LeanMachineLearning

Bandits.ClippedUCB.integrable_uncurry_ucb_comp๐Ÿ”—

Lemma

No docstring.

๐Ÿ”—theorem
Bandits.ClippedUCB.integrable_uncurry_ucb_comp.{u_1} {K : โ„•}
  {l u ฯƒ2 ฮด : โ„} {ฮฉ : Type u_1} {A : โ„• โ†’ ฮฉ โ†’ Fin K} {R : โ„• โ†’ ฮฉ โ†’ โ„}
  [MeasurableSpace ฮฉ] (hA : โˆ€ (t : โ„•), Measurable (A t))
  (hR : โˆ€ (t : โ„•), Measurable (R t)) {f : ฮฉ โ†’ Fin K} (hf : Measurable f)
  {g : ฮฉ โ†’ โ„•} (hg : Measurable g) {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsFiniteMeasure P] :
  MeasureTheory.Integrable (fun ฯ‰ => ucb A R l u ฯƒ2 ฮด (f ฯ‰) (g ฯ‰) ฯ‰) P
Bandits.ClippedUCB.integrable_uncurry_ucb_comp.{u_1}
  {K : โ„•} {l u ฯƒ2 ฮด : โ„} {ฮฉ : Type u_1}
  {A : โ„• โ†’ ฮฉ โ†’ Fin K} {R : โ„• โ†’ ฮฉ โ†’ โ„}
  [MeasurableSpace ฮฉ]
  (hA : โˆ€ (t : โ„•), Measurable (A t))
  (hR : โˆ€ (t : โ„•), Measurable (R t))
  {f : ฮฉ โ†’ Fin K} (hf : Measurable f)
  {g : ฮฉ โ†’ โ„•} (hg : Measurable g)
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsFiniteMeasure P] :
  MeasureTheory.Integrable
    (fun ฯ‰ =>
      ucb A R l u ฯƒ2 ฮด (f ฯ‰) (g ฯ‰) ฯ‰)
    P

Code

lemma integrable_uncurry_ucb_comp [MeasurableSpace ฮฉ] (hA : โˆ€ t, Measurable (A t))
    (hR : โˆ€ t, Measurable (R t)) {f : ฮฉ โ†’ Fin K} (hf : Measurable f) {g : ฮฉ โ†’ โ„•}
    (hg : Measurable g) {P : Measure ฮฉ} [IsFiniteMeasure P] :
    Integrable (fun ฯ‰ โ†ฆ ucb A R l u ฯƒ2 ฮด (f ฯ‰) (g ฯ‰) ฯ‰) P
Proof
by
  refine โŸจ(measurable_uncurry_ucb_comp hA hR hf hg).aestronglyMeasurable, ?_โŸฉ
  apply HasFiniteIntegral.of_bounded (C := max |l| |u|)
  filter_upwards with ฯ‰
  rw [Real.norm_eq_abs]
  unfold ucb
  grind

Actions: Source ยท Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 3th recorded change.

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: 4 project declarations, 62 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.