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caec230 | 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 | import OAI.MeasureTheory.DyadicAvoidance.FirstDefault
namespace OAI
universe u_K
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem310.FirstDefault
/-- Transport of the finite-depth survival law to actual finite spatial
selector tables. The center lookup need only select distinct coordinates;
unused table entries are integrated out automatically. -/
theorem measure_no_default_finite_table {M d : ℕ} {K : Type u_K} [Fintype K]
(ν : Measure Bool) [IsProbabilityMeasure ν]
(hfair : ∀ b, ν {b} = (2 : ENNReal)⁻¹)
(center : BoundedAddress M d → K) (hinj : Function.Injective center) :
(Measure.pi (fun _ : K => ν))
(⋃ f : Path M d, {ω | ∀ i : Tests f,
ω (center (boundedAddress f i)) = requiredBit f i}) =
(1 - ((2 : ENNReal)⁻¹) ^ M) ^ d := by
apply measure_no_default_bounded
(Measure.pi (fun _ : K => ν)) (fun a ω => ω (center a))
· intro a
exact measurable_pi_apply _
· have hi : iIndepFun (fun k (ω : K → Bool) => ω k)
(Measure.pi (fun _ : K => ν)) :=
iIndepFun_pi (fun _ => measurable_id.aemeasurable)
exact hi.precomp hinj
· intro a b
rw [(measurePreserving_eval (fun _ : K => ν) (center a)).measure_preimage
(measurableSet_singleton b).nullMeasurableSet]
exact hfair b
end Problem310.FirstDefault
end
end OAI
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