File size: 2,965 Bytes
9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import Mathlib
import OAI.Probability.Ballisticity.Walk.RetentionRowsAllHeights
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma normalizedMeasure_zero {α : Type*} [MeasurableSpace α] :
normalizedMeasure (0 : Measure α) = 0 := by simp [normalizedMeasure]
lemma bufferStage_normalized_stopped_rows {d : ℕ} (e f : Direction d)
(a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a))
(hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π)
{H : ℕ} (hH : 0 < H) (κ : ℝ≥0) (k : ℕ) :
@Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+k))) _
(fun ω => if bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω = k
then normalizedMeasure (bufferStageRetainedLaw e f a z₀ r ε α g π H ω κ) else 0) := by
have hm := measurable_normalizedMeasure.comp
(bufferStageRetainedLaw_stopped_rows e f a z₀ r ε α g π hπ hH κ k)
convert hm using 1
funext ω
dsimp only [Function.comp_def]
split <;> simp_all only [normalizedMeasure_zero]
lemma bufferStage_success_stopped_rows {d : ℕ} (e f : Direction d)
(a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a))
(hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π)
{H : ℕ} (hH : 0 < H) (k : ℕ) :
MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a+k))]
{ω | bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω = k ∧
BufferStageEvent (realPosition (step e)) f H z₀ r ε α g (π ω).val ω} := by
let ℓ := realPosition (step e)
by_cases h : k=H
· subst k
have heq := bufferFirstFailure_eq_measurable ℓ f a (z₀+(1-ε)*r) g π hπ H H
have hp : @Measurable _ _ (rowSigma (BelowHeight ℓ (a+H))) _ π :=
hπ.mono (rowSigma_below_height_mono ℓ a H) le_rfl
have hm := (measurableSet_bufferStageEvent_rows ℓ f hH z₀ r ε α g (PairAtHeight ℓ a)
(BelowHeight ℓ (a+H)) (fun x hx => strip_subset_below_height ℓ a x hx le_rfl)).preimage
(hp.prodMk measurable_id)
exact heq.inter hm
· have he : {ω | bufferFirstFailure ℓ f a (z₀+(1-ε)*r) g π H ω = k ∧
BufferStageEvent ℓ f H z₀ r ε α g (π ω).val ω} = ∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
rintro ω ⟨hk,hE⟩
have hlast := bufferFirstFailure_eq_top_of_pass ℓ f a (z₀+(1-ε)*r) g π hH ω
((bufferStageEvent_iff_last_tests ℓ f a z₀ r ε α g π hH ω).mp hE).1
exact h (hk.symm.trans hlast)
change MeasurableSet[rowSigma (BelowHeight ℓ (a+k))] _
rw [he]
exact @MeasurableSet.empty _ (rowSigma (BelowHeight ℓ (a+k)))
end DirectionalTransience
end
end
end OAI
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