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Learning.measurable_comap_indicator_stepsUntil_eq_zero🔗

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Minimal Lean file

measurable_comap_indicator_stepsUntil_eq_zero🔗

LemmaLearning.measurable_comap_indicator_stepsUntil_eq_zero
Details

No docstring.

🔗theorem
Learning.measurable_comap_indicator_stepsUntil_eq_zero.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐} [DecidableEq 𝓐]
  {A : ℕ → Ω → 𝓐} [MeasurableSingletonClass 𝓐] (a : 𝓐) (m : ℕ) :
  Measurable (Set.indicator {ω | stepsUntil A a m ω = 0} fun x => 1)
Learning.measurable_comap_indicator_stepsUntil_eq_zero.{u_1,
    u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  {m𝓐 : MeasurableSpace 𝓐} [DecidableEq 𝓐]
  {A : ℕ → Ω → 𝓐}
  [MeasurableSingletonClass 𝓐] (a : 𝓐)
  (m : ℕ) :
  Measurable
    (Set.indicator
      {ω | stepsUntil A a m ω = 0}
      fun x => 1)

Code

lemma measurable_comap_indicator_stepsUntil_eq_zero [MeasurableSingletonClass 𝓐] (a : 𝓐) (m : ℕ) :
    Measurable[MeasurableSpace.comap (A 0) inferInstance]
      ({ω | stepsUntil A a m ω = 0}.indicator fun _ ↦ 1)
Type uses (1)
Body uses (1)
Used by (1)

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Proof
by
  rw [measurable_indicator_const_iff]
  exact measurableSet_stepsUntil_eq_zero a m

Dependency graph

Type dependencies (1)

stepsUntil🔗

DefinitionLearning.stepsUntil
Details

Number of steps until action a was pulled exactly m times.

🔗def
Learning.stepsUntil.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (m : ℕ) (ω : Ω) : ℕ∞
Learning.stepsUntil.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (m : ℕ) (ω : Ω) : ℕ∞

Code

noncomputable
def stepsUntil (A : ℕ → Ω → 𝓐) (a : 𝓐) (m : ℕ) (ω : Ω) : ℕ∞ :=
  sInf ((↑) '' {s | pullCount A a (s + 1) ω = m})
Body uses (1)
Used by (46)

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All dependencies, transitively (1)

pullCount🔗

DefinitionLearning.pullCount
Details

Number of times action a was chosen up to time t (excluding t).

🔗def
Learning.pullCount.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ
Learning.pullCount.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (t : ℕ) (ω : Ω) : ℕ

Code

noncomputable
def pullCount (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ :=
  #(filter (fun s ↦ A s ω = a) (range t))
Used by (146)

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