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Bandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel๐Ÿ”—

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instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel๐Ÿ”—

InstanceBandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel
Details

No docstring.

๐Ÿ”—theorem
Bandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel.{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] :
  MeasureTheory.IsProbabilityMeasure (streamMeasure ฮฝ)
Bandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel.{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] :
  MeasureTheory.IsProbabilityMeasure
    (streamMeasure ฮฝ)

Code

instance (ฮฝ : Kernel ๐“ R) [IsMarkovKernel ฮฝ] : IsProbabilityMeasure (streamMeasure ฮฝ)
Type uses (1)
Used by (22)

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Proof
by
  unfold streamMeasure
  infer_instance

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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