LeanMachineLearning

ProbabilityTheory.Kernel.isSFinite_lift🔗

Lemma

No docstring.

🔗theorem
ProbabilityTheory.Kernel.isSFinite_lift.{x, y, w} {X : Type x}
  [MeasurableSpace X] {Y : Type y} [MeasurableSpace Y] {X' : Type w}
  [MeasurableSpace X'] {Y' : Type w} [MeasurableSpace Y'] (ex : X' ≃ᵐ X)
  (ey : Y' ≃ᵐ Y) (κ : Kernel X Y) :
  IsSFiniteKernel κ ↔ IsSFiniteKernel (lift κ)
ProbabilityTheory.Kernel.isSFinite_lift.{x,
    y, w}
  {X : Type x} [MeasurableSpace X]
  {Y : Type y} [MeasurableSpace Y]
  {X' : Type w} [MeasurableSpace X']
  {Y' : Type w} [MeasurableSpace Y']
  (ex : X' ≃ᵐ X) (ey : Y' ≃ᵐ Y)
  (κ : Kernel X Y) :
  IsSFiniteKernel κ ↔
    IsSFiniteKernel (lift κ)

Code

lemma isSFinite_lift (κ : Kernel X Y) :
    IsSFiniteKernel κ ↔ IsSFiniteKernel (κ.lift (ex := ex) (ey := ey))
Proof
by
  constructor
  · intro h
    simp only [lift]
    infer_instance
  · rintro ⟨κs, hfinite_κs, h⟩
    constructor
    let κs' (i : ℕ) := ((κs i).map ey).comap ex.symm ex.symm.measurable
    refine ⟨κs', ⟨fun i ↦ ?_, ?_⟩⟩
    · exact IsFiniteKernel.comap ((κs i).map ey) ex.symm.measurable
    · simp only [κs']
      ext a s hs
      replace h := DFunLike.congr (x := ey.symm '' s) (DFunLike.congr (x := ex.symm a) h rfl) rfl
      rw [sum_apply, Measure.sum_apply] at h ⊢
      · rw [lift_apply'] at h
        · convert h with x
          · simp
          · rw [image_symm]
            simp
          · simp only [coe_comap, Function.comp_apply]
            rw [map_apply' _ ey.measurable _ hs, image_symm]
        all_goals measurability
      all_goals measurability

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Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23).

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

Dependency graph

Audit surface: 1 project declarations, 11 external constants

✓ 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.