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Bandits.hasLaw_eval_eval_streamMeasure๐Ÿ”—

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hasLaw_eval_eval_streamMeasure๐Ÿ”—

LemmaBandits.hasLaw_eval_eval_streamMeasure
Details

No docstring.

๐Ÿ”—theorem
Bandits.hasLaw_eval_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] (n : โ„•) (a : ๐“) :
  ProbabilityTheory.HasLaw (fun h => h n a) (ฮฝ a) (streamMeasure ฮฝ)
Bandits.hasLaw_eval_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]
  (n : โ„•) (a : ๐“) :
  ProbabilityTheory.HasLaw
    (fun h => h n a) (ฮฝ a)
    (streamMeasure ฮฝ)

Code

lemma hasLaw_eval_eval_streamMeasure (ฮฝ : Kernel ๐“ R) [IsMarkovKernel ฮฝ] (n : โ„•) (a : ๐“) :
    HasLaw (fun h : โ„• โ†’ ๐“ โ†’ R โ†ฆ h n a) (ฮฝ a) (streamMeasure ฮฝ)
Type uses (1)
Body uses (2)
Used by (2)

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Proof
(hasLaw_eval_infinitePi ฮฝ a).comp (hasLaw_eval_streamMeasure ฮฝ n)

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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