File: Mathlib/MeasureTheory/MeasurableSpace/Card.lean

full_name: _aux_Mathlib_MeasureTheory_MeasurableSpace_Card___macroRules_termω₁_1
code: local notation "ω₁" => (WellOrder.α <| Quotient.out <| Cardinal.ord (aleph 1 : Cardinal))
goal: Lean.Macro

full_name: MeasurableSpace.generateMeasurableRec
code: /-- Transfinite induction construction of the sigma-algebra generated by a set of sets `s`. At each
step, we add all elements of `s`, the empty set, the complements of already constructed sets, and
countable unions of already constructed sets. We index this construction by an ordinal `< ω₁`, as
this will be enough to generate all sets in the sigma-algebra.

This construction is very similar to that of the Borel hierarchy. -/
def generateMeasurableRec (s : Set (Set α)) : (ω₁ : Type u) → Set (Set α)
  
goal: (Quotient.out (Cardinal.aleph 1).ord).α → Set (Set α)

full_name: MeasurableSpace.self_subset_generateMeasurableRec
code: theorem self_subset_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
    s ⊆ generateMeasurableRec s i 
goal: s ⊆ MeasurableSpace.generateMeasurableRec s i

full_name: MeasurableSpace.empty_mem_generateMeasurableRec
code: theorem empty_mem_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
    ∅ ∈ generateMeasurableRec s i 
goal: ∅ ∈ MeasurableSpace.generateMeasurableRec s i

full_name: MeasurableSpace.compl_mem_generateMeasurableRec
code: theorem compl_mem_generateMeasurableRec {s : Set (Set α)} {i j : ω₁} (h : j < i) {t : Set α}
    (ht : t ∈ generateMeasurableRec s j) : tᶜ ∈ generateMeasurableRec s i 
goal: tᶜ ∈ MeasurableSpace.generateMeasurableRec s i

full_name: MeasurableSpace.iUnion_mem_generateMeasurableRec
code: theorem iUnion_mem_generateMeasurableRec {s : Set (Set α)} {i : ω₁} {f : ℕ → Set α}
    (hf : ∀ n, ∃ j < i, f n ∈ generateMeasurableRec s j) :
    (⋃ n, f n) ∈ generateMeasurableRec s i 
goal: ⋃ n, f n ∈ MeasurableSpace.generateMeasurableRec s i

full_name: MeasurableSpace.generateMeasurableRec_subset
code: theorem generateMeasurableRec_subset (s : Set (Set α)) {i j : ω₁} (h : i ≤ j) :
    generateMeasurableRec s i ⊆ generateMeasurableRec s j 
goal: MeasurableSpace.generateMeasurableRec s i ⊆ MeasurableSpace.generateMeasurableRec s j

full_name: MeasurableSpace.cardinal_generateMeasurableRec_le
code: /-- At each step of the inductive construction, the cardinality bound `≤ (max #s 2) ^ ℵ₀` holds.
-/
theorem cardinal_generateMeasurableRec_le (s : Set (Set α)) (i : ω₁) :
    #(generateMeasurableRec s i) ≤ max #s 2 ^ aleph0.{u} 
goal: #↑(MeasurableSpace.generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀

full_name: MeasurableSpace.generateMeasurable_eq_rec
code: /-- `generateMeasurableRec s` generates precisely the smallest sigma-algebra containing `s`. -/
theorem generateMeasurable_eq_rec (s : Set (Set α)) :
    { t | GenerateMeasurable s t } =
        ⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i 
goal: {t | MeasurableSpace.GenerateMeasurable s t} = ⋃ i, MeasurableSpace.generateMeasurableRec s i

full_name: MeasurableSpace.cardinal_generateMeasurable_le
code: /-- If a sigma-algebra is generated by a set of sets `s`, then the sigma-algebra has cardinality at
most `(max #s 2) ^ ℵ₀`. -/
theorem cardinal_generateMeasurable_le (s : Set (Set α)) :
    #{ t | GenerateMeasurable s t } ≤ max #s 2 ^ aleph0.{u} 
goal: #↑{t | MeasurableSpace.GenerateMeasurable s t} ≤ max (#↑s) 2 ^ ℵ₀

full_name: MeasurableSpace.cardinalMeasurableSet_le
code: /-- If a sigma-algebra is generated by a set of sets `s`, then the sigma
algebra has cardinality at most `(max #s 2) ^ ℵ₀`. -/
theorem cardinalMeasurableSet_le (s : Set (Set α)) :
    #{ t | @MeasurableSet α (generateFrom s) t } ≤ max #s 2 ^ aleph0.{u} 
goal: #↑{t | MeasurableSet t} ≤ max (#↑s) 2 ^ ℵ₀

full_name: MeasurableSpace.cardinal_generateMeasurable_le_continuum
code: /-- If a sigma-algebra is generated by a set of sets `s` with cardinality at most the continuum,
then the sigma algebra has the same cardinality bound. -/
theorem cardinal_generateMeasurable_le_continuum {s : Set (Set α)} (hs : #s ≤ 𝔠) :
    #{ t | GenerateMeasurable s t } ≤ 𝔠 
goal: #↑{t | MeasurableSpace.GenerateMeasurable s t} ≤ 𝔠

full_name: MeasurableSpace.cardinal_measurableSet_le_continuum
code: /-- If a sigma-algebra is generated by a set of sets `s` with cardinality at most the continuum,
then the sigma algebra has the same cardinality bound. -/
theorem cardinal_measurableSet_le_continuum {s : Set (Set α)} :
    #s ≤ 𝔠 → #{ t | @MeasurableSet α (generateFrom s) t } ≤ 𝔠 
goal: #↑s ≤ 𝔠 → #↑{t | MeasurableSet t} ≤ 𝔠