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b317718 | 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 | import Mathlib.MeasureTheory.Function.FactorsThrough
import OAI.Probability.FreeIsing.Conditioning.DiscreteConditioning
import OAI.Probability.FreeIsing.Spins.ObservationFusion
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
local instance : MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩
theorem condExp_spin_comap_sup_eq_binaryFusion
{Ω A B : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
[mA : MeasurableSpace A] [mB : MeasurableSpace B]
{P : Measure Ω} [IsProbabilityMeasure P]
{S : Ω → IsingSpin} {X : Ω → A} {Y : Ω → B}
(hS : Measurable S) (hX : Measurable X) (hY : Measurable Y)
(hfair : ∀ a : IsingSpin, P {ω | S ω = a} = (1 / 2 : ENNReal))
(hInd : X ⟂ᵢ[S, hS; P] Y)
(hL : ∀ᵐ ω ∂P, |(P[fun ω => spinSign (S ω) | mA.comap X]) ω| < 1)
(hR : ∀ᵐ ω ∂P, |(P[fun ω => spinSign (S ω) | mB.comap Y]) ω| < 1) :
P[fun ω => spinSign (S ω) | mA.comap X ⊔ mB.comap Y] =ᵐ[P]
fun ω => binaryFusion (P[fun ω => spinSign (S ω) | mA.comap X] ω)
(P[fun ω => spinSign (S ω) | mB.comap Y] ω) := by
obtain ⟨q, hq, hqe⟩ :=
(show Measurable[mA.comap X]
(P[fun ω => spinSign (S ω) | mA.comap X]) from
stronglyMeasurable_condExp.measurable).exists_eq_measurable_comp
obtain ⟨r, hr, hre⟩ :=
(show Measurable[mB.comap Y]
(P[fun ω => spinSign (S ω) | mB.comap Y]) from
stronglyMeasurable_condExp.measurable).exists_eq_measurable_comp
have hpostX : P[fun ω => spinSign (S ω) | mA.comap X] =ᵐ[P] fun ω => q (X ω) :=
Filter.Eventually.of_forall (congrFun hqe)
have hpostY : P[fun ω => spinSign (S ω) | mB.comap Y] =ᵐ[P] fun ω => r (Y ω) :=
Filter.Eventually.of_forall (congrFun hre)
have h := condExp_spin_given_observation_pair_eq_binaryFusion hS hX hY hq hr
hpostX hpostY (by simpa only [hqe, Function.comp_apply] using hL)
(by simpa only [hre, Function.comp_apply] using hR) hfair
(fun a => indepFun_cond_fiber_of_condIndepFun hS hX hY hInd a)
simpa only [Prod.instMeasurableSpace, MeasurableSpace.comap_prodMk,
hqe, hre, Function.comp_apply] using h
end Problem367
end
end OAI
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