Bandits.probReal_sumRewards_le_sumRewards_le.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  [Nonempty 𝓐] [Fintype 𝓐] [MeasurableSingletonClass 𝓐]
  (h : Learning.IsAlgEnvSeq A R alg (Learning.stationaryEnv ν) P)
  (a : 𝓐) (n m₁ m₂ : ℕ) :
  MeasureTheory.Measure.real P
      {ω |
        Learning.pullCount A (bestArm ν) n ω = m₁ ∧
          Learning.pullCount A a n ω = m₂ ∧
            Learning.sumRewards A R (bestArm ν) n ω ≤
              Learning.sumRewards A R a n ω} ≤
    MeasureTheory.Measure.real (streamMeasure ν)
      {ω |
        ∑ i ∈ Finset.range m₁, ω i (bestArm ν) ≤
          ∑ i ∈ Finset.range m₂, ω i a}
Bandits.probReal_sumRewards_le_sumRewards_le.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  [Nonempty 𝓐] [Fintype 𝓐]
  [MeasurableSingletonClass 𝓐]
  (h :
    Learning.IsAlgEnvSeq A R alg
      (Learning.stationaryEnv ν) P)
  (a : 𝓐) (n m₁ m₂ : ℕ) :
  MeasureTheory.Measure.real P
      {ω |
        Learning.pullCount A (bestArm ν) n
              ω =
            m₁ ∧
          Learning.pullCount A a n ω =
              m₂ ∧
            Learning.sumRewards A R
                (bestArm ν) n ω ≤
              Learning.sumRewards A R a n
                ω} ≤
    MeasureTheory.Measure.real
      (streamMeasure ν)
      {ω |
        ∑ i ∈ Finset.range m₁,
            ω i (bestArm ν) ≤
          ∑ i ∈ Finset.range m₂, ω i a}