LeanMachineLearning

ProbabilityTheory.Kernel.hasLaw_trajMeasurešŸ”—

Lemma

Uniqueness of trajMeasure.

šŸ”—theorem
ProbabilityTheory.Kernel.hasLaw_trajMeasure.{u_1, u_2} {Ī© : Type u_1}
  {mĪ© : MeasurableSpace Ī©} {P : MeasureTheory.Measure Ī©}
  {X : ā„• ā†’ Type u_2} [(n : ā„•) ā†’ MeasurableSpace (X n)]
  {Īŗ : (n : ā„•) ā†’ Kernel ((i : ā†„(Finset.Iic n)) ā†’ X ā†‘i) (X (n + 1))}
  [āˆ€ (n : ā„•), IsMarkovKernel (Īŗ n)] {Ī¼ā‚€ : MeasureTheory.Measure (X 0)}
  [MeasureTheory.IsProbabilityMeasure Ī¼ā‚€]
  [MeasureTheory.IsFiniteMeasure P] {Y : (n : ā„•) ā†’ Ī© ā†’ X n}
  (hY_meas : āˆ€ (n : ā„•), Measurable (Y n)) (h0 : HasLaw (Y 0) Ī¼ā‚€ P)
  (h_condDistrib :
    āˆ€ (n : ā„•),
      HasCondDistrib (Y (n + 1)) (fun Ļ‰ i => Y (ā†‘i) Ļ‰) (Īŗ n) P) :
  HasLaw (fun Ļ‰ n => Y n Ļ‰) (trajMeasure Ī¼ā‚€ Īŗ) P
ProbabilityTheory.Kernel.hasLaw_trajMeasure.{u_1,
    u_2}
  {Ī© : Type u_1} {mĪ© : MeasurableSpace Ī©}
  {P : MeasureTheory.Measure Ī©}
  {X : ā„• ā†’ Type u_2}
  [(n : ā„•) ā†’ MeasurableSpace (X n)]
  {Īŗ :
    (n : ā„•) ā†’
      Kernel
        ((i : ā†„(Finset.Iic n)) ā†’ X ā†‘i)
        (X (n + 1))}
  [āˆ€ (n : ā„•), IsMarkovKernel (Īŗ n)]
  {Ī¼ā‚€ : MeasureTheory.Measure (X 0)}
  [MeasureTheory.IsProbabilityMeasure Ī¼ā‚€]
  [MeasureTheory.IsFiniteMeasure P]
  {Y : (n : ā„•) ā†’ Ī© ā†’ X n}
  (hY_meas : āˆ€ (n : ā„•), Measurable (Y n))
  (h0 : HasLaw (Y 0) Ī¼ā‚€ P)
  (h_condDistrib :
    āˆ€ (n : ā„•),
      HasCondDistrib (Y (n + 1))
        (fun Ļ‰ i => Y (ā†‘i) Ļ‰) (Īŗ n) P) :
  HasLaw (fun Ļ‰ n => Y n Ļ‰)
    (trajMeasure Ī¼ā‚€ Īŗ) P

Code

lemma hasLaw_trajMeasure [IsFiniteMeasure P]
    {Y : (n : ā„•) ā†’ Ī© ā†’ X n} (hY_meas : āˆ€ n, Measurable (Y n))
    (h0 : HasLaw (Y 0) Ī¼ā‚€ P)
    (h_condDistrib : āˆ€ n, HasCondDistrib (Y (n + 1)) (fun Ļ‰ ā†¦ fun i : Iic n ā†¦ Y i Ļ‰) (Īŗ n) P) :
    HasLaw (fun Ļ‰ n ā†¦ Y n Ļ‰) (trajMeasure Ī¼ā‚€ Īŗ) P where
  aemeasurable
Proof
by fun_prop
  map_eq := by
    refine IsProjectiveLimit.unique (P := fun (J : Finset ā„•) ā†¦ P.map (fun Ļ‰ (i : J) ā†¦ Y i Ļ‰)) ?_ ?_
    Ā· exact isProjectiveLimit_map (by fun_prop)
    rw [isProjectiveLimit_nat_iff]
    swap; Ā· exact isProjectiveMeasureFamily_map_restrict (by fun_prop)
    intro n
    rw [(hasLaw_Iic_of_forall_hasCondDistrib h0 h_condDistrib n).map_eq,
      trajMeasure_map_frestrictLe]

Actions: Source Ā· Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file Ā· open it in the Lean web editor.

Dependency graph

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited ā€” so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 28 external constants

āœ“ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.