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Bandits.iIndepFun_eval_streamMeasure๐Ÿ”—

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iIndepFun_eval_streamMeasure๐Ÿ”—

LemmaBandits.iIndepFun_eval_streamMeasure
Details

No docstring.

๐Ÿ”—theorem
Bandits.iIndepFun_eval_streamMeasure.{u_1, u_2} {๐“ : Type u_1}
  {R : Type u_2} {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] :
  ProbabilityTheory.iIndepFun (fun p ฯ‰ => ฯ‰ (Prod.fst p) (Prod.snd p))
    (streamMeasure ฮฝ)
Bandits.iIndepFun_eval_streamMeasure.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] :
  ProbabilityTheory.iIndepFun
    (fun p ฯ‰ =>
      ฯ‰ (Prod.fst p) (Prod.snd p))
    (streamMeasure ฮฝ)

Code

lemma iIndepFun_eval_streamMeasure (ฮฝ : Kernel ๐“ R) [IsMarkovKernel ฮฝ] :
    iIndepFun (fun (p : โ„• ร— ๐“) ฯ‰ โ†ฆ ฯ‰ p.1 p.2) (streamMeasure ฮฝ)
Type uses (1)
Used by (1)

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Proof
iIndepFun_uncurry_infinitePi' (X := fun _ _ โ†ฆ id) (fun _ โ†ฆ ฮฝ) (by fun_prop)

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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