LeanMachineLearning

Learning.measurable_rewardByCount๐Ÿ”—

Lemma

No docstring.

๐Ÿ”—theorem
Learning.measurable_rewardByCount.{u_1, u_2, u_3} {๐“ : Type u_1}
  {R : Type u_2} {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {R' : โ„• โ†’ ฮฉ โ†’ R} [MeasurableSingletonClass ๐“]
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n)) (a : ๐“) (m : โ„•) :
  Measurable fun ฯ‰ => rewardByCount A R' a m ฯ‰
Learning.measurable_rewardByCount.{u_1,
    u_2, u_3}
  {๐“ : Type u_1} {R : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R}
  {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {R' : โ„• โ†’ ฮฉ โ†’ R}
  [MeasurableSingletonClass ๐“]
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n))
  (a : ๐“) (m : โ„•) :
  Measurable fun ฯ‰ =>
    rewardByCount A R' a m ฯ‰

Code

lemma measurable_rewardByCount [MeasurableSingletonClass ๐“]
    (hA : โˆ€ n, Measurable (A n)) (hR' : โˆ€ n, Measurable (R' n)) (a : ๐“) (m : โ„•) :
    Measurable (fun ฯ‰ : ฮฉ ร— (โ„• โ†’ ๐“ โ†’ R) โ†ฆ rewardByCount A R' a m ฯ‰)
Proof
by
  simp_rw [rewardByCount_eq_ite]
  refine Measurable.ite ?_ ?_ ?_
  ยท exact (measurableSet_singleton _).preimage <| measurable_stepsUntil' hA a m
  ยท fun_prop
  ยท change Measurable ((fun p : โ„• ร— ฮฉ โ†ฆ R' p.1 p.2)
      โˆ˜ (fun ฯ‰ : ฮฉ ร— (โ„• โ†’ ๐“ โ†’ R) โ†ฆ ((stepsUntil A a m ฯ‰.1).toNat, ฯ‰.1)))
    have : Measurable fun ฯ‰ : ฮฉ ร— (โ„• โ†’ ๐“ โ†’ R) โ†ฆ ((stepsUntil A a m ฯ‰.1).toNat, ฯ‰.1) :=
      (measurable_stepsUntil' hA a m).toNat.prodMk (by fun_prop)
    refine Measurable.comp ?_ this
    refine measurable_from_prod_countable_right fun n โ†ฆ ?_
    simp only
    fun_prop

Actions: Source ยท Open Issue

Meaning last changed in v4.34.0-rc1-3-g2cf8f3b (2026-08-20).

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

Dependency graph

Audit surface: 3 project declarations, 37 external constants

โœ“ 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.