File: Mathlib/MeasureTheory/Constructions/Projective.lean

full_name: MeasureTheory.IsProjectiveMeasureFamily
code: /-- A family of measures indexed by finite sets of `ι` is projective if, for finite sets `J ⊆ I`,
the projection from `∀ i : I, α i` to `∀ i : J, α i` maps `P I` to `P J`. -/
def IsProjectiveMeasureFamily (P : ∀ J : Finset ι, Measure (∀ j : J, α j)) : Prop 
goal: Prop

full_name: MeasureTheory.IsProjectiveMeasureFamily.measure_univ_eq_of_subset
code: /-- Auxiliary lemma for `measure_univ_eq`. -/
lemma measure_univ_eq_of_subset (hP : IsProjectiveMeasureFamily P) (hJI : J ⊆ I) :
    P I univ = P J univ 
goal: (P I) Set.univ = (P J) Set.univ

full_name: MeasureTheory.IsProjectiveMeasureFamily.measure_univ_eq
code: lemma measure_univ_eq (hP : IsProjectiveMeasureFamily P) (I J : Finset ι) :
    P I univ = P J univ 
goal: (P I) Set.univ = (P J) Set.univ

full_name: MeasureTheory.IsProjectiveMeasureFamily.congr_cylinder_of_subset
code: lemma congr_cylinder_of_subset (hP : IsProjectiveMeasureFamily P)
    {S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (hT : MeasurableSet T)
    (h_eq : cylinder I S = cylinder J T) (hJI : J ⊆ I) :
    P I S = P J T 
goal: (P I) S = (P J) T

full_name: MeasureTheory.IsProjectiveMeasureFamily.congr_cylinder
code: lemma congr_cylinder (hP : IsProjectiveMeasureFamily P)
    {S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (hS : MeasurableSet S) (hT : MeasurableSet T)
    (h_eq : cylinder I S = cylinder J T) :
    P I S = P J T 
goal: (P I) S = (P J) T

full_name: MeasureTheory.IsProjectiveLimit
code: /-- A measure `μ` is the projective limit of a family of measures indexed by finite sets of `ι` if
for all `I : Finset ι`, the projection from `∀ i, α i` to `∀ i : I, α i` maps `μ` to `P I`. -/
def IsProjectiveLimit (μ : Measure (∀ i, α i))
    (P : ∀ J : Finset ι, Measure (∀ j : J, α j)) : Prop 
goal: Prop

full_name: MeasureTheory.IsProjectiveLimit.measure_cylinder
code: lemma measure_cylinder (h : IsProjectiveLimit μ P)
    (I : Finset ι) {s : Set (∀ i : I, α i)} (hs : MeasurableSet s) :
    μ (cylinder I s) = P I s 
goal: μ (MeasureTheory.cylinder I s) = (P I) s

full_name: MeasureTheory.IsProjectiveLimit.measure_univ_eq
code: lemma measure_univ_eq (hμ : IsProjectiveLimit μ P) (I : Finset ι) :
    μ univ = P I univ 
goal: μ Set.univ = (P I) Set.univ

full_name: MeasureTheory.IsProjectiveLimit.isFiniteMeasure
code: lemma isFiniteMeasure [∀ i, IsFiniteMeasure (P i)] (hμ : IsProjectiveLimit μ P) :
    IsFiniteMeasure μ 
goal: MeasureTheory.IsFiniteMeasure μ

full_name: MeasureTheory.IsProjectiveLimit.isProbabilityMeasure
code: lemma isProbabilityMeasure [∀ i, IsProbabilityMeasure (P i)] (hμ : IsProjectiveLimit μ P) :
    IsProbabilityMeasure μ 
goal: MeasureTheory.IsProbabilityMeasure μ

full_name: MeasureTheory.IsProjectiveLimit.measure_univ_unique
code: lemma measure_univ_unique (hμ : IsProjectiveLimit μ P) (hν : IsProjectiveLimit ν P) :
    μ univ = ν univ 
goal: μ Set.univ = ν Set.univ

full_name: MeasureTheory.IsProjectiveLimit.unique
code: /-- The projective limit of a family of finite measures is unique. -/
theorem unique [∀ i, IsFiniteMeasure (P i)]
    (hμ : IsProjectiveLimit μ P) (hν : IsProjectiveLimit ν P) :
    μ = ν 
goal: μ = ν