`Learning.randomSampling.image_action_tendsto_any`🔗

Lemma

The minimum distance from image of actions to any function value tends to zero.

theorem

Learning.randomSampling.image_action_tendsto_any.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure 𝓐} [MeasureTheory.IsProbabilityMeasure μ]
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨} {f : 𝓐 → 𝓨} [PseudoMetricSpace 𝓐]
  [SecondCountableTopology 𝓐] [OpensMeasurableSpace 𝓐]
  [MeasureTheory.Measure.IsOpenPosMeasure μ] [PseudoMetricSpace 𝓨]
  [BorelSpace 𝓨] (hfc : Continuous f)
  (h : IsAlgEnvSeq A Y (randomSampling μ) (evalEnv f ⋯) P) (a : 𝓐)
  {ε : ℝ} (hε : 0 < ε) :
  Filter.Tendsto
    (fun i =>
      P {x | ε ≤ Function.min fun j => dist (f (A (↑j) x)) (f a)})
    Filter.atTop (nhds 0)
Learning.randomSampling.image_action_tendsto_any.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure 𝓐}
  [MeasureTheory.IsProbabilityMeasure μ]
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨}
  {f : 𝓐 → 𝓨} [PseudoMetricSpace 𝓐]
  [SecondCountableTopology 𝓐]
  [OpensMeasurableSpace 𝓐]
  [MeasureTheory.Measure.IsOpenPosMeasure
      μ]
  [PseudoMetricSpace 𝓨] [BorelSpace 𝓨]
  (hfc : Continuous f)
  (h :
    IsAlgEnvSeq A Y (randomSampling μ)
      (evalEnv f ⋯) P)
  (a : 𝓐) {ε : ℝ} (hε : 0 < ε) :
  Filter.Tendsto
    (fun i =>
      P
        {x |
          ε ≤
            Function.min fun j =>
              dist (f (A (↑j) x)) (f a)})
    Filter.atTop (nhds 0)

Code

lemma image_action_tendsto_any
    (h : IsAlgEnvSeq A Y (randomSampling μ) (evalEnv f hfc.measurable) P)
    (a : 𝓐) {ε : ℝ} (hε : 0 < ε) :
    Tendsto (fun i => P {x | ε ≤ (fun (j : Iic i) ↦
      dist (f (A j.1 x)) (f a)).min}) atTop (𝓝 0)

Proof

by
  have hf := hfc.measurable
  rw [Metric.continuous_iff] at hfc
  obtain ⟨δ, hδ, hfc⟩ := hfc a ε hε
  refine action_tendsto_any h a hδ |> tendsto_zero_of_le <| ?_
  intro n
  refine measure_mono ?_
  simp only [Set.ofPred_subset_ofPred]
  intro ω hω
  rw [← argmin_spec]
  set j := argmin (fun (i : Iic n) ↦ dist (A i.1 ω) a)
  by_contra! h_contra
  specialize hfc (A j.1 ω) h_contra
  have := (fun (j : Iic n) ↦ dist (f (A (j) ω)) (f a)).min_le j
  linarith

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 10 project declarations, 92 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.randomSampling.image_action_tendsto_any🔗

`Learning.randomSampling.image_action_tendsto_any`🔗