Bandits.identDistrib_rewardByCount_eval.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] [StandardBorelSpace 𝓐]
  [Nonempty 𝓐] [StandardBorelSpace Ω] [Countable 𝓐]
  (h : Learning.IsAlgEnvSeq A R alg (Learning.stationaryEnv ν) P)
  (a : 𝓐) (n m : ℕ) (hn : n ≠ 0) :
  ProbabilityTheory.IdentDistrib (Learning.rewardByCount A R a n)
    (fun ω => ω m a) (MeasureTheory.Measure.prod P (streamMeasure ν))
    (streamMeasure ν)
Bandits.identDistrib_rewardByCount_eval.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} [DecidableEq 𝓐]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace Ω] [Countable 𝓐]
  (h :
    Learning.IsAlgEnvSeq A R alg
      (Learning.stationaryEnv ν) P)
  (a : 𝓐) (n m : ℕ) (hn : n ≠ 0) :
  ProbabilityTheory.IdentDistrib
    (Learning.rewardByCount A R a n)
    (fun ω => ω m a)
    (MeasureTheory.Measure.prod P
      (streamMeasure ν))
    (streamMeasure ν)