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Bandits.identDistrib_eval_eval_id_streamMeasure๐Ÿ”—

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identDistrib_eval_eval_id_streamMeasure๐Ÿ”—

LemmaBandits.identDistrib_eval_eval_id_streamMeasure
Details

No docstring.

๐Ÿ”—theorem
Bandits.identDistrib_eval_eval_id_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.IdentDistrib (fun h => h n a) id (streamMeasure ฮฝ)
    (ฮฝ a)
Bandits.identDistrib_eval_eval_id_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.IdentDistrib
    (fun h => h n a) id (streamMeasure ฮฝ)
    (ฮฝ a)

Code

lemma identDistrib_eval_eval_id_streamMeasure (ฮฝ : Kernel ๐“ R) [IsMarkovKernel ฮฝ] (n : โ„•) (a : ๐“) :
    IdentDistrib (fun h : โ„• โ†’ ๐“ โ†’ R โ†ฆ h n a) id (streamMeasure ฮฝ) (ฮฝ a) where
  aemeasurable_fst
Type uses (1)
Body uses (1)
Used by (7)

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Proof
Measurable.aemeasurable (by fun_prop)
  aemeasurable_snd := Measurable.aemeasurable (by fun_prop)
  map_eq := by
    rw [โ† (hasLaw_eval_eval_streamMeasure ฮฝ n a).map_eq,
      Measure.map_map (by fun_prop) (by fun_prop)]
    simp

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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