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Learning.IsBayesAlgEnvSeq.measurable_uncurry_actionMean_comp🔗

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measurable_uncurry_actionMean_comp🔗

LemmaLearning.IsBayesAlgEnvSeq.measurable_uncurry_actionMean_comp
Details

No docstring.

🔗theorem
Learning.IsBayesAlgEnvSeq.measurable_uncurry_actionMean_comp.{u_1, u_2,
    u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2} {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace Ω] [Countable 𝓐]
  [MeasurableSingletonClass 𝓐] {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ}
  {E : Ω → 𝓔} (hE : Measurable E) {f : Ω → 𝓐} (hf : Measurable f) :
  Measurable fun ω => actionMean κ E (f ω) ω
Learning.IsBayesAlgEnvSeq.measurable_uncurry_actionMean_comp.{u_1,
    u_2, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace Ω]
  [Countable 𝓐]
  [MeasurableSingletonClass 𝓐]
  {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ}
  {E : Ω → 𝓔} (hE : Measurable E)
  {f : Ω → 𝓐} (hf : Measurable f) :
  Measurable fun ω =>
    actionMean κ E (f ω) ω

Code

lemma measurable_uncurry_actionMean_comp [Countable 𝓐] [MeasurableSingletonClass 𝓐]
    {κ : Kernel (𝓔 × 𝓐) ℝ} {E : Ω → 𝓔} (hE : Measurable E) {f : Ω → 𝓐} (hf : Measurable f) :
    Measurable (fun ω ↦ actionMean κ E (f ω) ω)
Type uses (1)
Body uses (1)
Used by (3)

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Proof
by
  change Measurable ((fun aω ↦ actionMean κ E aω.1 aω.2) ∘ fun ω ↦ (f ω, ω))
  apply Measurable.comp _ (by fun_prop)
  exact measurable_from_prod_countable_right (fun _ ↦ measurable_actionMean hE)

Dependency graph

Type dependencies (1)

actionMean🔗

DefinitionLearning.IsBayesAlgEnvSeq.actionMean
Details

A random variable that gives the mean feedback of action a.

🔗def
Learning.IsBayesAlgEnvSeq.actionMean.{u_1, u_2, u_4} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {Ω : Type u_4} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ) (E : Ω → 𝓔) (a : 𝓐) (ω : Ω) :
  ℝ
Learning.IsBayesAlgEnvSeq.actionMean.{u_1,
    u_2, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ)
  (E : Ω → 𝓔) (a : 𝓐) (ω : Ω) : ℝ

Code

noncomputable
def actionMean (κ : Kernel (𝓔 × 𝓐) ℝ) (E : Ω → 𝓔) (a : 𝓐) (ω : Ω) : ℝ := (κ (E ω, a))[id]
Used by (12)

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