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Bandits.integral_eval_streamMeasure๐Ÿ”—

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integral_eval_streamMeasure๐Ÿ”—

LemmaBandits.integral_eval_streamMeasure
Details

No docstring.

๐Ÿ”—theorem
Bandits.integral_eval_streamMeasure.{u_1} {๐“ : Type u_1}
  {m๐“ : MeasurableSpace ๐“} (ฮฝ : ProbabilityTheory.Kernel ๐“ โ„)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] (n : โ„•) (a : ๐“) :
  โˆซ (h : โ„• โ†’ ๐“ โ†’ โ„), h n a โˆ‚streamMeasure ฮฝ = โˆซ (x : โ„), id x โˆ‚ฮฝ a
Bandits.integral_eval_streamMeasure.{u_1}
  {๐“ : Type u_1} {m๐“ : MeasurableSpace ๐“}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ โ„)
  [ProbabilityTheory.IsMarkovKernel ฮฝ]
  (n : โ„•) (a : ๐“) :
  โˆซ (h : โ„• โ†’ ๐“ โ†’ โ„),
      h n a โˆ‚streamMeasure ฮฝ =
    โˆซ (x : โ„), id x โˆ‚ฮฝ a

Code

lemma integral_eval_streamMeasure (ฮฝ : Kernel ๐“ โ„) [IsMarkovKernel ฮฝ] (n : โ„•) (a : ๐“) :
    โˆซ h, h n a โˆ‚(streamMeasure ฮฝ) = (ฮฝ a)[id]
Type uses (1)
Body uses (1)
Used by (1)

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Proof
by
  calc โˆซ h, h n a โˆ‚(streamMeasure ฮฝ)
  _ = โˆซ x, x โˆ‚((streamMeasure ฮฝ).map (fun h โ†ฆ h n a)) := by
    rw [integral_map (Measurable.aemeasurable (by fun_prop)) (by fun_prop)]
  _ = (ฮฝ a)[id] := by simp [(hasLaw_eval_eval_streamMeasure ฮฝ n a).map_eq]

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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