`Bandits.indepFun_eval_streamMeasure'`🔗

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Minimal Lean file

`indepFun_eval_streamMeasure'`🔗

LemmaBandits.indepFun_eval_streamMeasure'

Details

No docstring.

theorem

Bandits.indepFun_eval_streamMeasure'.{u_1, u_2} {𝓐 : Type u_1}
  {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] {a b : 𝓐} (h : a ≠ b) :
  ProbabilityTheory.IndepFun (fun ω n => ω n a) (fun ω n => ω n b)
    (streamMeasure ν)
Bandits.indepFun_eval_streamMeasure'.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν]
  {a b : 𝓐} (h : a ≠ b) :
  ProbabilityTheory.IndepFun
    (fun ω n => ω n a) (fun ω n => ω n b)
    (streamMeasure ν)

Code

lemma indepFun_eval_streamMeasure' (ν : Kernel 𝓐 R) [IsMarkovKernel ν] {a b : 𝓐} (h : a ≠ b) :
    IndepFun (fun ω n ↦ ω n a) (fun ω n ↦ ω n b) (streamMeasure ν)

Type uses (1)

streamMeasure

Body uses (1)

indepFun_proj_infinitePi_infinitePi

Used by (1)

probReal_sum_le_sum_streamMeasure

Actions: Source · Open Issue

Proof

indepFun_proj_infinitePi_infinitePi h

Dependency graph

Type dependencies (1)

`streamMeasure`🔗

DefinitionBandits.streamMeasure

Details

Measure of an infinite stream of rewards from each action.

def

Bandits.streamMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) : MeasureTheory.Measure (ℕ → 𝓐 → R)
Bandits.streamMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (ℕ → 𝓐 → R)

Code

noncomputable
def streamMeasure (ν : Kernel 𝓐 R) : Measure (ℕ → 𝓐 → R) :=
  Measure.infinitePi fun _ ↦ Measure.infinitePi ν

Used by (56)

Actions: Source · Open Issue

Bandits.indepFun_eval_streamMeasure'🔗

indepFun_eval_streamMeasure'🔗

streamMeasure🔗

`Bandits.indepFun_eval_streamMeasure'`🔗

`indepFun_eval_streamMeasure'`🔗

`streamMeasure`🔗