| 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 |
|
|
