheory.measurable_countableFiltration_countablePartitionSet
code: lemma measurable_countableFiltration_countablePartitionSet (α : Type*)
    [MeasurableSpace α] [CountablyGenerated α] (n : ℕ) :
    Measurable[countableFiltration α n] (countablePartitionSet n) 
goal: Measurable (MeasurableSpace.countablePartitionSet n)

full_name: ProbabilityTheory.measurable_countablePartitionSet
code: lemma measurable_countablePartitionSet (α : Type*) [MeasurableSpace α] [CountablyGenerated α]
    (n : ℕ) :
    Measurable (countablePartitionSet (α := α) n) 
goal: Measurable (MeasurableSpace.countablePartitionSet n)

full_name: ProbabilityTheory.iSup_countableFiltration
code: lemma iSup_countableFiltration (α : Type*) [m : MeasurableSpace α] [CountablyGenerated α] :
    ⨆ n, countableFiltration α n = m 
goal: ⨆ n, ↑(ProbabilityTheory.countableFiltration α) n = m