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Bandits.integrable_eval_streamMeasure๐Ÿ”—

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Minimal Lean file

integrable_eval_streamMeasure๐Ÿ”—

LemmaBandits.integrable_eval_streamMeasure
Details

No docstring.

๐Ÿ”—theorem
Bandits.integrable_eval_streamMeasure.{u_1} {๐“ : Type u_1}
  {m๐“ : MeasurableSpace ๐“} (ฮฝ : ProbabilityTheory.Kernel ๐“ โ„)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] (n : โ„•) (a : ๐“)
  (h_int : MeasureTheory.Integrable id (ฮฝ a)) :
  MeasureTheory.Integrable (fun h => h n a) (streamMeasure ฮฝ)
Bandits.integrable_eval_streamMeasure.{u_1}
  {๐“ : Type u_1} {m๐“ : MeasurableSpace ๐“}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ โ„)
  [ProbabilityTheory.IsMarkovKernel ฮฝ]
  (n : โ„•) (a : ๐“)
  (h_int :
    MeasureTheory.Integrable id (ฮฝ a)) :
  MeasureTheory.Integrable
    (fun h => h n a) (streamMeasure ฮฝ)

Code

lemma integrable_eval_streamMeasure (ฮฝ : Kernel ๐“ โ„) [IsMarkovKernel ฮฝ] (n : โ„•) (a : ๐“)
    (h_int : Integrable id (ฮฝ a)) :
    Integrable (fun h : โ„• โ†’ ๐“ โ†’ โ„ โ†ฆ h n a) (streamMeasure ฮฝ)
Type uses (1)
Body uses (2)

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Proof
Integrable.congr_identDistrib h_int (identDistrib_eval_eval_id_streamMeasure ฮฝ n a).symm

Dependency graph

Type dependencies (1)

streamMeasure๐Ÿ”—

DefinitionBandits.streamMeasure
Details

Measure of an infinite stream of rewards from each action.

๐Ÿ”—def
Bandits.streamMeasure.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R) : MeasureTheory.Measure (โ„• โ†’ ๐“ โ†’ R)
Bandits.streamMeasure.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R) :
  MeasureTheory.Measure (โ„• โ†’ ๐“ โ†’ R)

Code

noncomputable
def streamMeasure (ฮฝ : Kernel ๐“ R) : Measure (โ„• โ†’ ๐“ โ†’ R) :=
  Measure.infinitePi fun _ โ†ฆ Measure.infinitePi ฮฝ
Used by (56)

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